INFLUENCE OF RADIATIVE TRANSFER ON

RAYLEIGH-BÈNARD-MARANGONI CONVECTION

IN A COUPLE-STRESS FLUID SATURATED

POROUS MEDIUM

Dissertation submitted in partial fulfillment of the requirements for the award of the degree of

MASTER OF PHILOSOPHY IN MATHEMATICS

By

NINGTHOUJAM SHIBIRAJ SINGH Register No. 0935302

Supervisor Dr. S. MARUTHAMANIKANDAN

Department of Mathematics Christ University

Bangalore-560 029

HOSUR ROAD

BANGALORE-560 029

2010

DEDICATED TO

MY BELOVED PARENTS

N. NILLACHANDRA SINGH

&

N. SHANAHANBI DEVI

DECLARATION

I hereby declare that the dissertation entitled “Influence of Radiative

Transfer on Rayleigh-Bènard-Marangoni Convection in a Couple-Stress

Fluid Saturated Porous Medium” has been undertaken by me for the award

of M.Phil. degree in Mathematics. I have completed this under the guidance

of Dr. S. MARUTHAMANIKANDAN, Assistant Professor, Department of

Mathematics, Christ University, Bangalore-560 029. I also declare that this

dissertation has not been submitted for the award of any Degree, Diploma,

Associateship, Fellowship or other title.

Place:

Date:

NINGTHOUJAM SHIBIRAJ SINGH

Candidate

Dr. S. MARUTHAMANIKANDAN Assistant Professor Department of Mathematics Christ University Bangalore - 560 029.

CERTIFICATE This is to certify that the dissertation submitted by NINGTHOUJAM

SHIBIRAJ SINGH on the title “Influence of Radiative Transfer on

Rayleigh-Bènard-Marangoni Convection in a Couple-Stress Fluid

Saturated Porous Medium” is a record of research work done by him during

the academic year 2009 – 2010 under my guidance and supervision in partial

fulfillment of the requirements for the award of the degree of Master of

Philosophy in Mathematics. This dissertation has not been submitted for the

award of any Degree, Diploma, Associateship, Fellowship or other title.

Place:

Date:

Dr. S. MARUTHAMANIKANDAN

Supervisor

ACKNOWLEDGEMENT

At the outset, I would like to place on record my sincere gratitude to my

supervisor, Dr. S. Maruthamanikandan, for his patience, exceptional guidance,

and continued encouragement. He has been abundantly helpful and has offered

invaluable assistance and support.

I am indebted to Dr. S. Pranesh, Coordinator, Postgraduate Programme in

Mathematics, Christ University for his encouragement and support. I gratefully

acknowledge Prof. T.V. Joseph, HOD, Department of Mathematics, Christ

University and Mrs. Sangeetha George, Assistant Professor, Department of

Mathematics, Christ University for their whole-hearted support.

Many thanks to Prof. K.A. Chandrasekharan, the General Research

Coordinator and Prof. Dr. Nanje Gowda, The Dean of Science of Christ

University for their valuable advice and constant support.

Special thanks to the Vice–Chancellor, Dr. (Fr.) Thomas C. Mathew,

Pro-Vice-Chancellor and Director of Centre for Research and Consultancy,

Dr. (Fr.) Abraham V. M. of Christ University for the opportunity provided to do

this course.

I really need to acknowledge my friends who provided a stimulating and

fascinating environment. I am especially thankful to Chitra, Rashmi, Aparna, and

Deepika for sharing the good time.

Last but not least, I would like to thank my brother N. Shibadatta Singh and my

sister N. Shanjita Devi for their remarkable support and love.

NINGTHOUJAM SHIBIRAJ SINGH

ABSTRACT

The problem of Rayleigh-Benard-Marangoni convection in a

couple-stress fluid saturated porous medium with thermal radiation is

studied within the framework of linear stability analysis. Only infinitesimal

disturbances are considered. The linear stability analysis is based on the

normal mode technique. The Darcy law is used to model the momentum

equation. The fluid between the boundaries absorbs and emits thermal

radiation. The boundaries are treated as black bodies. The absorption

coefficient of the fluid is assumed to be the same at all wavelengths and to

be independent of the physical state. The principle of exchange of

stabilities is valid and the existence of oscillatory instability is ruled out.

The expression for the stationary Darcy-Rayleigh number is obtained as a

function of the governing parameters, viz., the wave number,

the couple-stress parameter, the conduction-radiation parameter,

the absorptivity parameter, the Marangoni number and the Biot number.

The Galerkin method is used to determine the eigenvalues. The effect of

various parameters on the stability of the fluid layer is discussed through

figures and tables.

CONTENTS

Page No.

CHAPTER I Introduction 1 1.1 Objective and Scope 1 CHAPTER II Literature Review 7

2.1 Rayleigh-Bénard Convection in Fluids 7 2.2 Marangoni Convection in Fluids 16 2.3 Convection with Thermal Radiation 26 2.4 Convection in a Porous Medium 35 2.5 Convection in Couple-Stress Fluids 50 2.6 Plan of Work 65 CHAPTER III Basic Equations, Boundary Conditions and

Dimensionless Parameters 66

3.1 Basic Equations 68 3.2 Boundary Conditions 72 3.3 Dimensionless Parameters 78 CHAPTER IV

Influence of Radiative Transfer on Rayleigh-Bènard-Marangoni Convection in a Couple-Stress Fluid Saturated Porous Medium

81

4.1 Introduction 81 4.2 Mathematical Formulation 83 4.3 Linear Stability Analysis 87 CHAPTER V

Results, Discussion and Concluding Remarks 93

5.1 Results and Discussion 93 5.2 Concluding Remarks 96 BIBLIOGRAPHY

107

1

CHAPTER I

INTRODUCTION 1.1 Objective and Scope

Fluid mechanics is the study of how fluids move and the forces on them. Fluids

include liquids and gases. Fluid mechanics can be divided into fluid statics, the

study of fluids at rest, and fluid dynamics, the study of fluids in motion. It is a

branch of continuum mechanics, a subject which models matter without using the

information that it is made out of atoms. Fluid mechanics, especially fluid

dynamics, is an active field of research with many unsolved or partly solved

problems. Fluid mechanics can be mathematically complex. Sometimes it can best

be solved by numerical methods, typically using computers. A modern discipline,

called Computational Fluid Dynamics (CFD), is devoted to this approach to solving

fluid mechanics problems. Also taking advantage of the highly visual nature of fluid

flow is Particle Image Velocimetry, an experimental method for visualizing and

analyzing fluid flow. Fluid mechanics is the branch of physics which deals with the

properties of fluids, namely liquids and gases, and their interaction with forces.

Rapid advancement in fluid mechanics began with Leonardo da Vinci

(observation and experiment), Evangelista Torricelli (barometer), Isaac Newton

(viscosity) and Blaise Pascal (hydrostatics), and was continued by Daniel Bernoulli

with the introduction of mathematical fluid dynamics in Hydrodynamica (1738).

Inviscid flow was further analyzed by various mathematicians (Leonhard Euler,

d'Alembert, Lagrange, Laplace, Poisson) and viscous flow was explored by a

multitude of engineers including Poiseuille and Gotthilf Heinrich Ludwig Hagen.

Further mathematical justification was provided by Claude-Louis Navier and

George Gabriel Stokes in the Navier-Stokes Equations, and boundary layers were

investigated (Ludwig Prandtl), while various scientists (Osborne Reynolds, Andrey

2

Kolmogorov, Geoffrey Ingram Taylor) advanced the understanding of fluid

viscosity and turbulence.

Rayleigh-Benard and Marangoni Convection Rayleigh-Benard convection (RBC) is the instability of a fluid layer which is

confined between two thermally conducting plates, and is heated from below to

produce a fixed temperature difference. Since liquids typically have positive

thermal expansion coefficient, the hot liquid at the bottom of the cell expands and

produces an unstable density gradient in the fluid layer. If the density gradient is

sufficiently strong, the hot fluid will rise, causing a convective flow which results in

enhanced transport of heat between the two plates. In order for convection to occur,

a small plume of hot fluid which begins to rise toward the top of the cell must grow

in strength, rather than fizzle out.

There are two processes which oppose this amplification. First, viscous damping

in the fluid directly opposes the fluid flow. In addition, thermal diffusion will

suppress the temperature fluctuation by causing the rising plume of hot fluid to

equilibriate with surrounding fluid, destroying the buoyant force. Convection occurs

if the amplifying effect exceeds the disippative effect of thermal diffusion and

buoyancy. This competition of forces is parameterized by the Rayleigh number,

which is the temperature difference, but appropriately normalized to take into

account the geometry of the convection cell and the physical properties of the fluid.

If the temperature difference is very large, then the fluid rises very quickly, and a

turbulent flow may be created. If the temperature difference is not far above the

onset, an organized flow resembling overturning of cylinders is formed. It is the

patterns created by these convection "rolls" that most people study.

3

Cross-sectional view of cell illustrating convection rolls

The first intensive experiments were carried out by Benard in 1900. He

experimented on a fluid of thin layer and observed appearance of hexagonal cells

when the instability in the form of convection developed. Rayleigh in 1916

developed the theory which found the condition for the instability with two free

surfaces. He showed that the instability would manifest if the temperature gradient

was large enough so that the so-called Rayleigh number exceeds a certain value

(critical value).

Experiments in the early stage were carried out with fluid heated from bottom

and the top surface is open to atmosphere. Thus the top surface is free to move and

deform. It was later (around 1960) realized that this could lead to another instability

mechanism (Benard-Marangoni convection) due to gradient in surface tension. This

mechanism coexists with the Rayleigh's mechanism but dominates in thin layer.

Most of the findings reported by Benard were actually due this latter instability

mechanism. The instabilities driven by surface tension decreases as the layer

becomes thicker. Later experiments on thermal convection (with or without free

upper surface) have obtained convective cells of many forms such as rolls, square

and hexagons.

In the case of Rayleigh-Benard convection, only buoyancy is responsible for the

appearance of convection cells. The initial movement is the upwelling of warmer

liquid from the heated bottom layer. In case of a free liquid surface in contact with

air surface tension effect will play a role besides buoyancy. It is known that liquids

flow from places of lower surface tension to places of higher surface tension. This

is called the Marangoni effect. When applying heat from below, the temperature at

4

the top layer will show temperature fluctuations. With increasing temperature,

surface tension decreases. Thus a lateral flow of liquid at the surface will take place,

from warmer areas to cooler areas. In order to preserve a horizontal (or nearly

horizontal) liquid surface, liquid from the cooler places on the surface have to go

down into the liquid. Thus the driving force of the convection cells is the

downwelling of liquid. Marangoni convection plays an important role in Benard

convection in shallow fluid layers, in chemical engineering as well as in crystal

growth and other materials processing technologies.

Assorted Constraints In quite a few heat transfer problems, suppressing or augmenting the convection

plays a vital role. There are several mechanisms that can be used effectively to

either delay or advance the convection, namely, by applying a magnetic/electric

field externally or by Coriolis force due to rotation or by maintaining non-uniform

temperature gradient across the porous layer. A non-uniform temperature gradient

can arise in various ways, notably by (i) transient heating or cooling at a boundary

(ii) volumetric distribution of heat sources (iii) radiative heat transfer (iv) thermal

modulation (v) vertical throughflow and (vi) chemical reaction.

Radiative Transfer Combined heat transfer processes such as convection-radiation play a significant

role in many chemical processes involving combustion, drying, fluidization, MHD

flows, and so forth (Pomraning, 1973). In general, the radiative process either

occurs at the boundaries or as a term in the energy equation. In the latter case, the

radiative term is usually approximated as a flux in such a way that the term

corresponding to radiation in the heat transfer equation appears as a gradient term

similar to Fourier’s conduction term. This method has found much favour among

many researchers. Alternatively, radiation effects can be incorporated at the

5

boundaries through appropriate assumption, such as the Stefan-Boltzman condition.

Free surface flows present a challenging problem to engineers as the combined

convection-radiation at the boundaries has major applications in many industries.

Couple-Stress Fluid Although the problem of Rayleigh–Benard convection and Marangoni

convection have been extensively investigated considering Newtonian fluids,

relatively little attention has been devoted to the thermal convection of non-

Newtonian fluids. With the growing importance of non-Newtonian fluids with

suspended particles in modern technology and industries, the investigation of such

fluids is desirable. The study of such fluids has applications in a number of

processes that occur in industry, such as the extrusion of polymer fluids,

solidification of liquid crystals, cooling of a metallic plate in a bath, exotic

lubrication, and colloidal and suspension solutions. In the category of non-

Newtonian fluids couple-stress fluids have distinct features such as polar effects.

Couple-stress fluid theory developed by Stokes (1966, 1984) is one among the polar

fluid theories which considers couple stresses in addition to the classical Cauchy

stress. It is the simplest generalization of the classical theory of fluids which allows

for polar effects such as the presence of couple stresses and body couples in the

fluid medium.

Goal The intent of this work is to examine the problem of the effect of thermal

radiation on Rayleigh–Benard-Marangoni convection in a couple-stress fluid

saturated densely packed horizontal porous layer with the following scope in mind:

6

The study of convection in a fluid saturated porous medium has attracted

considerable interest in recent years because of its significance in many practical

fields such as chemical engineering, geothermal activities, oil recovery techniques

and biological process. It is also of practical interest in the extraction of geothermal

energy and more specifically in understanding the mechanism of transfer of heat

from aquifers in the deep interior of earth to shallow depths (Wooding, 1960). The

effective mixing process in petroleum reservoirs, regarded as fixed bed reactor, is

achieved by thermal convection. On the other hand, the problems concerning couple

stress fluids in porous media have been extremely important owing to several

engineering applications within fields such as chemical engineering, thermal

insulation systems, and nuclear waste management.

The problem of convection in a couple-stress fluid saturated porous medium has

been extensively studied. However, attention has not been given to the study of

Rayleigh–Benard-Marangoni convection in a couple-stress fluid saturating porous

media with radiative transfer. Therefore, the objective of the work is to investigate

theoretically the effect of thermal radiation on Rayleigh–Benard-Marangoni

convection in a densely packed porous medium saturated with a couple-stress fluid

with emphasis on how the stability criterion for the onset of convection is modified

in the presence of thermal radiation.

7

CHAPTER II

LITERATURE REVIEW

The main objective of this work is to deal with Rayleigh–Benard-Marangoni

Convection in a couple-stress fluid saturated densely packed horizontal porous layer

in the presence of thermal radiation. Literature pertinent to this is classified as

follows.

• Rayleigh-Bénard convection (RBC) in fluids.

• Marangoni convection (MC) in fluids.

• Convection with thermal radiation.

• Convection in a porous medium.

• Convection in couple-stress fluids.

The relevant literature on the problem at hand is briefly discussed below in

keeping with the classifications above.

2.1 Rayleigh-Bénard Convection in Fluids

Natural convection in a horizontal layer of fluid heated from below and cooled

from above has been the subject of investigation for many decades owing to its

implications for the control and exploitation of many physical, chemical and

biological processes. We now make a brief review of the RBC problem keeping in

mind the objective and scope of the thesis.

The earliest experiment which called attention to the thermal instability was

briefly reported by Thompson (1882). Benard (1901) later presented a much more

complete description of the development of the convective flow. Lord Rayleigh

(1916) was the first to study the problem theoretically and aimed at determining the

8

conditions delineating the breakdown of the quiescent state. As a result, the thermal

instability situation described in the foregoing paragraph is referred to as Rayleigh-

Bénard convection (RBC). The Rayleigh theory was generalized and extended to

consider several boundary combinations by Jeffreys (1926), Low (1929) and

Sparrow et al. (1964). Chandra (1938) examined the RBC problem experimentally

for a gas. The most complete theory of the thermal instability problem was

presented by Pellew and Southwell (1940).

Malkus and Veronis (1958) investigated finite amplitude cellular convection

and determined the form and amplitude of convection by expanding the nonlinear

equations describing the fields of motion and temperature in a sequence of

inhomogeneous linear equations. Veronis (1959) studied finite amplitude cellular

convection in a rotating fluid and showed that the fluid becomes unstable to finite

amplitude disturbances before it becomes unstable to infinitesimal perturbations.

Palm (1960) showed that for a certain type of temperature-dependence of

viscosity, the critical Rayleigh number and the critical wavenumber are smaller than

those for constant viscosity and explained the observed fact that steady hexagonal

cells are formed frequently at the onset of convection.

Lorenz (1963) solved a simple system of deterministic ordinary nonlinear

differential equations representing cellular convection numerically. For those

systems with bounded solutions, it is found that non-periodic solutions are unstable

with respect to small modifications and that slightly differing initial states can

evolve into considerably different states.

Veronis (1966) analyzed the two-dimensional problem of finite amplitude

convection in a rotating layer of fluid by considering the boundaries to be free.

Using a minimal representation of Fourier series, he showed that, for a restricted

range of Taylor number, steady finite amplitude motions can exist for values of the

9

Rayleigh number smaller than the critical value required for overstability. Veronis

(1968) also examined the effect of a stabilizing gradient of solute on thermal

convection using both linear and finite amplitude analysis. It is found that the onset

of instability may occur as an oscillatory motion because of the stabilizing effect of

the solute in the case of linear theory and that finite amplitude instability may occur

first for fluids with a Prandtl number somewhat smaller than unity.

Krishnamurthy (1968a, b) presented a nonlinear theory of RBC problem and

discussed the formation of hexagonal cells and the existence of subcritical

instabilities. Torrance and Turcotte (1971) investigated the influence of large

variations of viscosity on convection in a layer of fluid heated from below.

Solutions for the flow and temperature fields were obtained numerically assuming

infinite Prandtl number, free-surface boundary conditions and two-dimensional

motion. The effect of temperature-dependent and depth-dependent viscosity was

studied motivated by the convective heat transport in earth’s mantle.

Busse (1975) considered the interaction between convection in a horizontal fluid

layer heated from below and an ambient vertical magnetic field. It is found that

finite amplitude onset of steady convection becomes possible at Rayleigh numbers

considerably below the values predicted by linear theory.

Booker (1976) investigated experimentally the heat transport and structure of

convection in a high Prandtl number fluid whose viscosity varies by up to a factor

of 300 between the boundary temperatures. Horne and Sullivan (1978) examined

the effect of temperature-dependent viscosity and thermal expansion coefficient on

the natural convection of water through permeable formations. They found that the

convective motion is unstable at even moderate values of the Rayleigh number and

exhibits a fluctuating convective state analogous to the case of a fluid with constant

viscosity and coefficient of thermal expansion.

10

Carey and Mollendorf (1980) presented a regular perturbation analysis for

several laminar natural convection flows in liquids with temperature-dependent

viscosity. Several interesting variable viscosity trends on flow and transport are

suggested by the results obtained. Stengel et al. (1982) obtained, using a linear

stability theory, the viscosity-ratio dependences of the critical Rayleigh number and

critical wave number for several types of temperature-dependence of viscosity.

Richter et al. (1983) showed, by an experiment with temperature-dependent

viscosity ratio as large as 106, the existence of subcritical convection of finite

amplitude near the critical Rayleigh number. Busse and Frick (1985) analyzed the

problem of RBC with linear variation of viscosity and showed an appearance of

square pattern for a viscosity ratio larger than 2.

White (1988) made an experiment for the fluid with Prandtl number of o(105)

and studied convective instability with several planforms for the Rayleigh number

up to 63000 and the temperature-dependent viscosity ratio up to 1000. He found

that if the viscosity ratio is 50 or 100 and the Rayleigh number is less than 25000,

stable hexagonal and square patterns are formed in a certain range of wavenumber

and that their wavenumbers increase with viscosity ratio. The possibility of multi-

valued solution in the thermal convection problem with temperature-dependent

viscosity has been examined numerically by Hirayama and Takaki (1993).

Tong and Shen (1992) studied high Rayleigh number turbulent convection using

the technique of photon-correlation homodyne spectroscopy to measure velocity

differences at various length scales. The measured power-law exponents are found

to be in excellent agreement with the theoretical predictions.

Massaioli et al. (1993) investigated the probability density function (pdf) of the

temperature field by numerical simulations of Rayleigh-Bénard convection in two

spatial dimensions. The Pdf of the temperature has been shown to have exponential

11

tails, consistently with previous laboratory experiments and numerical simulations.

They also offered a new theoretical explanation for the exponential tail of the Pdf.

Xi and Gunton (1993) presented a numerical study of the spontaneous formation

of spiral patterns in Rayleigh-Benard convection in non-Boussinesq fluids. They

solved a generalized two-dimensional Swift-Hohenberg equation that includes a

quadratic nonlinearity and coupling to mean flow. They showed that this model

predicts in quantitative detail many of the features observed experimentally in

studies of Rayleigh-Benard convection in CO2 gas. In particular, they studied the

appearance and stability of a rotating spiral state obtained during the transition from

an ordered hexagonal state to a roll state.

Mukutmoni and Yang (1994) reviewed the broad area of flow transitions of

Rayleigh-Benard convection in rectangular enclosures with sidewalls. They looked

into pattern selection for both small and intermediate enclosures.

Kafoussias and Williams (1995) studied, using an efficient numerical technique,

the effect of a temperature-dependent viscosity on an incompressible fluid in steady,

laminar, free-forced convective boundary layer flow over an isothermal vertical

semi-infinite flat plate. It is concluded that the flow field and other quantities of

physical interest are significantly influenced by the viscosity-temperature

parameter. Kafoussias et al. (1998) studied the combined free-forced convective

laminar boundary layer flow past a vertical isothermal flat plate with temperature-

dependent viscosity. The obtained results showed that the flow field is appreciably

influenced by the viscosity variation.

Severin and Herwig (1999) investigated the variable viscosity effect on the

onset of instability in the RBC problem. An asymptotic approach is considered

which provides results that are independent of specific property laws.

Kozhhoukharova et al. (1999) examined the influence of a temperature-dependent

12

viscosity on the axisymmetric steady thermocapillary flow and its stability with

respect to non-axisymmetric perturbations by means of a linear stability analysis.

The onset of oscillatory convection is studied numerically by a mixed Chebyshev-

collocation finite-difference method.

Rogers and Schatz (2000) reported the first observations of superlattices in

thermal convection. The superlattices are selected by a four-mode resonance

mechanism that is qualitatively different from the three-mode resonance responsible

for complex-ordered patterns observed previously in other nonequilibrium systems.

Numerical simulations quantitatively describe both the pattern structure and the

stability boundaries of superlattices observed in laboratory experiments. It is found

that, in the presence of inversion symmetry, superlattices numerically bifurcate

supercritically directly from conduction or from a striped base state.

Rogers et al. (2000) reported on the quantitative observations of convection in a

fluid layer driven by both heating from below and vertical sinusoidal oscillation.

Just above onset, convection patterns are modulated either harmonically or

subharmonically to the drive frequency. It is found that single frequency patterns

exhibit nearly solid-body rotations with harmonic and subharmonic states always

rotating in opposite directions. Further, flows with both harmonic and subharmonic

responses have been found near a co-dimension two point, yielding novel coexisting

patterns with symmetries not found in either single-frequency states.

You (2001) presented a simple method which can be applied to estimate the

onset of natural convection in a fluid with a temperature-dependent viscosity.

Straughan (2002) developed an unconditional nonlinear energy stability analysis for

thermal convection with temperature-dependent viscosity. The nonlinear stability

boundaries are shown to be sharp when compared with the instability thresholds of

linear theory.

13

Hossain et al. (2002) analyzed the effect of temperature-dependent viscosity on

natural convection flow from a vertical wavy surface using an implicit finite

difference method. They have focused their attention on the evaluation of local

skin-friction and the local Nusselt number. Chakraborty and Borkakati (2002)

studied the flow of a viscous incompressible electrically conducting fluid on a

continuously moving flat plate in the presence of uniform transverse magnetic field.

Assuming the fluid viscosity to be an inverse linear function of temperature, the

nature of fluid velocity and temperature is analyzed.

Abraham (2002) investigated the RBC problem in a micropolar ferromagnetic

fluid layer in the presence of a vertical uniform magnetic field analytically. It is

shown that the micropolar ferromagnetic fluid layer heated from below is more

stable as compared with the classical Newtonian ferromagnetic fluid.

Getling and Brausch (2003) studied numerically the evolution of three-

dimensional, cellular convective flows in a plane horizontal layer of Boussinesq

fluid heated from below. It is found that the flow can undergo a sequence of

transitions between various cell types. In particular, two-vortex polygonal cells may

form at some evolution stages, with an annular planform of the upflow region and

downflows localized in both central and peripheral regions of the cells. They also

showed that, if short-wave hexagons are stable, they exhibit a specific, stellate fine

structure.

Rudiger and Knobloch (2003) described the results of direct numerical

simulations of convection in a uniformly rotating vertical cylinder with no-slip

boundary conditions. They used these results to study the dynamics associated with

transitions between states with adjacent azimuthal wave numbers far from onset. In

certain regimes a novel burst-like state is identified and described.

14

Ma and Wang (2004) studied the bifurcation and stability of the solutions of the

Boussinesq equations, and the onset of the Rayleigh-Benard convection. A

nonlinear theory for this problem is established using a new notion of bifurcation

called attractor bifurcation and its corresponding theorem developed recently. This

theory includes the following three aspects. First, the problem bifurcates from the

trivial solution an attractor AR when the Rayleigh number R crosses the first critical

Rayleigh number Rc for all physically sound boundary conditions, regardless of the

multiplicity of the eigenvalue Rc for the linear problem. Second, the bifurcated

attractor AR is asymptotically stable. Third, when the spatial dimension is two, the

bifurcated solutions are also structurally stable and are classified as well. In

addition, the technical method developed provides a recipe, which can be used for

many other problems related to bifurcation and pattern formation.

Sprague et al. (2005) investigated pattern formation in a rotating Rayleigh-

Benard configuration for moderate and rapid rotation in moderate aspect-ration

cavities. While the existence of the Kuppers-Lortz rolls is predicted by the theory at

the onset of convection, square patterns have been observed in physical and

numerical experiments at relatively high rotation rates. In addition to presenting

numerical results produced from the direct numerical simulation of the full

Boussinesq equations, they derived a reduced system of nonlinear PDEs valid for

convection in a cylinder in the rapidly rotating limit.

Yanagisawa and Yamagishi (2005) carried out simulations of the Rayleigh-

Benard convection with infinite Prandtl number and high Rayleigh numbers in the

spherical shell geometry to understand the thermal structure of the mantle and the

evolution of the earth. The analysis reveals that the structural scale of convection

differs between the boundary region and the isothermal core region. The structure

near the boundary region is characterized by the cell type structure constructed by

15

the sheet-shaped downwelling and upwelling flows, and that of the core region by

the plume type structure which consists of the cylindrical flows.

Ma and Wang (2007) attempted at linking the dynamics of fluid flows with the

structure of these fluid flows in physical space and the transitions of this structure.

The two-dimensional Rayleigh-Bénard convection, which serves as a prototype

problem has been given attention and the analysis is based on two recently

developed nonlinear theories: geometric theory for incompressible flows and

bifurcation and stability theory for nonlinear dynamical systems (both finite and

infinite dimensional). They have shown that the Rayleigh-Bénard problem

bifurcates from the basic state to an attractor AR when the Rayleigh number R

crosses the first critical Rayleigh number Rc for all physically sound boundary

conditions, regardless of the multiplicity of the eigenvalue Rc for the linear

problem. In addition to a classification of the bifurcated attractor AR, the structure

of the solutions in physical space and the transitions of this structure are classified,

leading to the existence and stability of two different flows structures: pure rolls and

rolls separated by a cross the channel flow.

Zhou et al. (2007) presented an experimental study of the morphological

evolution of thermal plumes in turbulent thermal convection. They noted that as the

sheet-like plumes move across the plate, they collide and convolute into spiraling

swirls and that these swirls then spiral away from the plates to become

mushroomlike plumes which are accompanied by strong vertical vorticity. The

fluctuating vorticity is found to have the same exponential distribution and scaling

behaviour as the fluctuating temperature.

Barletta and Nield (2009) revisited the classical Rayleigh–Bénard problem in an

infinitely wide horizontal fluid layer with isothermal boundaries heated from below.

The effects of pressure work and viscous dissipation are taken into account in the

energy balance. A linear analysis is performed in order to obtain the conditions of

16

marginal stability and the critical values of the wave number and of the Rayleigh

number for the onset of convective rolls. Mechanical boundary conditions are

considered such that the boundaries are both rigid, or both stress-free, or the upper

stress-free and the lower rigid. It is shown that the critical value of Ra may be

significantly affected by the contribution of pressure work, mainly through the

functional dependence on the Gebhart number and on a thermodynamic Rayleigh

number. While the pressure work term affects the critical conditions determined

through the linear analysis, the viscous dissipation term plays no role in this

analysis being a higher order effect.

Song and Tang (2010) carried out a systematic study of turbulent Rayleigh-

Bénard convection in two horizontal cylindrical cells of different lengths filled with

water. Global heat transport and local temperature and velocity measurements are

made over varying Rayleigh numbers Ra. The scaling behavior of the measured

Nusselt number and the Reynolds number associated with the large-scale circulation

remains the same as that in the upright cylinders. The scaling exponent for the rms

value of local temperature fluctuations, however, is strongly influenced by the

aspect ratio and shape of the convection cell. The experiment clearly reveals the

important roles played by the cell geometry in determining the scaling properties of

convective turbulence.

We have so far reviewed the literature pertaining to Rayleigh-Benard

convection. In what follows we review the literature on Marangoni instability in

fluids.

2.2 Marangoni Convection in Fluids

Block (1956) was of the view that Bénard cells in shallow pools are actually

produced by variations in surface tension rather than due to buoyancy force which

are in turn due to non-uniformities in temperature over the free surface, which

17

would account for the surface depressions over upwelling hot liquid. This

mechanism is called the Marangoni effect.

Pearson (1958) theoretically demonstrated that surface tension force is sufficient

to cause hydrodynamic instability in a liquid layer with a free surface, provided

there is a temperature or concentration gradient of proper sense and of sufficient

magnitude across the layer. Thus, Pearson (1958) showed that, variation of surface

tension with temperature would drive steady Marangoni convection in a fluid layer

provided the non-dimensional Marangoni number is sufficiently large and positive.

The most significant limitation of Pearson’s (1958) work is that it considers only

the case of a non-deformable free surface corresponding to the limit of strong

surface tension.

Nield (1964) studied the effect of non-uniform temperature gradient on the

onset of Marangoni convection subject to the constant heat flux on the upper free

surface. Using a single-term Galerkin technique, he obtained critical Marangoni

number for different temperature profiles. Nield (1966) also studied the effect of

vertical magnetic field in an electrically conducting fluid on the onset of combined

RBC and MC problems. It is shown that, as the magnetic field strength increases,

the coupling between the two agencies causing instability becomes weaker so that

the values of Rayleigh number at which convection begins are independent of

surface-tension effect and similarly the critical Marangoni number is unaffected by

the buoyancy forces provided the Rayleigh number is less than critical.

Vidal and Acrivos (1966) and Takashima (1970) found that, when surface

deformation was neglected, the principle of exchange of stabilities is valid in the

case of Marangoni instability.

Davis (1969) examined the linear and nonlinear Rayleigh-Bénard-Marangoni

convection using energy method. The subcritical instability was found in a small

18

range of the Marangoni convection. It is shown that the equations governing the

energy theory are independent of the linear theory problem and that the surface

tension behaves like a bounded perturbation to the Bénard problem. The effects of

surface tension and buoyancy on the convective instability of a fluid layer with a

mean parabolic temperature distribution were examined by Debler and Wolf (1970).

Sarma (1979, 1981 and 1985) analyzed the effect of both uniform rotation and

magnetic field on the onset of steady Marangoni convection in a horizontal fluid

layer with a deformable free surface for a variety of combinations of thermal and

magnetic boundary conditions. He demonstrated the stabilizing effect of

rotation/magnetic field on the Marangoni instability of a fluid layer with a

deformable free surface. Takashima (1981a, 1981b) examined the effect of a free

surface deformation on the onset of stationary and oscillatory surface tension driven

instability using linear stability theory.

Lebon and Cloot (1984) studied the nonlinear analysis of the combined RBC

and MC in a horizontal fluid layer of infinite extent. They used the Gorkov-Malkus-

Veronis technique, which consists of developing the steady solution in terms of a

small parameter measuring the deviation from the marginal state and solved the

nonlinear equations describing the fields of temperature and velocity.

Lam and Bayazitoglu (1987) solved the problem of MC in a horizontal fluid

layer with internal heat generation using the sequential gradient-restoration

algorithm (SGRA) developed for optimal control problems. It is shown that the

temperature-dependent viscosity plays a larger role than the surface tension in

determining the critical conditions.

Maekawa and Tanasawa (1988) studied theoretically the onset of MC in an

electrically conducting fluid subject to a vertical magnetic field. It is found that

convection always sets in the form of longitudinal rolls whose axes are aligned with

19

the horizontal component of the magnetic field and that only the vertical component

of the magnetic field has any effect on the critical Marangoni number.

Benguria and Depassier (1989) studied linear stability of a fluid with a free

deformable upper surface. They found that when the heat flux on the upper and

lower surface is plane and isothermal, oscillatory instability occurs at lower values

of the Rayleigh number than the critical value for the onset of steady convection.

Gouesbet et al. (1990) investigated the overstability for the combined RBC and

MC problem by means of small disturbance analysis. The influence of Prandtl,

Bond and Crispation numbers, the modification induced by interfacial viscosities,

heat transfer at the free surface, buoyancy with respect to the pure Marangoni

mechanism and different thermal conditions at the rigid wall are discussed in the

analysis. Perez-Garcia and Carneiro (1991) analyzed the effects of surface tension

and buoyancy on the convective instability in a layer of fluid with a deformable free

surface. Their analysis is restricted to fixed values of a Prandtl number and Biot

number in order to determine the role of the Crispation number on convection.

Wilson (1993a, 1993b) investigated the effect of a uniform magnetic field on

the onset of Marangoni instability in a horizontal layer of quiescent electrically

conducting fluid in the presence as well as in the absence of buoyancy force using a

combination of analytical and numerical techniques. The vertical magnetic field

was found to have a stabilizing effect . Wilson (1994) studied the problem of MC in

an electrically conducting fluid with a uniform vertical temperature gradient

subjected to a prescribed heat flux at its rigid lower boundary. The critical

Marangoni number obtained was different from that of the isothermal case.

Char and Chiang (1994a) studied the problem of MC in fluids with internal heat

generation. Thess and Nitschke (1995) derived asymptotic expressions for the first

unstable mode of surface tension driven instability in an electrically conducting

20

fluid subjected to a strong magnetic field. The spatial structure of the velocity,

temperature and electric current density is characterized in terms of Hartmann

boundary layers.

Parmentier et al. (1996) performed the first weakly nonlinear analysis of the

combined RBC and MC problem without surface deformation in the case of finite

Prandtl number. They concluded that hexagons are preferred near the onset of

convection and found that the direction of motion in the cells depends on the value

of the Prandtl number.

Wilson (1997) used a combination of analytical and numerical techniques to

analyze the effect of uniform internal heat generation on the onset of steady

Marangoni convection. He obtained for the first time a closed form analytical

solution for the onset of steady Marangoni convection and presented

asymptotically- and numerically-calculated results for the linear growth rates of the

steady modes. Char et al. (1997) investigated the onset of oscillatory instability of

MC in a horizontal fluid layer subject to the Coriolis force and internal heat

generation. The upper surface is assumed to be deformably free and the lower

surface is rigid. The Crispation number is found to be significant for the occurrence

of oscillatory modes.

Selak and Lebon (1997) studied coupled surface tension and gravity driven

instabilities in fluids with variable thermophysical properties. Viscosity is assumed

to vary exponentially with temperature, while linear laws are assumed for the heat

capacity and thermal conductivity. Computations for glycerol and liquid potassium

show that temperature dependence of the thermophysical properties may have a

significant effect on the onset of convection.

Hashim and Wilson (1999a, b) have analyzed the effect of a uniform vertical

magnetic field on the onset of oscillatory MC and on the linear growth rates of

21

steady MC in a horizontal layer of electrically conducting fluid heated from below.

Their investigations showed that the presence of a magnetic field could cause the

preferred mode of instability to be oscillatory rather than steady and that the effect

of increasing the magnetic field strength was always to stabilize the layer by

decreasing the growth rate of the unstable modes. Hashim and Wilson (1999c) have

also investigated the effects of surface tension and buoyancy on the convective

instability in a planar horizontal layer of fluid in the most physically relevant case

when the non-dimensional Rayleigh and Marangoni numbers are linearly

dependent. The comprehensive asymptotic analysis of the marginal curves in the

limit of both long and short wavelength disturbances are studied.

Kozhoukharova and Roze (1999) studied stationary and oscillatory Marangoni

instability in a fluid layer with a deformable upper surface. The viscosity is assumed

to be temperature-dependent and the problem has been solved numerically by

Taylor series expansion method. The results of the study reveal that oscillatory

convection is possible only when the surface deformability is considered. Bau

(1999) demonstrated for the first time the critical Marangoni number for transition

from the conduction state to the motion state can be increased through the use of

feedback control strategies.

Hashim (2001) obtained for the first time an analytical description of the

marginal mode for the onset of oscillatory MC in the presence of uniform heat

sources in the asymptotic limit of short waves. Hashim and Arifin (2003) presented

numerically a necessary and sufficient condition for oscillatory MC in a horizontal

layer of electrically conducting fluid in the presence of a vertical magnetic field.

Borcia and Bestehorn (2006) presented a collection of methods for describing

Marangoni convection in two-layer systems in the phase field formalism. The

model consists of the Navier-Stokes equation with an extra term of the phase field,

the heat equation, and a spatiotemporal evolution equation for the phase field. It is

22

shown that the fluid-fluid interface becomes diffuse and free of interface conditions.

2D convection patterns are shown for incompressible, compressible, and

compressible evaporating fluids.

Murty (2006) studied the effect of throughtlow and a uniform magnetic field on

the onset of Marangoni convection in a horizontal layer of micropolar fluid bounded

below by a rigid isothermal surface and above by a non-deformable free adiabatic

surface is studied. It is observed that both stabilizing and destabilizing factors due to

constant vertical throughflow can be enhanced by magnetic field.

Simanovskii et al. (2006) investigated the influence of the horizontal component

of the temperature gradient on nonlinear regimes of oscillatory Marangoni

convection in a real symmetric three-layer system. The transitions between different

flow regimes have been studied. The general diagram of regimes is constructed.

Maroto et al. (2007) described experiments on Bénard–Marangoni convection

which permit a useful understanding of the main concepts involved in this

phenomenon such as Bénard cells, aspect ratio, Rayleigh and Marangoni numbers,

Crispation number and critical conditions. In spite of the complexity of convection

theory, they carried out a simple and introductory analysis which has the additional

advantage of providing very suggestive experiments.

Saravanan (2007) investigated the onset of Marangoni convection in a

horizontal Oldroyd-B fluid layer in the presence of a vertical throughflow using

linear stability analysis. An approximate solution of the resulting eigenvalue

problem is found by means of the Galerkin technique.

Guo and Narayanan (2007) examined the onset of Rayleigh-Benard-Marangoni

convection in a vertical annulus heated from below using linear stability analysis.

The results of the present study also show the pattern transitions as a function of

23

scaled gap width and aspect ratio. It is concluded that Marangoni convection can

change the fluid pattern in an otherwise pure Rayleigh problem. It is also concluded

that the gap width cannot significantly change the Marangoni effect as it is

essentially the depth of liquid and Biot number that play a dominant role.

Siri and Hashim (2008) analyzed the problem of the effect of feedback control

on the onset of steady and oscillatory Marangoni convection in a rotating horizontal

fluid layer. The role of the controller gain parameter on the Pr - Ta parameter space

is determined. Arifin and Bachok (2008) studied the onset of Marangoni convection

in a horizontal fluid layer with internal heat generation overlying a solid layer

heated from below. The effects of the thermal conductivity and the thickness of the

solid plate on the onset of convective instability with internal heating are discussed

in detail.

Abidin et al. (2008) studied the onset of Marangoni convection in a horizontal

fluid layer with a free surface overlying a solid layer heated from below. Problem is

focused on the effect of the solid layer depth or its conductivity. The viscosity

group, Biot number, depth ratio, and conductivity ratio, are significant on

determining the critical Marangoni number with the corresponding critical wave

number. The characteristics problem is solved numerically. Results show that the

temperature dependent viscosity destabilizes the fluid system, but it behaves

oppositely when a higher relative thermal conductivity ratio or higher depth ratio is

taken.

Hashim and Siri (2009) applied the linear stability theory to investigate the

effects of rotation and feedback control on the onset of steady and oscillatory

thermocapillary convection in a horizontal fluid layer heated from below with a

free-slip bottom. The thresholds and codimension-2 points for the onset of steady

and oscillatory convection are determined. Mahmud et al. (2009) investigated

thermocapillary instability in a horizontal liquid layer in the presence of a uniform

24

vertical magnetic field and suspended particles. The resulting eigenvalue problem is

solved for various basic temperature gradients.

Mahmud et al. (2009) investigated thermocapillary instability in a horizontal

liquid layer with the presence of a uniform vertical magnetic field and suspended

particles. The resulting eigenvalue is solved by the Galerkin technique for various

basic temperature gradients. It is found that the presence of magnetic field always

has a stability effect of increasing the critical Marangoni number.

Isa et al. (2009) considered the effect of non-uniform basic temperature gradients

on the onset of Marangoni convection in a horizontal layer with a free-slip bottom

heated from below and cooled from above. A linear stability analysis is performed

to undertake a detail investigation. The eigenvalues are obtained for lower rigid

isothermal and upper free adiabatic boundaries. The resulting eigenvalue problem is

solved using single-term Galerkin expansion procedure. The influence of various

parameters on the onset of convection has been analyzed. Six non-uniform basic

temperature profiles are considered and some general conclusions about their

destabilizing effects are presented.

Li et al. (2010) considered transition to chaos in double-diffusive Marangoni

convection in a rectangular cavity with horizontal temperature and concentration

gradients. Attention is restricted to the special case when the resultant thermal and

solutal Marangoni effects are equal and opposing. Direct numerical simulation is

used and some techniques from nonlinear dynamics are adopted to identify the

different dynamic regimes. It is found that the supercritical solution branch takes a

quasi-periodicity and phase locking route to chaos while the subcritical branch

follows the Ruelle–Takens–Newhouse scenario. Transient intermittency in the

supercritical branch is observed and physical instability mechanisms of the

subcritical branch are identified.

25

Mokhtar et al. (2010) applied linear stability analysis to investigate the effect of

internal heat generation on Marangoni convection in a fluid saturated porous

medium bounded below by an insulating rigid surface and above by a non-

deformable free surface. The Darcy law and the Brinkman model are used to

describe the flow in the porous medium heated from below. The conditions for the

onset of instability occurring via steady convective modes are solved exactly. The

asymptotic solution of the long wavelength is also obtained using regular

perturbation technique with wave number as a perturbation parameter. It is observed

that the critical Marangoni number decreases with an increase in the dimensionless

heat source strength.

Shklyaev et al. (2010) studied long-wave Marangoni convection in a layer

heated from below. Using the scaling k = O(Bi), where k is the wave number and Bi

is the Biot number, they derived a set of amplitude equations. Analysis of this set

shows presence of monotonic and oscillatory modes of instability. Oscillatory mode

has not been previously found for such direction of heating. Studies of weakly

nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be

found near the stability threshold.

For detailed descriptions of linear and nonlinear problems of both RBC and MC,

one may refer to the books of Chandrasekhar (1961), Gershuni and Zhukhovitsky

(1976), Kays and Crawford (1980), Ziener and Oertel (1982), Platten and Legros

(1984), Gebhart et al. (1988), Getling (1998), Colinet et al. (2001) and Straughan

(2004). Chapters on thermal convection are included in the books by Turner (1973),

Joseph (1976), Tritton (1979) and Drazin and Reid (1981). Reviews of research on

convective instability have been given by Normand et al. (1977), Davis (1987) and

Bodenschatz et al. (2000).

26

We have so far reviewed the literature pertaining to Rayleigh-Benard/

Marangoni convection. In what follows we review the literature concerning the

convective instability problems with radiative transfer.

2.3 Convection with Thermal Radiation Strictly speaking, due to interactions, the three heat transfer mechanisms, viz.,

conduction, convection and radiation are inseparable. In a physical sense, the

amount of heat transfer in conduction and convection depends upon the temperature

difference whereas that of radiation depends upon both the temperature difference

and the temperature level. Mathematically, heat transfer by conduction and

convection is governed by a set of partial differential equations, while the radiation

problem is quite complicated because it requires the solution of nonlinear

integro-differential equations.

There exist situations in which thermal radiation is important even though the

temperature may not be high. It is a fact that, even under some of the most

unexpected situations, the radiation heat transfer could account for a non-negligible

amount of total heat transfer. Earlier works on heat transfer in Newtonian fluids

considered convection and conduction and overlooked the effect of thermal

radiation (Siegel and Howell, 1992; Modest, 1993; Howell and Menguc, 1998). The

available literature barely delineates the part played by convection in a fluid

combined with radiation. The formulation of heat transfer by conduction and

convection leads to differential equations while that by radiation leads to integral

equations. Thus the complexity involved in the solution of the integro-differential

equations resulting from the combined convection and radiation problem warrants

the use of several simplifying assumptions. In this thesis we restrict our attention to

the case in which the absorption coefficient of the fluid is the same at all

wavelengths and is independent of the physical state (the so-called gray medium

27

approximation). The equation of radiative transfer is developed for optically thick

and thin approximations and the effect of scattering is ignored.

Goody (1956) investigated the RBC problem subject to radiative transfer using a

variational technique and a gray, two-stream radiative model to find the critical

conditions for linear stability. He considered the limits of optically thin and thick

fluids, and free-slip, optically black boundaries. Goody noted that radiative

damping tends to diminish temperature perturbations and it causes the basic

temperature profile in the interior of the domain to have a more stable lapse rate.

Spiegel (1960) considered the RBC problem in a radiating fluid layer for rigid

boundaries and for the entire range of optical thickness but neglected the effect of

conduction. The principle of exchange of stabilities is proved and the critical value

for instability is given as a function of the optical thickness of the layer. Following

Goody’s approach, Murgai and Khosla (1962) and Khosla and Murgai (1963),

respectively, included the effects of magnetic field and rotation. The principle of

exchange of stabilities and the concept of overstability have been discussed.

Christophorides and Davis (1970) added thermal conduction and Goody’s static

temperature profile to Spiegel’s integral formulation when limited to optically thin

media. An estimate for the convective heat transport in a transparent medium is

made using the shape factor assumption and compared with non-radiative

convection.

Arpaci and Gozum (1973) were the first to introduce the effects of fluid non-

grayness and boundary emissivities. Using the analysis of Arpaci and Gozum,

Onyegegbu (1980) added the rotation effect, and Yang (1990) introduced external

convective boundary conditions for rigid boundaries, which is suitable for solar

collector applications.

28

Thermal radiation from finite-conductivity boundaries can strongly affect the

stability of horizontally unbounded plane fluid layers heated from below. Lienhard

(1990) studied the role of thermal radiation in plane-layer instabilities under the

assumption that the fluid medium is transparent (as a model of IR transfer through

gas layers). The solution procedure modifies a previous formulation of the

conductive boundary problem to account for the gray radiant interchange between

boundaries. The non-isothermal character of the boundaries is shown to bias

instability toward higher wave numbers and to substantially increase the stability of

fluid layers between radiative non-conductive boundaries relative to layers having

non-radiative boundaries. A single layer is studied first, and then a case of parallel

interacting fluid layers is considered. Critical Rayleigh numbers are presented as

both tabulations and correlations. The implications for solar-collector design are

discussed.

Bdeoui and Soufiani (1997) provided a sophisticated treatment of nongray fluids

and also a short review of prior work. Mansour and Gorla (1999) presented a

regular perturbation analysis for the radiative effects on laminar natural convection

with temperature-dependent viscosity.

Mohammadein et al. (1998) presented a regular two-parameter perturbation

analysis based upon the boundary layer approximation to study the radiative effects

of both first- and second-order resistances due to a solid matrix on the natural

convection flows in porous media. Four different flows have been studied, those

adjacent to an isothermal surface, a uniform heat flux surface, a plane plume and the

flow generated from a horizontal line energy source on a vertical adiabatic surface.

The first-order perturbation quantities are presented for all these flows. Numerical

results for the four conditions with various radiation parameters are tabulated.

29

Mohammadien and El-Amin (2000) studied the effects of thermal dispersion and

thermal radiation on the non-Darcy natural convection over a vertical flat plate in a

fluid saturated porous medium. Forchheimer extension is considered in the flow

equations. The coefficient of thermal diffusivity has been assumed to be the sum of

molecular diffusivity and the dispersion thermal diffusivity due to mechanical

dispersion. Rosseland approximation is used to describe the radiative heat flux in

the energy equation. Similarity solution for the transformed governing equations is

obtained. Numerical results for the details of the velocity and temperature profiles

have been presented. The combined effect of thermal dispersion and thermal

radiation, for the two cases Darcy and non-Darcy porous medium, on the heat

transfer rate is discussed.

Larson (2001) studied linear and nonlinear stability properties of Goody’s model

analytically. When thermal diffusivity is zero, the energy method is used to rule out

subcritical instabilities. When thermal diffusivity is nonzero, the energy method is

used to find a critical threshold below which all infinitesimal and finite amplitude

perturbations are stable.

Mansour and El-Shaer (2001) presented a boundary layer solution to study the

effects of joule heating on magnetohydrodynamic natural convection flow. The

Rosseland approximation is used to describe the radiative heat flux in the energy

equation. Four different cases of flows have been studied namely an isothermal

surface, a uniform heat flux surface, a plane plume and flow generated from a

horizontal line energy source a vertical adiabatic surface. Numerical results

presented for the perturbation analysis four boundary conditions with various

parameters are tabulated

Seddeek and Aboeldahab (2001) investigated radiation effect on unsteady free

convection flow of an electrically conducting, gray gas near equilibrium in the

optically thin limit along an infinite vertical porous plate in the presence of strong

30

transverse magnetic field imposed perpendicularly to the plate, taking Hall currents

into account. Similarity equations are derived and solved numerically using the

shooting method. The numerical values of skin friction and the rate of heat transfer

are represented in a table. The effects of radiation parameter, Hall parameter, and

magnetic field parameters are discussed and shown graphically.

Taniguchi et al. (2002) attempted to introduce a simulation technique for

radiation–convection heat transfer in the high-temperature fields of industrial

furnaces, boilers, and gas turbine combustors. The authors introduced the zone

method and Monte-Carlo method for the integral equation of the radiation effect

and the finite difference method for the differential equation of the convection

effect. A three-dimensional analysis of the high-temperature furnace was performed

by this simulation technique to obtain its temperature distribution. Furthermore,

another radiation–convection heat transfer analysis in the low-temperature living

room was performed by the same technique. Finally, the authors tried to develop a

computer software for radiation–convection heat transfer and described their idea of

software construction for the above.

Lan et al. (2003) analyzed the stability of a fluid subject to combined natural

convection and radiation using a spectral method. Black boundaries and a gray

medium are prescribed. The influences of conduction-radiation parameter, Rayleigh

number and optical thickness on flow instabilities and bifurcations are discussed.

The knowledge of steady-state temperature distribution is known a priori for

natural convection stability analysis. The study of Liu (2003) determines the

temperature profiles in an infinite horizontal layer of fluid filled with saturated

porous medium confined between two convective surfaces subjected to external

radiative incidence and imposed downward convection. The investigation

concentrates on the influences of the thermal boundary condition (Biot number),

internal and external Rayleigh numbers, Peclet number, optical thickness, and

31

surface reflectivities. The implications of these factors on fluid stability are

addressed in detail.

Sunil et al. (2004) considered the thermosolutal convection in a ferromagnetic

fluid for a fluid layer heated and soluted from below in the presence of uniform

vertical magnetic field. For the case of two free boundaries, an exact solution is

obtained using a linear stability analysis. For the case of stationary convection,

magnetization has a destabilizing effect, whereas stable solute gradient has a

stabilizing effect on the onset of instability. The principle of exchange of stabilities

is found to hold true for the ferromagnetic fluid heated from below in porous

medium in the absence of stable solute gradient. The oscillatory modes are

introduced due to the presence of the stable solute gradient, which were non-

existent in its absence. A sufficient condition for non-existence of the overstability

is also obtained.

Osalusi (2007) studied the combined hydromagnetic and slip flow of a steady,

laminar conducting viscous fluid in the presence of thermal radiation due to an

impulsively started rotating porous disk taking into account the variable fluid

properties (density, ρ, viscosity, µ, and thermal conductivity, κ). These fluid

properties are taken to be dependent on temperature. The system of axisymmetric

nonlinear partial differential equations governing the MHD steady flow and heat

transfer are written in cylindrical polar coordinates and reduced to nonlinear

ordinary differential equations by introducing suitable similarity parameters. The

resulting steady equations are reduced to an initial valued problem and solved

numerically using a shooting method. A parametric study of all parameters involved

was conducted, and a representative set of results showing the effect of the

magnetic field, the radiation parameter, slip factor, the uniform suction parameter

and the relative temperature difference parameter on velocities, temperature,

skinfriction and Nusselt number are illustrated graphically.

32

Xu et al. (2007) studied the combined radiation and natural convection in an

open vertical cylinder. The cylinder is heated through its sidewall at a constant heat

flux and cooled at the top surface via radiation. The high order finite difference

method is used to simulate the fluid flow and heat transfer inside the cylinder and

the internal radiation is solved using discontinuous finite element method. The two

numerical methods are coupled through an iterative process. Based on the numerical

simulations, a linear stability analysis is carried out to investigate how the internal

radiation changes the stability of the flow.

Ghosh and Anwar (2008) investigated the unsteady free convective flow of a

viscous incompressible gray, absorbing-emitting but non-scattering, optically-thick

fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot

vertical plate with constant velocity. A Darcian viscous flow model is employed for

the porous medium. The momentum and thermal boundary layer equations are non-

dimensionalized using appropriate transformations and then solved subject to

physically realistic boundary conditions using the Laplace transform technique.

Thermal radiation effects are simulated via a radiation-conduction parameter based

on the Rosseland diffusion approximation. The influence of Grashof (free

convection) number, radiation-conduction parameter, inverse permeability

parameter and dimensionless time are studied graphically. The authors observed

that increasing thermal radiation parameter causes a noticeable increase in the flow

velocity. Temperature is significantly increased within the boundary layer with a

rise in radiation-conduction since the latter represents the relative contribution of

thermal radiation heat transfer to thermal conduction heat transfer. Increased

radiation therefore augments heat transfer, heats the fluid and increases the

thickness of the momentum and thermal boundary layers. This study has

applications in geological convection, forest fire propagation, glass heat treatment

processes at high temperature, metallurgical processing etc.

33

Rahman and Sultana (2008) analyzed a two-dimensional steady convective flow

of a micropolar fluid past a vertical porous flat plate in the presence of radiation

with variable heat flux numerically. Using Darcy-Forchheimer model the

corresponding momentum, microrotation and energy equations have been solved

numerically. The local similarity solutions for the flow, microrotation and heat

transfer characteristics are illustrated graphically for various material parameters.

The effects of the pertinent parameters on the local skin friction coefficient, plate

couple stress and the heat transfer are also calculated. It was shown that large Darcy

parameter leads to decrease the velocity while it increases the angular velocity as

well as temperature of the micropolar fluids. The rate of heat transfer in weakly

concentrated micropolar fluids is higher than strongly concentrated micropolar

fluids.

Anwar et al. (2008) studied the effects of thermal radiation and porous drag

forces on the natural convection heat and mass transfer of a viscous,

incompressible, gray, absorbing, emitting fluid flowing past an impulsively started

moving vertical plate adjacent to a non-Darcian porous regime. The Rosseland

diffusion approximation is employed to analyze the radiative heat flux and is

appropriate for non-scattering media. The model is non-dimensionalized and solved

with the network simulation model. The authors studied the influence of Prandtl

number, radiation-conduction parameter, thermal Grashof number, species Grashof

number, Schmidt number, Darcy number and Forchheimer number on the

dimensionless velocity, temperature and species function distributions. Additionally

they computed the variation of the local skin friction, Nusselt number and

Sherwood number for selected thermophysical parameters. Increasing Darcy

number is seen to accelerate the flow; the converse is apparent for an increase in

Forchheimer number. Thermal radiation is seen to reduce both velocity and

temperature in the boundary layer. The interactive effects of second order porous

34

drag and thermal radiation are also considered. The model finds applications in

solar energy collection systems, porous combustors, transport in fires in porous

media (forest fires) and also the design of high temperature chemical process

systems.

Ishak (2009) investigated the effects of thermal radiation and thermal buoyancy

on the steady, laminar boundary layer flow over a horizontal plate. The plate

temperature is assumed to be inversely proportional to the square root of the

distance from the leading edge. The set of similarity equations is solved

numerically, and the solutions are given for some values of the radiation and

buoyancy parameters for Prandtl number unity. It is found that dual solutions exist

for negative values of the buoyancy parameter, up to certain critical values. Beyond

these values, the solution does no longer exist. Moreover, it is found that there is no

local heat transfer at the surface except in the singular point at the leading edge. The

radiation parameter is found to increase the local Stanton number.

Chauhan and Rastogi (2010) investigated the unsteady natural convection MHD

flow of a rotating viscous electrically conducting fluid in a vertical channel partially

filled by a porous medium with high porosity in the presence of radiation effects. It

is assumed that the conducting fluid is gray, emitting-absorbing radiation, and non-

scattering medium. The two infinite vertical porous plates of the channel are

subjected to a constant injection velocity at the one plate and the same constant

suction velocity at the other plate. The entire system rotates about the axis normal to

the plates with a uniform angular velocity. The analytic expressions for velocity and

temperature field are obtained and effects of the radiation-conduction parameter

(Stark number), Prandtl number, Grashof number, rotation parameter, magnetic

field, and permeability of the porous medium on the velocity field, temperature field

and Nusselt number have been discussed in the analysis.

35

Khaleque and Samad (2010) analyzed the radiation and viscous dissipation

effects on a steady two-dimensional magneto-hydrodynamics free convection flow

along a stretching sheet with heat generation. The nonlinear partial differential

equations, governing the flow field under consideration have been transformed by a

similarity transformation into a system of nonlinear ordinary differential equations

and then solved numerically by applying Nachtsheim-Swigert shooting iteration

technique together with sixth order Runge-Kutta integration scheme. Resulting non-

dimensional velocity, temperature and concentration profiles are then presented

graphically for different values of the parameters of physical and engineering

interest.

Shateyi et al. (2010) sought to investigate the influence of a magnetic field on

heat and mass transfer by mixed convection from vertical surfaces in the presence

of Hall, radiation, Soret thermal diffusion, and Dufour-diffusion-thermo effects.

The similarity solutions were obtained using suitable transformations. The

similarity ordinary differential equations were then solved by MATLAB routine

bvp4c. The numerical results for some special cases were compared with the exact

solution and were found to be in good agreement. A parametric study illustrating

the influence of the magnetic strength, Hall current, Dufour, and Soret, Eckert

number, thermal radiation, and permeability parameter on the velocity, temperature,

and concentration was investigated.

We next review the literature pertaining to convective instability problems of

porous media.

2.4 Convection in a Porous Medium The early pioneering work concerning flow through porous media began about

one and a half centuries ago (Darcy, 1856). Later Muskat (1937) achieved a high

degree of correlation between theory and experiment and thus paved the way for a

36

rapid development of studies in flow through porous media. Since then extensive

investigations have been conducted by hydrologists, petroleum geologists, chemical

engineers, geologists, geophysicists, and applied mathematicians on flow

characteristics and heat transfer in a porous medium which covers a broad range of

different fields with wide applications.

The first attempt to derive Darcy’s law from the basic principles of fluid

mechanics was given by Hubbert (1956). Following Hubbert’s approach, Whittekar

(1966) attempted to derive the Darcy’s law by an integration of the Navier-Stokes’

equations about a local representative volume in a porous medium, An important

advance in this approach was made independently by Slattery (1967) and Whittekar

(1967). Both developed a theorem concerning the volume average. This average

theorem served as the basis for a rigorous derivation of macroscopic equations for a

porous medium flow from microscopic equations, which would help in gaining an

insight into the assumptions involved.

The motivation for the study of convective instability in a porous medium, a

widespread phenomenon in nature and with rich technological applications, seems

to have emerged from the similarity of this flow with the usual Rayleigh-Benard

convection. In fact, there is a fairly close analogy between these two from a

phenomenological viewpoint.

The basic assumption involved in the analytical treatment of the study of free

convection in a porous medium is that the Darcy flow model applies. Coupled with

the Boussinesq incompressible fluid model, the Darcy flow assumption leads to a

set of linear momentum equations. However, the mathematical problem remains

weakly nonlinear due to the convective heat transport term present in the energy

equation. The Darcy law is valid only when the permeability parameter of the

porous medium is very small ( 35 1010 −− − ). Nevertheless, for values of the

37

permeability parameter in the range 23 1010 −− − , Darcy model is not valid and

one has to consider the Brinkman (1947) model.

The infinitesimal convection occurring in a horizontal fluid saturated porous

layer heated from below has been extensively studied since the first papers by

Horton and Rogers (1945) and Lapwood (1948). Although a study related to the so-

called geothermal or hot spring areas was made earlier by Einarsson (1942), the

possibility of free convection in a porous medium heated uniformly from below and

its similarity with the Rayleigh-Benard problem was pointed out by Horton and

Rogers (1945) and Lapwood (1948). The much more speculative possibility is that

the earth’s mantle behaves like a porous medium. This idea has been used in a

discussion of earth-quake sources by Frank (1965) and in a model of volcanism by

Elder (1966). Lapwood (1948), using linear theory, determined the criterion for the

onset of convection.

Early experimental studies of Morrison (1947), Morrison et al. (1949), Rogers

and Schilberg (1951) and Rogers et al. (1951) exhibited quantitative disagreement

with the theoretical predictions of Horton and Rogers (1945) and Lapwood (1948)

in the sense that the observed critical temperature gradients were less, by an order

of magnitude, than the predicted results. This gap between theoretical and

experimental results was reduced to some extent by Rogers and Morrison (1950),

Rogers et al. (1951), Morrison and Rogers (1952), Elder (1958) and List (1965) by

means of some adhoc patching up of the theory to allow property variations such as

the variation of viscosity with temperature, nonlinear temperature distribution etc.

(Nield, 1968). List (1965) studied the effect of uniformly distributed heat sources

while Gheorghitza (1961) restricted his analysis to infinitesimal disturbances in an

inhomogeneous porous layer. Lapwood’s (1948) problem was greatly extended by

Wooding (1957, 1958, 1959, 1960, 1963) both theoretically and experimentally.

38

Wooding (1959) investigated the problem of the stability of a viscous liquid, the

density of which increases with height, in a vertical tube with insulating wall

containing porous material. This problem is analogous to that of Taylor (1954) and

analytically simpler than Taylor’s but many of the qualitative physical

characteristics are similar. Wooding (1959) found that the equilibrium of the liquid

is stable provided that the density gradient does not exceed certain value.

The investigation carried out by Wooding (1960) on the onset of convection in a

Hele-Shaw cell with viscous fluid is analogous to the two-dimensional convection

in a horizontal porous layer. Several other experimental studies (Schneider, 1963;

Elder, 1967; Katto and Masuoka, 1967) have been carried out as a satisfactory test

of the Horton-Rogers-Lapwood theory and a good conformity is achieved.

Elder (1967) investigated the problem of steady free convection in a porous

medium experimentally using Schmidt-Miltoverton heat transfer technique and

numerically by a method proposed by himself in an earlier study. He observed that

as in the case of ordinary viscous fluid, the motion exists only when the Rayleigh

number exceeds its critical value and is of multicellular nature which would be

considerably affected by the end effects. In fact, he studies the flow for variety of

boundary conditions. Further he modified the result when the boundary layer

thickness was comparable to the grain size and found a linear asymptotic relation

between the Nusselt number and the Rayleigh number. This relation means that the

amount of heat transferred is independent of the layer thickness and thermal

conductivity of the porous medium. Elder (1967) showed that this relation holds

only when certain condition is satisfied. As the Rayleigh number becomes large, the

sharp temperature gradient regions began to be confined to just near the boundary

wall, there may appear an influence of the characteristic dimension of the porous

medium as pointed out by Elder (1967).

39

The Benard problem from the viscous flow limit to the Darcy’s law limit has

been considered both theoretically and experimentally by Katto and Masuoka

(1967) and theoretically by Walker and Homsy (1977). Katto and Masuoka (1967)

and Mausoka (1972) pointed out that the region over which the criterion for the

onset of convection is applicable is valid up to the magnitude of 32 10−≅

dk . In

particular, they suggested that the thermal diffusivity k which appears in the

Rayleigh number should be treated as conductivity of the medium divided by the

heat capacity per unit mass of the fluid and not that of the medium. They have also

predicted the criterion for the onset of convection in a fluid layer with spherical

fillings.

Mausoka (1972) investigated the heat transfer by free convection in a horizontal

porous layer composed of glass balls and water both theoretically and

experimentally. The theoretical investigation was carried out in a region which

slightly exceeds the critical condition for the onset of convection using the

eigenfunction expansion. The experimental study covers a wide region. The heat

transfer above the critical state is divided into two regions. He showed that the low

Rayleigh number region is a process of transition to the high Rayleigh number

region.

The effect of vertical through flow on the onset of convection in a two

dimensional porous layer bounded laterally by insulated walls was studied by

Sutton (1970) using series expansion method. He showed that at large values of

aspect ratio the critical Rayleigh number approaches the value of 24π as obtained

by Lapwood (1948). Further, he found that at large aspect ratio, the critical

Rayleigh number increases with increasing values of dimensionless flow strength.

40

Beck (1972) examined the effect of lateral walls on the onset of convection in an

enclosed three-dimensional porous medium with heating from below using both the

approaches (Joseph, 1965; Westbrook, 1969) of energy method. Beck (1972)

showed that the lateral walls have little effect on the critical Rayleigh number

except for in very narrow tall boxes. Moreover, he noted that two-dimensional rolls

are invariably preferred whenever the height is not the smallest dimension.

Furthermore, when rolls do form, they are not necessarily parallel to the shortest

side in the case of viscous flow (Davis, 1967).

Natural convection is an important heat transfer mechanism in the technology of

building insulation. From the viewpoint of basic research in heat transfer, the

phenomenon is being studied mainly in terms of simple models of free convection

in rectangular enclosures filled with either Newtonian fluid or with a fluid-saturated

porous medium. The subject of free convection in enclosures is extensive and has

numerous applications to practical engineering situations. A comprehensive review

of free convection heat transfer in enclosures filled with fluid was presented by

Ostrach (1972).

The effect of inclination angle and the aspect ration on the onset of free

convection and heat transfer in an inclined porous layer with differentially heated

side walls was studied both theoretically and experimentally by Bories and

Combarnous (1973). They determined the critical conditions for the transition

between unicellular and polycellular flows.

The equivalent of Gill’s (1966) theory for convection in a vertical porous slot

was reported a few years later by Weber (1975) whose theoretical result for Nusselt

number agreed fairly with the experimental data reported by Schneider (1963) and

Klarsfeld (1970).

41

Bejan (1979) modified the Weber theory fitting the boundary layer solution with

average zero energy flux conditions along the top and bottom walls. The Nusselt

number predicted by the modified theory agreed extremely well with the

experimental data of Schneider (1963) and Klarsfeld (1970) as well as the

numerical heat transfer calculations reported by Bankvall (1974) and Burns et al.

(1977).

Heat transfer by natural convection in a vertical porous layer with horizontal

heat flow has been studied both theoretically and experimentally by Masuoka et al.

(1981). A boundary layer analysis is extended to take account of both the vertical

temperature gradient in the core of the porous layer and the apparent wall film

thermal resistance which is caused by a local increase in porosity in the vicinity of

the wall.

Griffiths (1981) studied the layered double diffusive convection in a porous

medium consisting of glass spheres experimentally. He showed that layered double

diffusive convection of a fluid within a porous medium is possible. A thin diffusive

interface was observed in a Hele-Shaw cell and in a laboratory porous medium,

despite the lack of inertial forces, with salt and sugar or heat and salt as the

diffusing components.

Finite amplitude thermohaline convection in a horizontal porous layer has been

studied by Srimani (1981) using the power integral technique which is based on the

Stuart’s shape assumption. By comparing the results of two-dimensional and three-

dimensional analysis, she concluded that the only stable finite amplitude solutions

of the infinite number of possible steady solutions is the two-dimensional rolls.

The stability of a fluid saturated porous layer subject to sudden rise in surface

temperature was investigated by Caltagirone (1980) using linear theory, energy

method and a two-dimensional numerical method. He concluded that linear theory

42

gives a sufficient condition for instability and the energy method gives a sufficient

condition for stability. Further, he found a good agreement between numerical and

energy method.

Pattern formation in convection in a porous medium is less easily visualized by

shadowgraph techniques because of the difficulties of transmitting light through the

porous medium. Howle et al. (1993) showed how these difficulties can be overcome

by constructing porous media in which the interfaces between solid and liquid are

either parallel or perpendicular to the confining boundaries of the experimental

system. Convection in such a medium can be visualized using conventional

shadowgraph methods, and they compared the stationary flow patterns observed

against measurements of heat transport.

Herron (2000) treated the problem of Rayleigh-Benard convection with internal

heat source and a variable gravity field. For the case of stress-free boundary

conditions, it is proved that the principle of exchange of stabilities holds as long as

the product of gravity field and the integral of the heat source is nonnegative

throughout the layer.

Siddheshwar and Sri Krishna (2001) studied the qualitative effect of nonuniform

temperature gradient on the linear stability analysis of the Rayleigh-Benard

convection problem in a Boussinesquian, viscoelastic fluid-filled, high-porosity

medium numerically using the single-term Galerkin technique. The eigenvalue is

obtained for free-free, free-rigid, and rigid-rigid boundary combinations with

isothermal temperature conditions. Thermodynamics and also the present stability

analysis dictates the strain retardation time to be less than the stress relaxation time

for convection to set in as oscillatory motions in a high-porosity medium.

43

Malashetty and Basavaraja (2002) investigated the effect of time-periodic

temperature/gravity modulation at the onset of convection in a Boussinesq fluid-

saturated anisotropic porous medium using a linear stability analysis. Brinkman

flow model with effective viscosity larger than the viscosity of the fluid is

considered to give a more general theoretical result. The perturbation method is

applied for computing the critical Rayleigh and wave numbers for small amplitude

temperature/gravity modulation. The shift in the critical Rayleigh number is

calculated as a function of frequency of the modulation, viscosity ratio, anisotropy

parameter and porous parameter.

Degan and Vasseur (2003) investigated the effect of anisotropy on the onset of

natural convection heat transfer in a fluid saturated porous horizontal cavity

subjected to nonuniform thermal gradients analytically. The porous layer is heated

from the bottom by a constant heat flux while the other surfaces are being insulated.

The horizontal boundaries are either rigid/rigid or stress-free/stress-free. The

hydrodynamic anisotropy of the porous matrix is considered. The principal

directions of the permeability are oriented in a direction that is oblique to the

gravity. Based on a parallel flow assumption, closed-form solution for the flow and

heat transfer variables, valid for the onset of convection corresponding to

vanishingly small wave number, is obtained in terms of the Darcy–Rayleigh number

Ra, the Darcy number Da, and the anisotropic parameters K* and θ.

Aniss et al. (2005) investigated the convective instability of a horizontal Hele–

Shaw liquid layer subject to a time-varying gradient of temperature. The stationary

component of the temperature gradient is considered either different or equal to

zero. The aspect ratio of the cell is considered smaller than unity. They examined

the effects of temperature oscillations on the onset of convective instability for these

two asymptotic cases. They have shown that for the first regime, modulation of

temperature has no effect on the convective threshold and that in contrast, the

44

second regime presents a competition between the harmonic and subharmonic

modes at the onset of convection.

Malashetty et al. (2005) examined analytically the stability of a horizontal fluid

saturated sparsely packed porous layer heated from below and cooled form above

when the solid and fluid phases are not in local thermal equilibrium. The Lapwood–

Brinkman model is used for the momentum equation and a two-field model is used

for energy equation each representing the solid and fluid phases separately.

Although the inertia term is included in the general formulation, it does not affect

the stability condition since the basic state is motionless. The linear stability theory

is employed to obtain the condition for the onset of convection. The effect of

thermal non-equilibrium on the onset of convection is discussed. It is shown that the

results of Darcy model for the non-equilibrium case can be recovered in the limit as

Darcy number Da → 0. Asymptotic analysis for both small and large values of the

inter phase heat transfer coefficient H is also presented. An excellent agreement is

found between the exact solutions and asymptotic solutions when H is very small.

Hill (2005) performed linear and nonlinear stability analyses of double-diffusive

convection in a fluid saturated porous layer with a concentration based internal heat

source using Darcy’s law.

Zhao and Bau (2006) studied theoretically the ability of linear controllers to

stabilize the conduction state of a saturated porous layer heated from below and

cooled from above. Proportional, suboptimal robust and linear quadratic Gaussian

controllers are considered. As a model system, they examined two-dimensional

convection in a box containing a saturated porous medium, heated from below and

cooled from above. The heating is provided by a large number of individually

controlled heaters. It is found that, by appropriate selection of a controller, one can

minimize, but not eliminate, the controlled linear system’s non-normality.

45

Nield and Kuznetsov (2006) applied the classical Rayleigh–Bénard theory to a

bidisperse porous medium. The linear stability analysis leads to an expression for

the critical Rayleigh number as a function of a Darcy number, two volume

fractions, a permeability ratio, a thermal capacity ratio, a thermal conductivity ratio,

an inter-phase heat transfer parameter and an inter-phase momentum transfer

parameter.

Straughan (2006) showed that the global nonlinear stability threshold for

convection with a thermal nonequilibrium model is exactly the same as the linear

instability boundary. This result is shown to hold for the porous medium equations

of Darcy, Forchheimer or Brinkman. This optimal result is important because it

shows that linearized instability theory has captured completely the physics of the

onset of convection. The equivalence of the linear instability and nonlinear stability

boundaries is also demonstrated for thermal convection in a non-equilibrium model

with the Darcy law, when the layer rotates with a constant angular velocity about an

axis in the same direction as gravity.

Nield (2007) examined the impracticality of MHD convection in a porous

medium. Nield and Kuznetsov (2007) investigated the effects of both horizontal and

vertical hydrodynamic and thermal heterogeneity on the onset of convection in a

horizontal layer of a saturated bidisperse porous medium uniformly heated from

below using linear stability theory for the case of weak heterogeneity. It is found

that the effect of such heterogeneity on the critical value of the Rayleigh number Ra

based on mean properties is of second order if the properties vary in a piecewise

constant or linear fashion.

Bhadauria (2007) studied the effect of temperature modulation on the onset of

double diffusive convection in a sparsely packed porous medium using linear

stability analysis and Brinkman-Forchheimer extended Darcy model. The effect of

permeability and thermal modulation on the onset of double diffusive convection

46

has been studied using Galerkin method and Floquet theory. The critical Rayleigh

number is calculated as a function of frequency and amplitude of modulation,

Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number.

Bhadauria and Sherani (2008) studied the effect of temperature modulation on

the onset of thermal convection in an electrically conducting-fluid-saturated porous

medium, which is heated from below and cooled from above. The correction in the

value of the critical Darcy Rayleigh number is calculated as function of amplitude

and frequency of modulation, Darcy Chandrasekhar number, thermal Prandtl

number, magnetic Prandtl number and the Vadasz number Va. It is found that the

effect of temperature modulation on the onset of convection is to advance or delay

the convection, depending on the proper tuning of the frequency of modulation.

Rapaka et al. (2008) described and used the recently developed non-modal

stability theory to compute maximum amplifications possible, optimized over all

possible initial perturbations. The details of three-dimensional spectral calculations

of the governing equations are presented.

Motsa (2008) addressed the problem of double-diffusive convection in a

horizontal layer filled with a fluid in the presence of temperature gradients (Soret

effects) and concentration gradients (Dufour effects). The onset of convection is

studied using linear stability analysis. The critical Rayleigh numbers for the onset of

convection are determined in terms of the governing parameters.

Malashetty et al. (2008) studied double diffusive convection in a fluid-saturated

porous layer heated from below and cooled from above when the fluid and solid

phases are not in local thermal equilibrium, using both linear and nonlinear stability

analyses. The Darcy model with time derivative term is employed as momentum

equation. A two-field model that represents the fluid and solid phase temperature

fields separately is used for energy equation. The onset criterion for stationary,

47

oscillatory and finite amplitude convection is derived analytically. It is found that

small inter-phase heat transfer coefficient has significant effect on the stability of

the system. There is a competition between the processes of thermal and solute

diffusion that causes the convection to set in through either oscillatory or finite

amplitude mode rather than stationary. The nonlinear theory based on the truncated

representation of Fourier series method predicts the occurrence of subcritical

instability in the form of finite amplitude motions. The effect of thermal non-

equilibrium on heat and mass transfer is also brought out.

Malashetty and Heera (2008) studied double diffusive convection in a fluid-

saturated rotating porous layer heated from below and cooled from above, when the

fluid and solid phases are not in local thermal equilibrium, using both linear and

non-linear stability analyses. A two-field model that represents the fluid and solid

phase temperature fields separately is used for energy equation. The onset criterion

for stationary, oscillatory and finite amplitude convection is derived analytically. It

is found that small inter-phase heat transfer coefficient has significant effect on the

stability of the system. There is a competition between the processes of thermal and

solute diffusions that causes the convection to set in through either oscillatory or

finite amplitude mode rather than stationary. The effect of solute Rayleigh number,

porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz

number and Taylor number on the stability of the system is investigated. The non-

linear theory, based on the truncated representation of Fourier series method,

predicts the occurrence of subcritical instability in the form of finite amplitude

motions. The effect of thermal non-equilibrium on heat and mass transfer is also

brought out.

Postelnicu (2008) performed a linear stability analysis in order to analyze the

onset of Darcy–Brinkman convection in a fluid-saturated porous layer heated from

below, by considering the case when the fluid and solid phases are not in local

48

thermal equilibrium. The problem is transformed into an eigenvalue equation which

is solved in a first step by using a one-term Galerkin approach: an explicit

relationship between the Darcy–Rayleigh number based on the fluid properties R

and the horizontal wave number k is obtained. Minimization of R over k is

performed analytically and finally, critical values for R and k are obtained for

various values of the three parameters of the problem, namely the Darcy number D,

the porosity-scaled conductivity ratio γ and the scaled inter-phase heat transfer

coefficient H. In a second step, a general N-terms Galerkin approach is used and

finally comparisons are performed between the results given by these two

approaches.

Zeng et al. (2009) investigated numerically the problem of natural convection in

an enclosure filled with a diamagnetic fluid-saturated porous medium under strong

magnetic field. The Brinkman-Forchheimer extended Darcy model is used to solve

the momentum equations, and the energy equations for fluid and solid are solved

with the Local Thermal Non-Equilibrium (LTNE) model. The results show that the

magnetic force has significant effect on the flow field and heat transfer in a

diamagnetic fluid-saturated porous medium.

Wang and Tan (2009) studied on the basis of Brinkman model, the onset of

double-diffusive (thermosolutal) convection with a reaction term in a horizontal

sparsely packed porous media using the normal mode analysis. Some results of

Darcy model have been recovered as limiting cases.

Mokhtar et al. (2009) studied the problem of Bénard convection in a fluid

saturated porous medium heated from below with non-uniform temperature gradient

under magnetic field. A linear stability analysis is performed to undertake a detailed

investigation. We found that it is possible to delay the onset of Bénard convection

in saturated porous medium with the effect of a cubic state temperature profile and

also by increasing the magnetic field.

49

Falsaperla et al. (2010) considered the problem of thermal convection in a

rotating horizontal layer of porous medium. The porous medium is described by the

equations of Darcy. A novel aspect of this work is to consider boundary conditions

for the temperature of Newton–Robin type with heat flux prescribed as a limiting

case. The effect of rotation is found to be crucial. For the Taylor number small

enough the critical wave number is zero, but it is found that a threshold such that for

Taylor numbers beyond this non-zero critical wave numbers are found. The

threshold is verified via a weakly nonlinear analysis. Finally, a sharp global

nonlinear stability analysis is given.

Vanishree and Siddheshwar (2010) performed a linear stability analysis for

mono-diffusive convection in an anisotropic rotating porous medium with

temperature-dependent viscosity. The Galerkin variant of the weighted residual

technique is used to obtain the eigenvalue of the problem. The effect of Taylor–

Vadasz number and the other parameters of the problem are considered for

stationary convection in the absence or presence of rotation. Oscillatory convection

seems highly improbable. Some new results on the parameters’ influence on

convection in the presence of rotation, for both high and low rotation rates, are

presented.

A comprehensive review of the literature concerning convection in a fluid

saturated porous medium can be found in the books of Ingham and Pop (1998),

Nield and Bejan (2006) and Vafai (2000; 2005).

In what follows we review the literature pertaining to convective instability

problems of couple-stress fluids.

50

2.5 Convection in Couple-Stress Fluids Couple-stress fluid theory developed by Stokes (1966, 1984) is one among the

polar fluid theories which considers couple stresses in addition to the classical

Cauchy stress. It is the simplest generalization of the classical theory of fluids

which allows for polar effects such as the presence of couple stresses and body

couples in the fluid medium. Srivastava (1986) investigated the problem of peristaltic transport of a couple-

stress fluid under a zero Reynolds number and long wavelength approximation. A

comparison of the results with those for a Newtonian fluid model shows that the

magnitude of the pressure rise under a given set of conditions is greater in the case

of the couple-stress fluid. The pressure rise increases as the couple-stress parameter

decreases. The difference between the pressure rise for a Newtonian and a couple-

stress fluid increases with increasing amplitude ratio at zero flow rate. However, it

is found that increasing the flow rate reduces this difference.

Jaw-Ren Lin (1998) presented a theoretical study of squeeze film behaviour for a

finite journal bearing lubricated with couple stress fluids. On the basis of the

microcontinuum theory, the modified Reynolds equation is obtained by using the

Stokes equations of motion to account for the couple stress effects due to the

lubricant blended with various additives. With the Conjugate Gradient Method of

iteration the built-up pressure is calculated, and then applied to predict the squeeze

film characteristics of the system. Compared with the case of a Newtonian

lubricant, the couple stress effects increase the load-carrying capacity significantly

and lengthen the response time of the squeeze film behaviour. On the whole, the

presence of couple stresses improves the characteristics of finite journal bearings

operating under pure squeeze film motion. It is found that the rheological effects of

couple stress fluids agree with previous works.

51

Jaw-Ren Lin (2000) presented, on the basis of the microcontinuum theory, a

theoretical analysis of the effects of couple stresses on the squeeze film behavior

between a sphere and a flat plate. The modified Reynolds equation governing the

squeeze film pressure is derived by using the Stokes constitutive equations to take

an account of the couple stress effects due to the lubricant blended with various

additives. According to the results obtained, the influence of couple stresses

signifies an enhancement in the film pressure. On the whole, the couple stress

effects characterized by the couple stress parameter produce an increase in value of

the load-carrying capacity and the response time as compared to the classical

Newtonian-lubricant case. It improves the squeeze film characteristics of the

system.

Sharma et al. (2000) considered a layer of a couple-stress fluid heated from

below in a porous medium to include the effect of uniform rotation. For the case of

stationary convection, the couple stress may hasten the onset of convection in the

presence of rotation, while in the absence of rotation, it always postpones the onset

of convection. For the case of stationary convection, rotation postpones the onset of

convection. Graphs have been plotted by giving numerical values to the parameters

to depict the stability characteristics. Rotation is found to introduce oscillatory

modes in the system, which were non-existent in its absence. A sufficient condition

for the non-existence of overstability is obtained.

Sharma and Thakur (2000) considered a layer of electrically conducting couple-

stress fluid heated from below in a porous medium in the presence of magnetic

field. For stationary convection, the couple-stress and magnetic field postpone the

onset of convection, whereas the medium permeability hastens the onset of

convection. The magnetic field introduces oscillatory modes in the system, which

were non-existent in its absence. A sufficient condition for the non-existence of

overstability is obtained.

52

Sharma and Shivani (2001a) considered a layer of couple-stress fluid heated

from below in a porous medium. Using linearized stability theory and normal mode

analysis, the dispersion relation is obtained. For stationary convection, the couple-

stress postpones the onset of convection, whereas the medium permeability hastens

the onset of convection. The principle of exchange of stabilities is valid for the

couple-stress fluid heated from below in porous medium.

Sharma and Shivani (2001b) considered a layer of electrically conducting

couple-stress fluid heated from below in porous medium in the presence of uniform

horizontal magnetic field. They found that the medium permeability hastens the

onset of convection, whereas the magnetic field and couple-stress postpone the

onset of convection for the case of stationary convection. The oscillatory modes are

introduced by the magnetic fields which were not present in the absence of the

magnetic field. The overstable case has been considered and a sufficient condition

for the non-existence of overstability is obtained.

Sunil et al. (2002) considered a layer of couple-stress fluid heated from below in

a porous medium in the presence of a uniform vertical magnetic field and uniform

vertical rotation. For the case of stationary convection, the rotation postpones the

onset of convection. The magnetic field and couple-stress may hasten the onset of

convection in the presence of rotation, while in the absence of rotation, they always

postpone the onset of convection. The medium permeability hastens the onset of

convection in the absence of rotation, while in the presence of rotation, it may

postpone the onset of convection. Graphs have been plotted by giving numerical

values to the parameters to depict the stability characteristics. The rotation and

magnetic field are found to introduce oscillatory modes in the system, which were

nonexistent in their absence. A sufficient condition for the nonexistence of

ovestabllity is also obtained.

53

Zakaria (2002) cast the equations of a polar fluid of hydromagnetic fluctuating

through a porous medium into matrix form using the state space and Laplace

transform techniques. The resulting formulation is applied to a variety of problems.

The solution to a problem of an electrically conducting polar fluid in the presence of

a transverse magnetic field and to a problem for the flow between two parallel fixed

plates is obtained. The inversion of the Laplace transforms is carried out using a

numerical approach. Numerical results for the velocity, angular velocity distribution

and the induced magnetic field are illustrated graphically for each problem.

Elsharkawy (2004) presented a mathematical model for the hydrodynamic

lubrication of misaligned journal bearings with couple stress lubricants. A modified

form for Reynolds equation in which the effects of couple stresses arising from the

lubricant blended with various additives is used. The journal misalignment is

allowed to vary in magnitudes as well as in direction with respect to the bearing

boundaries. The flexibility of the bearing liner was incorporated into the analysis by

using the thin liner model. A numerical solution for the mathematical model using a

finite difference scheme is introduced. The predicted performance characteristics

are compared with available theoretical and experimental results. The effects of the

degree and angle of misalignment, the length-to-diameter ratio, and the couple

stress parameter on static performance such as pressure distribution, load-carrying

capacity, friction coefficient, and side leakage flow and misalignment moment are

presented and discussed.

Sharma and Sharma (2004a) considered the thermal instability of a couple-stress

fluid with suspended particles. Following the linear stability analysis and normal

mode analysis, the dispersion relation is obtained. For the case of stationary

convection, couple-stress is found to postpone the onset of convection, whereas

suspended particles hasten it. It is found that the principle of exchange of stabilities

is valid. The thermal instability of a couple-stress fluid with suspended particles, in

54

the presence of rotation and magnetic field, is also considered. The magnetic field

and rotation are found to have stabilizing effects on the stationary convection and

introduce oscillatory modes in the system. A sufficient condition for the

nonexistence of ovestabllity is also obtained.

Sharma and Sharma (2004b) considered a layer of couple-stress fluid, permeated

with suspended particles, heated and soluted from below in a porous medium. The

couple-stress and stable solute gradient postpone the onset of convection, whereas

the medium permeability and suspended particles hasten the onset of convection.

The principle of exchange of stabilities is valid for the couple-stress fluid permeated

with suspended particles heated from below in porous medium. The oscillatory

modes are introduced due to the presence of stable solute gradient.

Siddheshwar and Pranesh (2004) investigated the effect of Raleigh–Benard

situation in Boussinesq–Stokes suspensions using both linear and nonlinear stability

analyses. The linear and nonlinear analyses are based on a normal mode solution

and minimal representation of double Fourier series respectively. The effect of

suspended particles on convection is delineated against the background of the

results of the clean fluid. The realm of nonlinear convection warrants the

quantification of heat transfer and this has been achieved on the Rayleigh–Nusselt

plane. Possibility of aperiodic convection is also discussed.

Sunil et al. (2004) considered a layer of couple-stress fluid permeated with

suspended particles, heated and soluted from below in a porous medium. For the

case of stationary convection, the stable solute gradient and couple-stress have

stabilizing effect on the onset of convection, whereas the suspended particles and

medium permeability have destabilizing effect on the couple-stress fluid permeated

with suspended particles. Graphs have been plotted by giving numerical values to

the parameters to depict the stability characteristics. The stable solute gradient is

55

found to introduce oscillatory modes in the system, which are nonexistent in its

absence. A sufficient condition for the nonexistence of overstability is obtained.

Liao et al. (2005) attempted to provide the dynamic characteristics of long

journal bearings lubricated with couple stress fluids. Based upon the micro-

continuum theory generated by Stokes, the dynamic Reynolds-type equation

governing the film pressure is derived to account for the couple stress effects

resulting from the non-Newtonian behavior of complex fluids. By applying the

linear stability theory to the non-linear equations of motion the journal rotor, the

equilibrium positions and dynamic characteristics of the system are evaluated. As

compared to the classical Newtonian model, the effects of couple stresses signify

enhanced stiffness and damping coefficients at moderate values of the steady

eccentricity ratio. It is found that long bearings lubricated with couple stress fluids

under small disturbance results in a higher stability threshold speed than that of the

Newtonian-lubricant case.

Sarangi et al. (2005a) extended conventional elastohydrodynamic lubrication

(EHL) analysis of point contacts to include couple-stress effects in lubricants

blended with polymer additives. A transient pressure differential equation, generally

referred to as a modified Reynolds equation, is derived from the Stokes

microcontinuum theory and solved using the finite difference method with a

successive over-relaxation scheme. The solution is obtained under isothermal

conditions, assuming a suitable exponential relation of pressure-viscosity variation.

A non-dimensional couple-stress parameter, which can be considered the molecular

length of the additives in the lubricant, is used in the analysis. From the results

obtained, the influence of the couple-stress parameter on the EHL point contacts is

apparent and cannot be neglected. Lubricants with couple stresses provide an

increase in the load-carrying capacity and reduction in friction coefficient as

compared to Newtonian lubricants. Empirical formulas for the calculation of central

and minimum film thicknesses of lubricated point contacts with couple-stress fluids

56

are derived with the nonlinear least-squares curve-fitting technique using different

numerically evaluated data.

Sarangi et al. (2005b) evaluated numerically stiffness and damping coefficients

of isothermal elastohydrodynamically lubricated point-contact problems

numerically with couple-stress fluids. A set of equations under steady-state and

dynamic conditions is derived from the modified Reynolds equation using a

linearized perturbation method. This paper is the second part of the present study;

the modified Reynolds equation derived from the Stokes micro-continuum theory is

used in the previous article. Dynamic pressures are found after solving the set of

perturbed equations using the previously obtained steady-state pressure from the

modified Reynolds equation. The stiffness and damping coefficients of the film are

determined using the dynamic pressures. Then the overall stiffness and damping

matrices of the ball bearing are obtained from load distribution, coordinate

transformation, and compatibility relations. The bearing coefficients are introduced

into a rotor system to simulate the response. It has been observed that the influence

of couple-stress fluids on the dynamics of a rotor supported on lubricated ball

bearings is marginal; hence, Newtonian theory can be used instead for simplicity.

However, with increasing content of polymer additives in lubricant, for an accurate

analysis the effect of couple stresses in a fluid should not be neglected.

Sharma and Mehta (2005) paid attention to a layer of compressible, rotating,

couple-stress fluid heated and soluted from below. For the case of stationary

convection, the compressibility, stable solute gradient and rotation postpone the

onset of convection, whereas the couple-stress viscosity postpones as well as

hastens the onset of convection depending on rotation parameter. The case of

overstability is also studied wherein a sufficient condition for the non-existence of

overstability is found.

57

Ezzat et al. (2006) introduced a magnetohydrodynamic model of boundary-layer

equations for a perfectly conducting couple-stress fluid. This model is applied to

study the effects of free convection currents with thermal relaxation on the flow of a

polar fluid through a porous medium, which is bounded by a vertical plane surface.

The state space formulation is introduced. The resulting formulation, together with

the Laplace transform technique, are applied to a variety of problems. The solution

to a thermal shock problem and to the problem of the flow in the whole space with a

plane distribution of heat sources are obtained. It is also applied to a semi-space

problem with a plane distribution of heat sources located inside the fluid. A

numerical method is employed for the inversion of the Laplace transforms. The

effects of Grashof number, material parameters, Alfven velocity, relaxation time,

Prandtl number and the permeability parameter on the velocity, the temperature and

the angular velocity distributions are discussed. The effects of cooling and heating

of a couple-stress fluid have also been discussed.

Malashetty et al. (2006) examined the double diffusive convection in a two-

component couple stress liquid layer with Soret effect using both linear and non-

linear stability analyses. The linear theory is based on normal mode technique and

the non-linear analysis is based on a minimal representation of double Fourier

series. The effect of couple stress parameter, the Soret parameter, the solute

Rayleigh number, the Prandtl number and the diffusivity ratio on the stationary,

oscillatory and finite amplitude convection are shown graphically. It is found that

the effects of couple stress are quite large and the positive Soret number enhances

the stability, while the negative Soret number enhances the instability. The

nonlinear theory predicts that finite amplitude motions are possible only for

negative Soret parameter. The transient behaviour of thermal and solute Nusselt

numbers has been investigated by solving numerically a fifth order Lorenz model

using Runge–Kutta method.

58

Naduvinamani and Kashinath (2006) studied the effect of surface roughness on

the performance of curved pivoted slider bearings. A more general type of surface

roughness is mathematically modelled by a stochastic random variable with nonzero

mean, variance and skewness. The averaged modified Reynolds type equation is

derived on the basis of Stokes microcontinuum theory for couple stress fluids. The

closed-form expressions for the mean pressure, load-carrying capacity, frictional

force and the centre of pressure are obtained. Numerical computations show that the

performance of the slider bearing is improved by the use of lubricants with

additives (couple stress fluid) as compared to Newtonian lubricants. Further, it is

observed that the negatively skewed surface roughness increases the load-carrying

capacity and frictional force and reduces the coefficient of friction, whereas the

positively skewed surface roughness on the bearing surface adversely affects the

performance of the pivoted slider bearings.

Gaikwad et al. (2007) studied the onset of double diffusive convection in a two

component couple stress fluid layer with Soret and Dufour effects using both linear

and non-linear stability analysis. The linear theory depends on normal mode

technique and non-linear analysis depends on a minimal representation of double

Fourier series. The effect of couple stress parameter, the Soret and Dufour

parameters, and the Prandtl number on the stationary and oscillatory convection are

presented graphically. The Dufour parameter enhances the stability of the couple

stress fluid system in case of both stationary and oscillatory mode. The effect of

positive Soret parameter is to destabilize the system in case of stationary mode

while it stabilizes the system in case of oscillatory mode. The negative Soret

parameter enhances the stability in both stationary and oscillatory mode. The couple

stress parameter enhances the stability of the system in both stationary and

oscillatory modes. The Dufour parameter increases the heat transfer, while the

couple stress parameter has reverse effect. The Soret parameter has negligible

influence on heat transfer. Both Dufour and Soret parameters increase the mass

59

transfer, while the couple stress parameter has dual effect depending on the value of

the Rayleigh number.

Lewicka (2007) investigated the Stokes–Boussinesq equations in a slanted (that

is, not aligned with gravity's direction) cylinder of any dimension and with an

arbitrary Rayleigh number. The author proved the existence of a non-planar

traveling wave solution, propagating at a constant speed, and satisfying the

Dirichlet boundary condition in the velocity and the Neumann condition in the

temperature.

Jaw-Ren Lin and Chi-Ren Hung (2007) presented on the basis of the Stokes

micro-continuum theory together with the averaged inertia principle, the combined

effects of non-Newtonian couple stresses and convective fluid inertia forces on the

squeeze film motion between a long cylinder and an infinite plate. A closed-form

solution has been derived for squeeze film characteristics including the film

pressure, the load capacity and the response time. Comparing with the Newtonian-

lubricant non-inertia case, the combined effects of couple stresses and convective

inertia forces provide an increase in the film pressure, the load capacity and the

response time. In addition, the quantitative effects of couple stresses and convective

inertia forces are more pronounced for cylinder–plate system operating at a larger

couple stress parameter and film Reynolds number, as well as a smaller squeeze

film height. To guide the use of the present study, a numerical example is also

illustrated for engineers when considering both the effects of non-Newtonian couple

stresses and fluid convective inertia forces.

Naduvinamani and Siddangouda (2007) presented the theoretical study of the

effect of couple stresses on the optimum lubrication characteristics of porous

Rayleigh step bearings. The lubricant with additives in the film region and also in

the porous region is modelled as the Stokes couple-stress fluid. Modified Darcy

type equations accounting for the polar effects in the porous region is considered

60

and the generalized Reynolds type equation is derived for the lubrication of the

porous Rayleigh step bearings. Exact solution of the generalized Reynolds type

equation is obtained and the closed form expressions for the bearing characteristics

are presented. Performance characteristics of the porous Rayleigh step bearing are

presented for the various values of the non-dimensional parameters such as the

couple-stress parameter and the permeability parameter. It is observed that the

couple-stress fluid lubricants provide the increased load carrying capacity and the

decreased coefficient of friction as compared to the corresponding Newtonian case.

This theory suggests that, adverse effects of the presence of porous facing on the

stator could be compensated with proper selection of lubricants with proper

additives.

Devakar and Iyengar (2008) considered Stokes’ first and second problems for an

incompressible couple-stress fluid under isothermal conditions. The problems are

solved through the use of Laplace transform technique. Inversion of the Laplace

transform of the velocity component in each case is carried out using a standard

numerical approach. Velocity profiles are plotted and studied for different times and

different values of couple stress Reynolds number. The results are presented

through graphs in each case.

Yan-yan (2008) derived, to take into account the couple stress effects, a

modified Reynolds equation for dynamically loaded journal bearings with the

consideration of the elasticity of the liner. The numerical results show that the

influence of couple stresses on the bearing characteristics is significant. Compared

with Newtonian lubricants, lubricants with couple stresses increase the fluid film

pressure, as a result enhance the load-carrying capacity and reduce the friction

coefficient. However, since the elasticity of the liner weakens the couple stress

effect, elastic liners yield a reduction in the load-carrying capacity and an increase

in the friction coefficient. The elastic deformation of the bearing liner should be

considered in an accurate performance evaluation of the journal bearing.

61

Indira et al. (2008) considered flow of couple-stress fluid flowing in an eccentric

annulus. This study has its importance whenever simultaneous flow of two fluids

has to be considered. The eccentric annulus in the domain D bounded internally by

C1 and externally by C2 is mapped onto a concentric annulus bounded internally by

1 and externally by 2 using a conformal mapping. A closed form solution is

obtained. Two dimensional velocity profile is plotted for different couple-stress

parameters, area of cross section and eccentricity parameter. Numerical

computation reveals that the use of eccentric annulus facilitates transport of more

fluid. The rate of flow increases as eccentricity increases. Rate of flow increases

with decrease in the couple stress parameter.

Aggarwal and Suman Makhija (2009) examined theoretically the thermal

stability of a couple-stress fluid in the presence of magnetic field and rotation.

Following the linear stability theory and normal mode analysis, the dispersion

relation is obtained. For stationary convection, rotation has stabilizing effect

whereas couple stresses in fluid and magnetic field have stabilizing effect under

certain conditions. It is found that principle of exchange of stabilities is satisfied in

the absence of magnetic field and rotation. The sufficient conditions for the non

existence of overstability are also obtained.

Crosby and Chetti (2009) studied static and dynamic characteristics of two-lobe

journal bearings lubricated with couple-stress fluids. The load-carrying capacity, the

stiffness and damping coefficients, the non-dimensional critical mass, and the whirl

ratio are determined for various values of the couple stress parameter l. The results

obtained are compared with the characteristics of two-lobe bearings lubricated with

Newtonian fluids. It is found that the effect of the couple stress parameter is very

significant on the performance of the journal bearing and that the stability is

improved compared to bearings lubricated with Newtonian fluids.

62

Malashetty et al. (2009) investigated the stability of a couple stress fluid

saturated horizontal porous layer heated from below and cooled from above when

the fluid and solid phases are not in local thermal equilibrium. The Darcy model is

used for the momentum equation and a two-field model is used for energy equation

each representing the solid and fluid phases separately. The linear stability theory is

employed to obtain the condition for the onset of convection. The effect of thermal

non-equilibrium on the onset of convection is discussed. It is shown that the results

of the thermal non-equilibrium Darcy model for the Newtonian fluid case can be

recovered in the limit as couple stress parameter C→0. They also presented

asymptotic analysis for both small and large values of the inter phase heat transfer

coefficient H. They found an excellent agreement between the exact solutions and

asymptotic solutions when H is very small.

Pardeep Kumar and Mahinder Singh (2009) considered the thermosolutal

instability of couple-stress fluid in the presence of uniform vertical rotation.

Following the linear stability theory and normal mode analysis, the dispersion is

obtained. For the case of stationary convection, the stable solute gradient and

rotation have stabilizing effects on the system, whereas the couple-stress has both

stabilizing and destabilizing effects. The dispersion relation is also analyzed

numerically. The stable solute gradient and the rotation introduce oscillatory modes

in the system, which did not occur in their absence. The sufficient conditions for the

non-existence of overstability are also obtained.

Srinivasacharya et al. (2009) considered an incompressible laminar flow of a

couple-stress fluid in a porous channel with expanding or contracting walls.

Assuming symmetric injection or suction along the uniformly expanding porous

walls and using similarity transformations, the governing equations are reduced to

nonlinear ordinary differential equations. The resulting equations are then solved

numerically using quasi-linearization technique. The graphs for velocity

63

components and temperature distribution are presented for different values of the

fluid and geometric parameters.

Umavathi et al. (2009) presented an analytical solution for fully developed

laminar flow between vertical parallel plates filled with 2 immiscible viscous and

couple stress fluids in a composite porous medium. The flow in the porous medium

is modeled using the Brinkman equation. The viscous and Darcy dissipation terms

are included in the energy equation. The transport properties of the fluids in both

regions are assumed to be constant. The continuity conditions for the velocity,

temperature, shear stress, and the heat flux at the interfaces between the couple

stress permeable fluid layer and the viscous fluid layer are assumed. The influence

of physical parameters on the flow, such as the couple stress parameter, porous

parameter, Grashof number, viscosity ratio and conductivity ratio, are evaluated,

and a set of graphical results is presented. An interesting and new approach is

incorporated to analyze the flow for strong, weak, and comparable porosity

conditions with the couple stress fluid parameter.

Devakar and Iyengar (2010) considered the flow of an incompressible fluid

between two parallel plates, initially induced by a constant pressure gradient. After

steady state is attained, the pressure gradient is suddenly withdrawn, while the

plates are impulsively started simultaneously. The arising flow is referred to as run

up flow and the present paper aims at studying this flow in the context of a couple

stress fluid. Using Laplace transform technique, the expression for velocity is

obtained in Laplace transform domain which is later inverted to the space time

domain using a numerical approach. The variation of velocity with respect to

various flow parameters is presented through graphs.

Mahinder Singh and Pardeep Kumar (2010) considered the problem of thermal

instability of compressible, electrically conducting couple-stress fluids in the

presence of a uniform magnetic field. Following the linear stability theory and

64

normal mode analysis, the dispersion relation is obtained. For stationary convection,

the compressibility, couple-stress, and magnetic field postpone the onset of

convection. Graphs have been plotted by giving numerical values of the parameters

to depict the stability characteristics. The principle of exchange of stabilities is

found to be satisfied. The magnetic field introduces oscillatory modes in the system

that were non-existent in its absence. The case of overstability is also studied

wherein a sufficient condition for the non-existence of overstability is obtained.

Malashetty et al. (2010) studied the onset of double-diffusive convection in a

couple-stress fluid-saturated horizontal porous layer using linear and weak

nonlinear stability analyses. The modified Darcy equation that includes the time

derivative term and the inertia term is used to model the momentum equation. The

expressions for stationary, oscillatory and finite-amplitude Rayleigh number are

obtained as a function of the governing parameters. The effect of couple-stress

parameter, solute Rayleigh number, Vadasz number and diffusivity ratio on

stationary, oscillatory and finite-amplitude convection is shown graphically. It is

found that the couple-stress parameter and the solute Rayleigh number have a

stabilizing effect on stationary, oscillatory and finite-amplitude convection. The

diffusivity ratio has a destabilizing effect in the case of stationary and finite-

amplitude modes, with a dual effect in the case of oscillatory convection. The

Vadasz number advances the onset of oscillatory convection. The heat and mass

transfer decreases with an increase in the values of couple-stress parameter and

diffusivity ratio, while both increase with an increase in the value of the solute

Rayleigh number.

65

2.6 Plan of Work The dissertation is organized as follows:

The Third Chapter consists of basic equations, approximations, boundary

conditions and a discussion of the dimensionless parameters. In Chapter IV, linear

stability analysis of Rayleigh–Benard-Marangoni convection in a couple-stress fluid

saturated densely packed horizontal porous layer in the presence of thermal

radiation is studied. The linear stability analysis is based on a normal mode

technique. The Galerkin technique is used to solve the variable coefficient system

of differential equations. The results are discussed in Chapter V with the help of

figures and some of the numerical results are tabulated. An exhaustive bibliography

follows this last chapter.

66

CHAPTER III

BASIC EQUATIONS, BOUNDARY CONDITIONS AND DIMENSIONLESS PARAMETERS

Like any mathematical model of the real world, fluid mechanics makes some

basic assumptions about the materials being studied. These assumptions are turned

into equations that must be satisfied if the assumptions are to hold true. For

example, consider an incompressible fluid in three dimensions. The assumption that

mass is conserved means that for any fixed closed surface (such as a sphere) the rate

of mass passing from outside to inside the surface must be the same as rate of mass

passing the other way. (Alternatively, the mass inside remains constant, as does the

mass outside). This can be turned into an integral equation over the surface.

Fluid mechanics assumes that every fluid obeys the following:

• Conservation of mass

• Conservation of momentum

• The continuum hypothesis, detailed below.

Further, it is often useful (and realistic) to assume a fluid is incompressible - that

is, the density of the fluid does not change. Liquids can often be modelled as

incompressible fluids, whereas gases cannot. Similarly, it can sometimes be

assumed that the viscosity of the fluid is zero (the fluid is inviscid). Gases can often

be assumed to be inviscid. If a fluid is viscous, and its flow contained in some way

(e.g. in a pipe), then the flow at the boundary must have zero velocity. For a viscous

fluid, if the boundary is not porous, the shear forces between the fluid and the

boundary results also in a zero velocity for the fluid at the boundary. This is called

the no-slip condition. For a porous media otherwise, in the frontier of the containing

vessel, the slip condition is not zero velocity, and the fluid has a discontinuous

67

velocity field between the free fluid and the fluid in the porous media (this is related

to the Beavers and Joseph condition).

Fluids are composed of molecules that collide with one another and solid

objects. The continuum assumption, however, considers fluids to be continuous.

That is, properties such as density, pressure, temperature, and velocity are taken to

be well-defined at "infinitely" small points, defining a REV (Reference Element of

Volume), at the geometric order of the distance between two adjacent molecules of

fluid. Properties are assumed to vary continuously from one point to another, and

are averaged values in the REV. The fact that the fluid is made up of discrete

molecules is ignored.

The continuum hypothesis is basically an approximation, in the same way

planets are approximated by point particles when dealing with celestial mechanics,

and therefore results in approximate solutions. Consequently, assumption of the

continuum hypothesis can lead to results which are not of desired accuracy. That

said, under the right circumstances, the continuum hypothesis produces extremely

accurate results.

Those problems for which the continuum hypothesis does not allow solutions of

desired accuracy are solved using statistical mechanics. To determine whether or

not to use conventional fluid dynamics or statistical mechanics, the Knudsen

number is evaluated for the problem. The Knudsen number is defined as the ratio of

the molecular mean free path length to a certain representative physical length

scale. This length scale could be, for example, the radius of a body in a fluid. (More

simply, the Knudsen number is how many times its own diameter a particle will

travel on average before hitting another particle). Problems with Knudsen numbers

at or above unity are best evaluated using statistical mechanics for reliable

solutions.

68

3.1 Basic Equations

When a fluid permeates a porous material, the actual path of an individual fluid

particle cannot be followed analytically. The gross effect, as the fluid slowly

percolates the pores of the medium, must be represented by a macroscopic law

which is applicable to masses of fluid compared with the dimension of the porous

structure of a medium and this is the basis for the Darcy’s law. The original from of

Darcy law states that the mean filter velocity q is proportional to the sum of the

gradient of pressure and the gravitational force. That is,

( )constantq p gρµ

= −∇ + (3.1)

The modified Darcy law due to Muskat (1937) is

( )kq p gρµ

= −∇ + (3.2)

where k is the permeability of the porous medium which determines the intrinsic

properties of the medium. This law is generally accepted as the macroscopic

equation for Newtonian fluids. The flow governed by this law, in the case of

homogeneous isotropic porous medium, is of potential type rather than boundary

layer nature. In other words, Darcy model takes into account only the frictional

force offered by the solid particles to the fluid rather than usual viscous shear.

To study convection in a porous medium, in addition to Darcy law, we have to

use the continuity and energy equations. As shown by Lapwood (1948), the set of

basic equations results in a fourth order differential equation governing the onset of

convection. However, using physical arguments, six boundary conditions based on

no-slip condition can be defined. In mathematical sense, the problem is therefore

not properly posed. All the earlier works have sidestepped the problem by simply

ignoring the no-slip conditions. However, from the physical point of view, the no-

69

slip boundary conditions are as much valid as other conditions and there appears no

a priori reason to reject them. In fact, Morels et al. (1951) have shown that

boundary conditions resembling no-slip do exist in a packed bed reactor, which is a

porous medium device. Beavers and Joseph (1967) postulated that a boundary layer

exists in a porous medium. This is further corroborated by Brenner (1970). Taylor

(1971) and Vafai and Tien (1981) and their work suggest that the boundary layer in

a porous medium is of order k . On the other hand, the experiments of Elder

(1967) and Katto ans Masuoka (1967) suggest that the boundary layer in a

horizontal porous layer is not so important in determining the point of the onset of

convection. However, it must be borne in mind that free convective heat transfer is

essentially a boundary value problem and hence the boundary layer must play a role

in governing the magnitude of the heat transfer after the onset of convection.

Therefore, one of the approximate boundary layer types of equation in a porous

medium is the Brinkman model. This model, first postulated by Brinkman (1947)

and later used by many, consists of viscous force term q2∇µ in addition to the

Darcy resistance term qkµ −

in the momentum equation. Thermal convective

instability in a layer of porous medium has received extensive attention over the

years and has now emerged as an important field of study in the broader area of

fluid dynamics and in the area of heat transfer in particular. The growing volume of

work involving this field is well documented by Ingham and Pop (1998), Vafai

(2000, 2005) and Nield and Bejan (2006).

On the other hand, albeit numerous studies have been undertaken in the past to

understand convective instability of fluids, most of the investigations have been

limited to Newtonian fluids. Nevertheless, the growing importance of non-

Newtonian fluids in modern technology has impressed researchers because the

conventional Newtonian fluids cannot precisely describe the characteristics of the

70

fluid flow encountered in many practical situations such as the extrusion of polymer

fluids, solidification of liquid crystals, cooling of metallic plates in a bath, exotic

lubricants and colloidal fluids to mention a few.

These fluids deform and produce a spin field due to the microrotation of

suspended particles forming a micropolar fluid. The theory of micropolar fluids was

developed by Eringen (1966) which takes care of local effects arising from the

microstructure and as well as the intrinsic motions of microfluidics. The spin field

due to microrotation of freely suspended particles sets up an anti-symmetric stress,

known as couple stress, and thus forming a couple stress fluid. Thus couple-stress

fluid, according to Eringen (1966), is a variant of micropolar fluid when

microrotation balances with the natural vorticity of the fluid. The couple-stress fluid

has distinct features such as polar effects and whose microstructure is mechanically

significant. The constitutive equations for couple-stress fluids are proposed by

Stokes (1966, 1984). The theory proposed by Stokes is the simplest one, which

allows for polar effects such as the presence of couple stress and body couple. The

theory of couple stress fluids has several industrial and scientific applications,

which comprise pumping fluids such as synthetic fluids, polymer thickened oils,

liquid crystal, animal blood, synovial fluid present in synovial joints and the theory

of lubrication (Naduvinamani et al. 2001; 2002; 2003a; 2003b; 2005).

To obtain the basic equations the following approximations have been made use of:

1. The saturated fluid and the porous layer are as if incompressible except that

variations in density, brought about by heating, is taken into account only in the

term gρ of the momentum equation. This is valid only when the speed of the

fluid is much less than that of sound and all accelerations are slow compared

with those associated with sound waves. This is the well-known Boussinesq

approximation.

71

2. At any point in the fluid, the temperature of the solid ( )ST and that of the fluid

( )FT are the same. That is, the thermal behaviour of the medium is described by

a single condition for the average temperature FS TTT == . This most

commonly used approach is valid when the flow velocity is not too high and if

both phases (solid and fluid) are well dispersed.

3. The physical properties, namely, thermal conductivity, viscosity and permeability

are assumed to be constants.

4. It is assumed that the fluid is emitting and absorbing radiation, and non-scattering

medium. We also restrict our attention to the case in which the absorption

coefficient of the fluid is the same at all wavelengths and is independent of the

physical state (the so-called gray medium approximation).

Under these approximations, the basic equations are the following:

Conservation of momentum

( ) ( )2o 1 1. cρ q q q p ρ g µ µ qε t ε k

∂+ = −∇ + − − ∇ ∂

∇ (3.3)

Conservation of energy

( ) 2.r

T Gγ q T χ Tt s

∂+ = ∇ +

∂∇ (3.4)

Conservation of mass

0. =∇ q (3.5) Equation of state

( )o o1ρ ρ α T T = − − (3.6)

72

where ( )wvuq ,,= is the mean filter velocity, t is the time, p is the pressure, ρ is

the fluid density, oρ is the reference density, g is the acceleration due to gravity,

µ is the fluid viscosity, cµ is the couple-stress viscosity, ε is the porosity, k is the

permeability of the porous medium, χ is the effective thermal diffusivity, γ is the

ratio of the specific heat of the solid due to porous medium and that of the fluid at

constant pressure, α is the thermal expansion coefficient, T is the temperature,

oT is the reference temperature, G is the rate of radiative heating per unit volume,

rs is the heat content of the fluid per unit volume, ∇ is the vector differential

operator and ( )zyx ,, are the spatial coordinates.

The expression for temperature dependent surface tension ( )s Tσ for a single

component fluid can be written

( ) ( )o o= − −Ts T T Tσ σ σ , (3.7)

where

o( )= − ∂ ∂T Tsσ σ T .

3.2 Boundary Conditions

(i) Boundary conditions on velocity The boundary conditions on velocity are obtained from mass balance, the no-slip

condition and the stress principle of Cauchy depending on the fact that the fluid

layer is bounded by free or rigid surfaces.

If Darcy law is used, the normal component of velocity must vanish at the

impermeable surfaces while slip boundary conditions are allowed. This is because

Darcy’s equation is of lower order than Navier-Stokes’ equations. If Brinkman

equation is used, then no-slip conditions at the impermeable surfaces can be

imposed because a thin boundary layer inevitably arises at the boundaries. As noted

73

by Beck (1972) convective term cannot be added to Darcy law simply when the

basic state is not quiescent because the addition of convective term would raise the

order of the equation and the additional boundary conditions required are presently

unknown. In the case of quiescent state such difficulty will not arise (Beck, 1972).

Thus for Darcy equation

0ˆ.q n = (3.8) where n̂ is the unit vector normal to the surface. For Brinkman equation

0=== wvu (3.9) at the impermeable surfaces. The following combinations of boundary surfaces are

considered in the convective instability problems:

(i) Both lower and upper boundary surfaces are rigid.

(ii) Both lower and upper boundary surfaces are free.

(iii) Lower surface is rigid and upper surface is free.

a) Rigid surfaces

If the fluid layer is bounded above and below by rigid surfaces, then the viscous

fluid adheres to its bounding surface; hence the velocity of the fluid at a rigid

boundary surface is that of the boundary. This is known as the no-slip condition and

it indicates that the tangential components of velocity in the x and y directions are

zero, i.e. u = 0, v = 0. If the boundary surface is fixed or stationary, then in addition

to u = 0, v = 0, the normal component of velocity .q n∧→ is also zero, i.e., w = 0.

Hence at the rigid boundary we have

u = v = w = 0. (3.10)

74

Since u = v = 0 for all values of x and y at the boundary, we have 0ux∂

=∂

and

0vx

∂=

∂, and hence from the continuity equation subject to the Boussinesq

approximation, it follows that

0wz

∂=

∂

at the boundaries. Thus, in the case of rigid boundaries, the boundary conditions for

the z-component of velocity are

0wwz

∂= =∂

. (3. 11)

b) Free surfaces

In the case of a free surface the boundary conditions for velocity depend on

whether we consider the surface-tension or not. If there is no surface-tension at the

boundary, i.e., the free surface does not deform in the direction normal to itself, we

must require that

w = 0. (3.12) We have taken the z-axis perpendicular to the xy plane, therefore w does not vary

with respect to x and y, i.e.

0wx

∂=

∂ and 0w

y∂

=∂

. (3.13)

In the absence of surface tension, the non-deformable free surface (assumed

flat) is free from shear stresses so that

0u vz z

∂ ∂= =

∂ ∂. (3.14)

From the equation of continuity subject to the Boussinesq approximation, we have

75

0u v wx y z

∂ ∂ ∂+ + =

∂ ∂ ∂. (3.15)

Differentiating this equation with respect to ‘z’ and using Eq. (3.14) yields

2

2 0wz

∂=

∂. (3.16)

Thus, in the absence of surface-tension, the conditions for the z-component of

velocity at the free surfaces are

2

2 0wwz

∂= =∂

. (3.17)

This condition is the stress-free condition.

In the presence of surface-tension, the boundary conditions can be obtained by

equating the shear stresses at the surface to the variations of surface-tension, i.e.,

xzsu w

z x xστ µ

∂∂ ∂= + = ∂ ∂ ∂

(3.18)

and

y zsv w

z y yστ µ

∂∂ ∂= + = ∂ ∂ ∂

, (3.19)

where µ is assumed constant. The expression for the surface-tension sσ , appearing

in the equation above, is given by Eq. (3.7). Using Eq. (3.12), differentiating Eq.

(3.18) with respect to ‘x’ and Eq. (3.19) with respect to ‘y’ and adding the two, we

obtain

21 s

u vz x y

µ σ ∂ ∂ ∂

+ = ∇ ∂ ∂ ∂ . (3.20)

76

In arriving at Eq. (3.20) we have used the fact that x and y variations of w cease

to exist at the boundaries.

Eq. (3.20), using the continuity equation (3.15), reduces to

2212 s

wz

µ σ∂− = ∇

∂. (3.21)

Equation (3.21), on using Eq. (3.7), becomes

2

212

∂= ∇

∂T

w Tz

µ σ . (3.22)

Thus the boundary conditions for a free surface in the presence of surface-

tension are

w = 0 and 2

212

∂= ∇

∂T

w Tz

µ σ . (3.23)

(ii) Thermal boundary conditions The thermal boundary conditions depend on the nature of the boundaries

(Sparrow et al., 1964). Four different types of thermal boundary conditions are

discussed below.

(a) Fixed surface temperature If the bounding wall of the fluid layer has high heat conductivity and large heat

capacity, the temperature in this case would be spatially uniform and independent of

time, i.e. the boundary temperature would be unperturbed by any flow or

temperature perturbation in the fluid. Thus

T = 0 (3.24)

77

at the boundaries. The effect is to maintain the temperature and this boundary

condition is known as isothermal boundary condition or boundary condition of the

first kind, which is the Dirichlet type boundary condition.

(b) Fixed surface heat flux Heat exchange between the free surface and the environment takes place in the

case of free surfaces. According to Fourier’s law, the heat flux TQ passing through

the boundary per unit time and area is

1TTQ kz

∂= −

∂ (3.25)

where Tz

∂∂

is the temperature gradient of the fluid at the boundary. If TQ is

unperturbed by thermal or flow perturbations in the fluid, it follows that

Tz

∂∂

= 0 (3.26)

at the boundaries. This thermal boundary condition is known as adiabatic boundary

condition or insulating boundary condition or boundary condition of the second

kind which is the Neumann type boundary condition.

(c) Boundary condition of the third kind This is a general type of boundary condition on temperature which is given by

T Bi Tz

∂= −

∂. (3.27)

When Bi→ ∞ , we are led to the isothermal boundary condition T = 0 and when

0Bi → , we obtain the adiabatic boundary condition 0Tz

∂=

∂.

78

3.3 Dimensionless parameters Exact solutions are rare in many branches of fluid mechanics because of

nonlinearities and general boundary conditions. Hence to determine approximate

solutions of the problem, numerical techniques or analytical techniques or a

combination of both are used. The key to tackle modern problems is mathematical

modelling. This process involves keeping certain elements, neglecting some, and

approximating yet others. To accomplish this important step one needs to decide the

order of magnitude, i.e., smallness or largeness of the different elements of the

system by comparing them with one another as well as with the basic elements of

the system. This process is called non-dimensionalization or making the variables

dimensionless. Expressing the equations in dimensionless form brings out the

important dimensionless parameters that govern the behaviour of the system. The

first method used to make the equations dimensionless is by introducing the

characteristic quantities and the other is by comparing similar terms. We use the

former method of introducing characteristic quantities. The following are the

important dimensionless parameters arising in the present study.

(i) Rayleigh number The thermal Rayleigh number R is defined as

3

o g T dR ∆=αρ

µ χ,

where T∆ is the temperature difference between the boundaries and d is the

thickness of the fluid layer. The thermal Rayleigh number plays a significant role in

fluid layers where the buoyancy forces are predominant. Physically it represents the

balance of energy released by the buoyancy force and the energy dissipation by

viscous and thermal effects. We observe from the expression of R that the terms in

the numerator drive the motion and the terms in the denominator oppose the motion.

Mathematically, this number denotes the eigenvalue in the study of stability of

79

thermal convection. The critical thermal Rayleigh number is the value of the

eigenvalue at which the conduction state breaks down and convection sets in.

(ii) Marangoni number

The Marangoni number M is defined as

T T dM ∆=σµ χ

,

where Tσ represents the rate of decrease of surface tension with increase in

temperature. The Marangoni number plays a significant role in fluid layers where

the surface tension force is predominant. It represents the relative importance of

surface tension forces to viscous forces. At a critical value of the Marangoni

number convection sets in. The Marangoni number M is an eigenvalue in the study

of problems of thermal convection driven by surface tension (Pearson, 1958).

(iii) Biot number The Biot number Bi is defined as

CH dBi =χ

,

where CH is the convective heat transfer coefficient. Biot number represents the

ratio of conductive resistance within the fluid layer to convective resistance at the

free surface of the fluid layer. The Biot number plays a significant role in the heat

transfer problems where the fluid layer has a free surface. As has been mentioned

earlier, small enough values of Bi describe the insulating boundary condition on

temperature, while large values signify an isothermal boundary condition.

80

(iv) Conduction-radiation parameter

We define ψ to be

43 a r

QK s

=πψ

χ,

where 3

oc4Q S T π= , aK is the absorption coefficient of the fluid, cS is the

Stefan-Boltzmann constant and oT is the reference temperature. The conduction-

radiation parameter ψ is indicative of the temperature in the equilibrium state. The

results pertaining to non-radiating fluids can be obtained in the limit as 0→ψ

(Khosla and Murgai, 1963).

(v) Absorptivity parameter

The absorptivity parameter λ is defined as 3 (1 )aλ K d ψ= + . The dimensionless parameter λ is the characteristic of absorption coefficient

and distance between the horizontal planes. The results relating to non-radiating

fluids can be obtained in the limit as 0→λ (Khosla and Murgai, 1963).

(vi) Couple-stress parameter – C

The couple-stress parameter C is defined as

2cµC

µ d= (0 )C m≤ ≤ ,

where m is a finite, positive real number according to the Clasusius-Duhem

inequality. As →∞µ , we find that 0C → . This is the Stokesian description of

suspension.

81

CHAPTER IV

INFLUENCE OF RADIATIVE TRANSFER

ON RAYLEIGH-BÈNARD-MARANGONI CONVECTION

IN A COUPLE-STRESS FLUID SATURATED

POROUS MEDIUM 4.1 Introduction

The convective instability problems for radiating fluids received great attention

in the past due to their implications in astrophysical and geophysical applications,

and in other applications such as solar collectors (Bdeoui and Soufiani, 1997). The

Rayleigh-Bénard and Marangoni instability problems involve only two modes of

heat transfer, viz., conduction and convection. Radiative heat transfer is important in

physical systems that have less convective motions because of its stabilizing effect

(Siegel and Howell, 1992; Modest, 1993; Howell and Menguc, 1998). Heat transfer

problems involving conduction, convection and radiation are difficult to solve as the

momentum and heat transport equations are coupled and the latter equation is an

integro-differential equation.

Goody (1956) estimated the radiative transfer effects in the natural convection

problem with free boundaries using a variational method. He solved the problem for

optically thin and optically thick cases and showed that there could be very large

variations near the boundaries. Goody’s radiative transfer model has been extended

and modified by many researchers by taking into account the effects of magnetic

field, rotation and fluid non-grayness (Spiegel, 1960; Murgai and Khosla, 1962;

Khosla and Murgai, 1963; Christophorides and Davis, 1970; Arpaci and Gozum,

1973; Yang, 1990; Bdeoui and Soufiani, 1997; Yan and Li, 2001 and references

therein). Motivated by the meteorological applications, Larson (2001) studied the

82

linear and nonlinear stability of an idealized radiative-convective model due to

Goody (1956). More recently, Lan et al. (2003) analyzed, assuming the boundaries

to be black and the fluid medium to be gray, the stability of a Newtonian fluid

subject to combined natural convection and radiation using a spectral method.

Osalusi (2007) studied the combined hydromagnetic and slip flow of a steady,

laminar conducting viscous fluid in the presence of thermal radiation due to an

impulsively started rotating porous disk taking into account the variable fluid

properties. Xu et al. (2007) studied the combined radiation and natural convection

in an open vertical cylinder. The cylinder is heated through its sidewall at a constant

heat flux and cooled at the top surface via radiation.

Ghosh and Anwar (2008) investigated the unsteady free convective flow of a

viscous incompressible gray, absorbing-emitting but non-scattering, optically-thick

fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot

vertical plate with constant velocity. Rahman and Sultana (2008) analyzed a two-

dimensional steady convective flow of a micropolar fluid past a vertical porous flat

plate in the presence of radiation with variable heat flux.

Ishak (2009) investigated the effects of thermal radiation and thermal buoyancy

on the steady, laminar boundary layer flow over a horizontal plate. Chauhan and

Rastogi (2010) investigated the unsteady natural convection MHD flow of a

rotating viscous electrically conducting fluid in a vertical channel partially filled by

a porous medium with high porosity in the presence of radiation effects. Shateyi et

al. (2010) sought to investigate the influence of a magnetic field on heat and mass

transfer by mixed convection from vertical surfaces in the presence of Hall,

radiation, Soret thermal diffusion, and Dufour-diffusion-thermo effects.

On the other hand, the problem of convection in a couple-stress fluid saturated

porous medium has attracted considerable interest over the past few decades

83

because of its wide range of applications, which comprise pumping fluids such as

synthetic fluids, polymer thickened oils, liquid crystal, animal blood, synovial fluid

present in synovial joints and the theory of lubrication (Naduvinamani et al. 2005).

Thermal process control of couple-stress fluids saturated porous media through

radiative heat transfer is also important in quite a few engineering applications. The

problem of convection in a couple-stress fluid saturated porous medium has been

extensively studied. However, attention has not been paid to the effect of thermal

radiation on Rayleigh-Benard-Marangoni convection in a couple-stress fluid

saturated porous medium. We therefore study in this dissertation the problem of

Rayleigh-Benard-Marangoni convection in a couple-stress fluid saturated porous

medium with thermal radiation using linear stability analysis with emphasis on how

the stability criterion for the onset of convection would be modified in the presence

of radiative transfer.

4.2 Mathematical Formulation Consider a horizontal porous layer saturated with a couples-stress fluid confined

between two parallel, infinite, boundaries heated from below. The upper surface is

free and non-deformable. A Cartesian coordinate system is used with the z-axis

vertically upward. We assume that the Oberbeck-Boussinesq approximation

(Rajagopal et al., 1996) is valid and that the flow in the porous medium is governed

by the Darcy’s law. Moreover, it is assumed that local thermal equilibrium exists

between the solid matrix and the saturated fluid. The fluid between the boundaries

absorbs and emits thermal radiation. We treat the two boundaries as black bodies.

The absorption coefficient of the fluid is assumed to be the same at all wavelengths

and to be independent of the physical state. The system of equations describing the

problem at hand is the following

0. =∇ q , (4.1)

84

( ) ( )2o 1 1. cq q q p g qt k

∂+ = −∇ + − − ∇ ∂

∇ρ ρ µ µε ε

, (4.2)

( ) 2.r

T Gγ q T χ Tt s

∂+ = ∇ +

∂∇ , (4.3)

( )o o1 T T = − − ρ ρ α , (4.4)

( ) ( )o o= − −Ts T T Tσ σ σ , (4.5)

where ( )wvuq ,,= is the mean filter velocity, t is the time, p is the pressure, ρ is

the fluid density, oρ is the reference density, g is the acceleration due to gravity,

µ is the fluid viscosity, cµ is the couple-stress viscosity, ε is the porosity, k is the

permeability of the porous medium, χ is the effective thermal diffusivity, γ is the

ratio of the specific heat of the solid due to porous medium and that of the fluid at

constant pressure, α is the thermal expansion coefficient, T is the temperature,

oT is the reference temperature, G is the rate of radiative heating per unit volume,

rs is the heat content of the fluid per unit volume, sσ is the temperature dependent

surface tension, oσ is the reference surface tension at oT T= , o

( )= − ∂ ∂T Tsσ σ T ,

∇ is the vector differential operator and ( )zyx ,, are the spatial coordinates.

The equation of radiative heat transfer is

( ) ( )a BdI r K P I r

d s

→→ = −

, (4.6)

85

where ( )I r→ is the intensity of radiation along the direction of the vector r→

, ds is

an infinitesimal displacement in the r→

direction, aK is the absorption coefficient

of the fluid and BP is the Planck black-body intensity. The radiative heating rate is

( )s

dI rG dds

ω→

= − ∫ , (4.7)

where the integral is taken over the solid angle 4π and sω is the element of solid

angle. In the quiescent basic state, the equation of radiative transfer (4.6) takes the

form

[ ]3 a BdI K P Idz

µ = − , (4.8)

where 3µ is the directional cosine of r→

in the z-direction. Equation (4.8) is

indicative of the fact that the intensity of radiation is increased by emission and

decreased by absorption.

Iterative solutions of one-dimensional radiative equilibrium problems all show

that remarkably accurate results can be obtained by assuming a simple form for the

angular distribution of radiative intensity. One of the simplest assumptions is the

Milne-Eddington approximation (Goody, 1956)

3 3

3 3

( , ) ( ) , for 0 < 1 ,

( , ) ( ) , for 1 0 .

I z I z

I z I z

µ µ

µ µ

+

−

= ≤

= − ≤ ≤ (4.9)

The energy equation (4.3) in the basic state becomes

2

2 0b b

r

G d Ts dz

+ =χ . (4.10)

86

Equation (4.10) is suggestive of the fact that the heat transfer in the basic state is

essentially by conduction and radiation. If zF is the z-component of the radiative

heat flux, then we may write

bzdFG

dz= − (4.11)

and we may write Eq. (4.10) in the integrated form

1z rF s C+ =χ β , (4.12)

where bdTdz

β = − and 1C is the constant of integration.

Assuming the Milne-Eddington approximation (4.9), and using the radiative

heat transfer equation (4.8), the differential equation associated with the heat flux

zF can be obtained in the form (Goody, 1956)

2

2 212 1*

zz

d F F Cdz

− = −+ψλ λψ

, (4.13)

where ( )2 2 2 4* , 3 1 ,3a

a r

z Qz K dd K s

= = + =πλ ψ ψ

χ, 3

o4 cSQ T=π

and cS is the

Stefan-Boltzmann constant. Solving Eq. (4.13) using the following dimensionless

radiative boundary conditions

2 at * 1 2 ,*

2 at * 1 2 ,*

a z

a z

z

z

dF K F d zd zdF K F d zd z

= − = + =+ = −

(4.14)

we obtain

( ) ( )1 2* cosh *f z L z L= = +β λβ

, (4.15)

where

87

( ) ( )1

12 1 3 3 sinh cosh2 22

L−

= + + + ψ λ λψ ψλ

,

( ) ( )12

1 3 3 sinh cosh2 22LL = + +

λ λψψ

,

and β is the mean value of β throughout the medium. The radiative boundary

conditions (4.14) are obtained using the fact that the molecular conduction ensures

continuity of temperature at the two surfaces. It is advantageous mentioning that

( *)f z tends to unity if either λ or ψ tends to zero independently. Moreover, if λ

and ψ are both greater than unity, the variation of the basic state temperature is

exponential. In other words, the basic state temperature is no longer linear if the

radiation effect is accounted for. In what follows we study the stability of the

quiescent state within the framework of the linear theory.

4.3 Linear Stability Analysis

Let the components of the perturbed physical quantities be

( ), ,

( ) ( , , , )

( ) ( , , , )

( ) ( , , , )

( ) ( , , , )

b

b

b

b

q u v w

p p z p x y z t

z x y z t

T T z T x y z t

G G z G x y z t

′ ′ ′ =

′= + ′= + ′= + ′= +

ρ ρ ρ , (4.16)

where the primes indicate infinitesimally small perturbations. Substituting (4.16)

into the governing equations, neglecting the nonlinear terms, incorporating the

quiescent state solutions and eliminating the pressure term gives the following

equations

88

( )2 2 2 4o 1

cw g T w wt k k

ο ∂ ′ ′ ′ ′∇ = ∇ − ∇ + ∇∂

ρ µµα ρε

, (4.17)

2

r

T Gw Tt s′ ′∂ ′ ′− = ∇ +

∂γ β χ , (4.18)

where 2 2

21 2 2x y

∂ ∂∇ = +

∂ ∂. Since Eq. (4.18) is an integro-differential equation, we

adopt two approximations, one is valid when the fluid medium is optically thin

(known as transparent approximation) and the other is applicable when the fluid

medium is optically thick (known as opaque approximation). For the transparent

approximation (Goody, 1956), the relation becomes

2 21 14 aG Q K T' '∇ = − ∇π (4.19)

and for the optically thick one (Goody, 1956), the relation is

( )2 2 21 1

43 a

QG TK

' 'π∇ = ∇ ∇ . (4.20)

Equation (4.18), corresponding to the transparent approximation, after making

use of the expression in Eq. (4.19), becomes

( ) ( )2 2 2 2 21 1 1 1

4 a

r

Q KT w T Tt s∂ ′ ′ ′ ′∇ − ∇ = ∇ ∇ − ∇∂

πγ β χ . (4.21)

Similarly Eq. (4.18), relating to the opaque approximation, turns out to be

( ) ( ) ( )2 2 2 2 2 21 1 1 1

43

a

a r

Q KT w T Tt K s∂ ′ ′ ′ ′∇ − ∇ = ∇ ∇ + ∇ ∇∂

πγ β χ . (4.22)

We next make Eqs. (4.17), (4.21) and (4.22) dimensionless by considering the

following transformations

89

( )* * * * * *2; ; ; , , , ,d T x y zw w T t t x y z

d d d dd

′ ′= = = =

χχ β

, (4.23)

where the quantities with asterisks are dimensionless. Eqs. (4.17), (4.21) and (4.22),

using (4.23), can be written (after dropping the asterisks)

2 2 2

11 1 0C w R TVa tγ

∂+ − ∇ ∇ − ∇ = ∂

, (4.24)

( ) ( ) ( )2

2 2 2 2 21 1 1 11

T f z w T Tt

λ ψψ

∂∇ − ∇ = ∇ ∇ − ∇

∂ +, (4.25)

( ) ( ) ( ) ( )2 2 2 2 2 21 1 1 1T f z w T T

tψ∂

∇ − ∇ = ∇ ∇ + ∇ ∇∂

, (4.26)

where 2

o

dVak

ε µρ χ

= is the Vadasz number, 2

o g d kR αρ βµ χ

= is the Darcy-Rayleigh

number, 43 a r

QK s

=πψ

χ is the conduction-radiation parameter, 3 (1 )aλ K d ψ= +

is the absorptivity parameter and 2cµC

µ d= is the couple-stress parameter and

( )f z is given in Eq. (4.15).

It should be noted that Eq. (4.25) corresponds to the transparent medium and

Eq. (4.26) has to do with the opaque medium. It is known that, for the problem

under consideration, oscillatory motions can never occur and that the stationary

instability is the preferred mode. In other words, the principle of exchange of

stabilities is valid for the problem at hand (Goody, 1956; Siddheshwar and Pranesh,

2004). Thus we confine our attention to the study of stationary convection.

90

As for the stationary instability, Eqs. (4.24) – (4.26) now read as

( )2 2 211 0C w R T− ∇ ∇ − ∇ = , (4.27)

( ) ( )2

2 2 2 21 1 1 0

1T T f z wλ ψ

ψ∇ ∇ − ∇ + ∇ =

+, (4.28)

( ) ( ) ( )2 2 21 11 0T f z wψ+ ∇ ∇ + ∇ = . (4.29)

As is customary in the linear stability analysis we employ the normal mode

technique. To this end, we assume that the solutions of Eqs. (4.27) - (4.29) have the

form

( )( )

( ) ,i l x m yW zwe

T z+

= Θ (4.30)

where l and m are the horizontal wave numbers in the x and y directions

respectively. Substitution of (4.30) into Eqs. (4.27) - (4.29) gives the following

equations

( ) ( )2 2 2 2 21 0C D a D a W R a − − − + Θ = , (4.31)

( ) ( )2

2 2 01

D a f z Wλ ψψ

− Θ − Θ+ =+

, (4.32)

( )( ) ( )2 21 0D a f z Wψ+ − Θ + = , (4.33)

where 2 2 2a l m= + and D d d z= . Equations (4.31) - (4.33) are solved subject to

the following boundary conditions (with vanishing couple stress on each boundary)

91

2 2 0 at 1 20 at 1 2

0 at 1 2

W D W M a zD Bi zW DW z

= + Θ = =Θ + Θ = = = = Θ = = −

, (4.34)

where M is the Marangoni number and Bi is the Biot number. Each of the systems

comprising Eqs. (4.31) and (4.32) corresponding to the transparent medium and the

system of equations (4.31) and (4.33) having to do with the opaque medium

together with the boundary conditions (4.34) is an eigenvalue problem with R

being the eigenvalue. Since both systems consist of space varying coefficients, it is

no longer possible to obtain a closed form solution of the problem. We therefore use

the Galerkin method (Finlayson, 1972) to solve the eigenvalue problem described

by both systems. We choose suitable trial functions for the z-component of velocity

and temperature that satisfy some of the given boundary conditions, but may not

exactly satisfy the differential equations. This results in residuals when the trial

functions are substituted into the differential equations. The Galerkin method

warrants the residuals be orthogonal to each trial function. It should be made

explicit that the Galerkin method is an invariant of the weighted-residual method

(Finlayson, 1972).

Eigenvalue expression for the transparent medium Applying the Galerkin method to the system of equations (4.31) and (4.32)

coupled with the boundary conditions (4.34) leads to the following equations

( ) ( ) ( )2 21 2 1 1 11 2 1 2 0X C X A CM a DW R a Y B + + Θ − = , (4.35)

( )2

223 2 3 1 1 2 0

1X A Y a Y Bi Bλ ψ

ψ

− + + + Θ = +

, (4.36)

where

92

( )2 2 21 1 1X DW a W= + , ( ) ( )

2 22 4 2 22 1 1 12X D W a W a DW= + + ,

( )3 1 1X f z W= Θ , 1 1 1Y W= Θ , ( )22 1Y D= Θ , 23 1Y = Θ .

Here 1W and 1Θ are the trial functions and 1 2

1 2( )f f z dz

−

= ∫ . On using the

criterion for the existence of the unique solution of the system of Eqs. (4.35) and

(4.36), we obtain the following eigenvalue expression

( ) ( ) ( ) ( )2 22 2

3 1 1 1 2 2 3 1

21 3

1 2 1 2 1 21

C a M X DW X C X Y a Y Bi

Y a XR

λ ψψ

Θ + + + + + Θ

+ = .

(4.37)

Eigenvalue expression for the opaque medium

On the other hand, application of the Galerkin method to the system of

Eqs. (4.31) and (4.33) coupled with the boundary conditions (4.34) results in the

following eigenvalue expression

( ) ( ) ( ) ( ) ( )22 23 1 1 1 2 2 3 1

21 3

1 2 1 2 1 1 2C a M X DW X C X Y a Y Bi

Y a XR

ψ Θ + + + + + Θ = .

(4.38) Keeping in mind the chosen boundary conditions in Eq. (4.34), we deal with the

following trial functions

2

11 12 2

W z z = − +

and 2

112

z Θ = +

.

93

CHAPTER V

RESULTS, DISCUSSION AND

CONCLUDING REMARKS

5.1 Results and Discussion

The problem of Rayleigh-Benard-Marangoni convection in a couple-stress fluid

saturated porous medium with thermal radiation is studied within the framework of

linear stability analysis. Only infinitesimal disturbances are considered. The linear

stability analysis is based on the normal mode technique. The Darcy law is used to

model the momentum equation. The fluid between the boundaries absorbs and emits

thermal radiation. The boundaries are treated as black bodies. Moreover, the

absorption coefficient of the fluid is assumed to be the same at all wavelengths and

to be independent of the physical state. This assumption, having to do with a gray

medium, allows us to consider two asymptotic cases, namely, optically thin fluid

medium (transparent medium) and optically thick fluid medium (opaque medium).

The latter approximation is appropriate for shallow layers in the laboratory.

It is well known that the principle of exchange of stabilities is valid for the

problem at hand, meaning the stationary instability is the preferred mode. In view of

this, the expression for the stationary Darcy-Rayleigh number R is obtained as a

function of the governing parameters, namely, the wave number a , the couple-

stress parameter C , the conduction-radiation parameter ψ , the absorptivity

parameter λ , the Marangoni number M and the Biot number Bi . The Galerkin

invariant of the weighted-residual method is used to determine the eigenvalues.

Neutral stability curves in the ( ),R a plane are plotted for different values of

94

various parameters involved in the problem. The coordinates of the lowest point on

these curves designate the critical values cR and ca .

As for the chosen values of the radiation parameters, we would like to point out

that large radiative effects are more likely if a gas rather than a liquid is used as a

fluid (Howell and Menguc, 1998).

In order to understand better the results arrived at in the problem, we analyze the

nonlinear basic temperature distribution which throws some light on the effect of

radiative heat transfer on the stability of the fluid layer. Figs. 5.2 and 5.3 are plots of

z versus β β for different values of the radiation parameters ψ and λ

respectively. It is clear that 1β β → as either 0ψ → or 0λ → . We also notice

that as ψ or λ increases, the basic temperature profile is exponential. We further

note that the effect of radiative heat transfer is prominent in the boundary layer

regions at the two bounding surfaces. Moreover, the basic temperature distribution

is symmetric about the line z = 0. This symmetric property is largely responsible

for the stabilizing effect of ψ and λ in the radiative heat transfer problem.

Fig. 5.4 depicts the variation of the thermal Rayleigh number R as a function of

the wave number a for different values of the couple-stress parameter C and for

fixed values of ψ , λ , M and Bi . The couple-stress parameter C is indicative of

the concentration of suspended particles. It is seen that R increases with an increase

in C , indicating that the effect of C is to delay the onset of convection. The effect

of conduction-radiation parameter ψ on the onset of convection for fixed values of

C , λ , M and Bi is shown in Fig. 5.5. The parameter ψ is indicative of the

temperature in the equilibrium state. The stabilizing effect of ψ on the system is

obvious from Fig. 5.5.

95

The variation of R with wave number a for different values of λ and for fixed

values of C , ψ , M and Bi is shown in Fig. 5.6. The parameter λ is the

characteristic of the absorption coefficient and the distance between horizontal

planes. We see that the radiation parameter λ has a dual effect on the stability of the

system. This dual effect of λ has been reported by other researchers as well

(Maruthamanikandan, 2003).

The destabilizing influence of the surface-tension mechanism is obvious from

Fig. 5.7. On the other hand, Fig. 5.8 shows the effect of Biot number Bi on the

onset of convection. Biot number Bi is the ratio of conductive resistance within the

fluid layer to convective resistance at the free surface of the fluid layer. When

= 0Bi , an insulated surface retains more energy within the fluid layer and thus the

system is less stable. Once the heat release to the gas is allowed, i.e., 0Bi ≠ , the

fluid becomes more stable. The stabilizing influence of Bi is quite evident from

Fig. 5.8.

A close examination of Figs. 5.4 through 5.8 reveals that the critical wave

number ca is sensitive to the variation in C , ψ , λ , M and Bi , meaning the

parameters C , ψ , λ , M and Bi all have a say on the size of the convection cells.

The results pertaining to the opaque medium are shown in Tables 5.1 through

5.5. The effect of the parameters C , ψ , λ , M and Bi is qualitatively similar to

that corresponding to the transparent medium. It is worth mentioning that the

critical Rayleigh number cR pertaining to the opaque medium is much higher than

that of the transparent medium, meaning maximum stabilization is achieved when

the fluid is optically thick. The effect of ψ and λ on ca , however, needs special

mention. We observe that the radiative heat transfer has no effect on the convection

cell size if the fluid medium is opaque.

96

5.2 Concluding Remarks

The effect of radiative heat transfer on Rayleigh-Benard-Marangoni convection

in a couple-stress fluid saturated porous medium is examined. The eigenvalue

problem is solved numerically using the Galerkin method. The following

conclusions are drawn:

1. The principle of exchange of stabilities is valid and the existence of oscillatory

instability is ruled out.

2. The presence of couple stresses is to delay the onset of convection. In other

words, the effect of suspended particles whose spin matches with the vorticity of

the fluid is to enhance the stability of the system.

3. The radiative transfer tends to damp out any motions which may arise due to the

heat transfer from hotter to colder parts of the fluid.

4. The absorptivity parameter λ has a dual effect on the onset of convection.

5. The opaque medium is shown to release heat for convection more slowly than the

transparent medium.

6. The dimension of the convection cells is not influenced by the radiation

parameters as long as the fluid medium is opaque.

7. The Marangoni number and the Biot number have mutually antagonistic

influence on the onset of convection.

Figure 5.1: Configuration of the problem.

z = d/2

Radiating couple-stress fluid

in a porous medium

x

z

T = T0

T = T0 + ∆T

y

z = – d/2

g

-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5

-25 0 25 50 75 100 125 150 175 200

Figure 5.2: β β as a function of the vertical coordinate z for different values

of the conduction-radiation parameter ψ and for λ = 10.

z

β β

ψ = 105

103

10 –1

101

-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5

-2 0 2 4 6 8 10 12

Figure 5.3: β β as a function of the vertical coordinate z for different values

of the absorptivity parameter λ and for ψ = 210 .

β β

10 –1 100 λ = 102 101

z

1 2 3 4 5 6 7 8

0

3000

6000

9000

12000

15000

18000

Figure 5.4: Plot of Rayleigh number R as a function of the wave number a for

ψ = 1, λ = 1, M = 10, Bi = 1 and for different values of the couple-stress parameter C pertaining to the transparent case.

R

a

C = 1 C = 0. 5

C = 0

1 2 3 4 5 6 7 8

0

3000

6000

9000

12000

Figure 5.5: Plot of Rayleigh number R as a function of the wave number a for

C = 0.5, λ = 1, M = 10, Bi = 1 and for different values of the conduction-radiation parameter ψ pertaining to the transparent case.

R

a

ψ = 1, 5, 10

1 2 3 4 5 6 7 8

0

5000

10000

15000

20000

25000

Figure 5.6: Plot of Rayleigh number R as a function of the wave number a for

C = 0.5, ψ = 1, M = 10, Bi = 1 and for different values of the absorptivity parameter λ pertaining to the transparent case.

R

a

λ = 5 λ = 1

λ = 3

1 2 3 4 5 6 7 80

3000

6000

9000

Figure 5.7: Plot of Rayleigh number R as a function of the wave number a for

C = 0.5, ψ = 1, λ = 1, Bi = 1 and for different values of the Marangoni number M pertaining to the transparent case.

R

a

M = 10, 50, 100

1 2 3 4 5 6 7 8

0

5000

10000

15000

20000

25000

30000

35000

Figure 5.8: Plot of Rayleigh number R as a function of the wave number a for C = 0.5, ψ = 1, λ = 1, M = 10 and for different values of the Biot number Bi pertaining to the transparent case.

R

a

Bi = 0, 5, 10

Table 5.1 : Critical Rayleigh number and critical wave number for different values of the couple-stress parameter C pertaining to the opaque case with 1=ψ ,

1=λ , M = 10 and Bi = 1.

C cR ca

0 0.5 1

236.0574075.5847909.030

3.575 3.051 3.041

Table 5.2 : Critical Rayleigh number and critical wave number for different values of

the conduction-radiation parameter ψ pertaining to the opaque case with C = 0.5, 1=λ , M = 10 and Bi = 1.

ψ cR ca

1 5 10

4075.58415413.78529978.022

3.051 3.051 3.051

Table 5.3 : Critical Rayleigh number and critical wave number for different values of

the absorptivity parameter λ pertaining to the opaque case with C = 0.5, 1=ψ , M = 10 and Bi = 1.

λ cR ca

1 5 10

4075.5842642.1212803.410

3.051 3.051 3.051

Table 5.4 : Critical Rayleigh number and critical wave number for different values of the Marangoni number M pertaining to the opaque case with C = 0.5,

1=ψ , 1=λ and Bi = 1.

M cR ca

10 50 100

4075.5843475.5802725.580

3.051 3.051 3.051

Table 5.5 : Critical Rayleigh number and critical wave number for different values of

the Biot number Bi pertaining to the opaque case with C = 0.5, 1=ψ , 1=λ and M = 10.

Bi cR ca

0 5 10

3023.5907820.780

12211.200

2.755 3.572 3.844

107

BIBLIOGRAPHY Abidin N.Z, Arifin N.M. and Noorani M.S.M. (2008). Boundary effect on

Marangoni convection in a variable viscosity fluid layer. Journal of Mathematics and Statistics, Vol. 4, pp. 1 – 8.

Abraham A. (2002). Rayleigh-Bénard convection in a micropolar ferromagnetic

fluid. Int. J. Engg. Sci., Vol. 40, pp. 449–460. Aggarwal A.K. and Suman Makhija. (2009). Thermal stability of Couple-Stress

Fluid in presence of Magnetic Field and Rotation. Indian Journal of Biomechanics: Special Issue, NCBM 7-8 March 2009.

Aniss S., Belhaq M., Souhar M. and Velarde M.G. (2005). Asymptotic study of

Rayleigh-Benard convection under time periodic heating in Hele-Shaw cell. Phys. Scr., Vol. 71, pp. 395 – 401.

Anwar O., Zueco J., Takhar H.S. and B´eg T.A. (2008). Network Numerical

Simulation of Impulsively-Started Transient Radiation-Convection Heat and Mass Transfer in a Saturated Darcy-Forchheimer Porous Medium. Nonlinear Analysis: Modelling and Control, Vol. 13, pp. 281–303.

Arifin N.Md. and Bachok N. (2008). Boundary effect of on the onset of

Marangoni convection with internal heat generation. Proc. of World Academy of Science, Engineering and Technology, Vol. 28, pp. 20 – 23.

Arpaci V.S. and Gozum D. (1973). Thermal stability of radiating fluids: The

Bénard problem. Phys. Fluids, Vol. 16, pp. 581–588. Bachok N., Ishak A., and Pop I. (2010). Mixed Convection Boundary Layer Flow

over a Permeable Vertical Flat Plate Embedded in an Anisotropic Porous Medium. Mathematical Problems in Engineering, Vol. 2010, pp. 659023 – 659034.

Bankvall C.C. (1974). Convection in a vertical permeable space. Warme-und-

Stoffubertregung, Vol. 7, p. 22. Barletta A. and Nield D.A. (2009). Effect of pressure work and viscous dissipation

in the analysis of the Rayleigh-Benard problem. Int. J. Heat and Mass Transfer, Vol. 52, pp. 3279 – 3289.

108

Bau H.H. (1999). Control of Marangoni-Bénard convection. Int. J. Heat Mass Trans., Vol. 42, pp. 1327–1341.

Baytas A.C. (2003). Thermal non-equilibrium natural convection in a square

enclosure filled with a heat-generating solid phase, non-Darcy porous medium. Int. J. Energy Research, Vol. 27, pp. 975 – 988.

Bdeoui F. and Soufiani A. (1997). The onset of Rayleigh-Bénard instability in

molecular radiating gases. Phys. Fluids, Vol. 9, pp. 3858–3872. Beavers G.S. and Joseph D.D. (1967). Boundary conditions at a naturally

permeable wall. J. Fluid Mech., Vol. 30, p. 197. Beck J.L. (1972). Convection in a box of porous material saturated with fluid.

Phys. Fluids, Vol. 15, p. 1377. Bejan A. (1979). On the boundary layer regime in a vertical enclosure filled with a

porous medium. Letters in Heat and Mass Transfer, Vol. 6, p. 93. Bénard H. (1901). Les tourbillions cellularies dans une nappe liquide transportant

de la chaleut par convection en regime permanent. Ann. d. Chimie et de Physique, Vol. 23, pp. 32–40.

Benguria R.D. and Depassier M.C. (1989). On the linear stability of Bénard-

Marangoni convection. Phys. Fluids A, Vol. 1, pp. 1123–1127. Bhadauria B.S. (2007). Double-diffusive convection in a porous medium with

modulated temperature on the boundaries. Trans. Porous Media, Vol. 70, pp. 191 – 211.

Bhadauria B.S. and Sherani A. (2008). Onset of Darcy convection in a magnetic

fluid saturated porous medium subject to temperature modulation of the boundaries. Trans. Porous Media, Vol. 73, pp. 349 – 368.

Block M.J. (1956). Surface tension as the cause of Bénard cells and surface

deformations in a liquid film. Nature, Vol. 178, pp. 650–651. Bodenschatz E., Pesch W. and Ahlers G. (2000). Recent developments in

Rayleigh-Bénard convection. Ann. Rev. Fluid Mech., Vol. 32, pp. 709–778. Booker J.R. (1976). Thermal convection with strongly temperature-dependent

viscosity. J. Fluid Mech., Vol. 76, pp. 741–754.

109

Borcia R. and Bestehorn M. (2006). Phase field models for Marangoni convection in planar layers. J. Optoelectronics and Advanced Materials, Vol. 8, pp. 1037 – 1039.

Bories S.A. and Combarnous M.A. (1973). Natural convection in a sloping porous

layer. J. Fluid Mech., Vol. 57, p. 63. Bragard J. and Velarde M.G. (1998). Bénard–Marangoni convection: Planforms

and related theoretical predictions. Journal of Fluid Mechanics, Vol. 368, pp. 165-194.

Brenner H. (1970). Rheology of two-phase systems. Ann. Rev. Fluid Mech.,

Vol. 2, p. 137. Brinkman H.C. (1947). Calculation of the viscous force exerted by a flow in a

fluid on dense swarm of particles. Appl. Sci. Res., Vol. 1, P. 27. Buchlin J.M. (2010). Convective heat transfer and infrared thermography (IRTh).

Journal of Applied Fluid Mechanics, Vol. 3, pp. 55-62. Burns P.J., Chow L.C. and Tien C.L. (1977). Convection in a vertical slot filled

with porous insulation. Int. J. Heat Mass Transfer, Vol. 20, p. 919. Busse F.H. (1975). Nonlinear interaction of magnetic field and convection. J. Fluid

Mech., Vol. 71, pp. 193–206. Busse F.H. and Frick H. (1985). Square-pattern convection in fluids with strongly

temperature-dependent viscosity. J. Fluid Mech., Vol. 150, pp. 451–465. Caltagirone J.P. (1980). Stability of a saturated porous layer subject to a sudden

rise in surface temperature: comparison between the linear and energy method. Quart. Jl. Mech. Appl. Math., Vol. 33, p. 47.

Carey V.P. and Mollendorf J.C. (1980). Variable viscosity effects in several

natural convection flows. Int. J. Heat Mass Trans., Vol. 23, pp. 95–108. Chakrabarti A. and Gupta A.S. (1981). Nonlinear thermohaline convection in a

rotating porous media. Mech. Research Comm., Vol. 8, p. 9. Chakraborty S. and Borkakati A.K. (2002). Effect of variable viscosity on

laminar convection flow of an electrically conducting fluid in uniform magnetic field. Theoret. Appl. Mech., Vol. 27, pp. 49–61.

110

Chandra K. (1938). Instability of fluids heated from below. Proc. R. Soc. Lond. A, Vol. 164, pp. 231–242.

Chandrasekhar S. (1961). Hydrodynamic and hydromagnetic stability. Oxford

University Press, Oxford. Char M.I. and Chiang K.T. (1994). Stability analysis of Benard-Marangoni

convection in fluids with internal heat generation. J. Phys. D: Appl. Phys., Vol. 27, pp. 748 – 755.

Char M.I., Chiang, K.T. and Jou, J.J. (1997). Oscillatory instability analysis of

Bénard-Marangoni convection in a rotating fluid with internal heat generation. Int. J. Heat Mass Trans., Vol. 40, pp. 857–867.

Chauhan D.S. and Rastogi P. (2010). Radiation Effects on Natural Convection

MHD Flow in a Rotating Vertical Porous Channel Partially Filled with a Porous Medium. Applied Mathematical Sciences, Vol. 4, pp. 643 – 655.

Christophorides C. and Davis S. H. (1970). Thermal instability with radiative

transfer. Phys. Fluids, Vol. 13, pp. 222–226. Churchill S.W. and Yu B. (2006). The effect of an energetic chemical reaction on

convective heat transfer. Annals of the Assembly for International Heat Transfer Conference 13.

Colinet P., Legros J.C. and Velarde M.G. (2001). Nonlinear dynamics of surface-

tension-driven instabilities. Wiley-VCH, Berlin. Crosby W.A. and Chetti B. (2009). The Static and Dynamic Characteristics of a

Two-Lobe Journal Bearing Lubricated with Couple-Stress Fluid. Tribology Transactions, Vol. 52, pp. 262 – 268.

Darcy H. (1856). Les fontaines publiques de la ville de dijon. V. Dalmont, Paris. Davis S.H. (1967). Convection in a box: Linear theory. J. Fluid Mech.,

Vol. 30, p. 465. Davis S.H. (1969). Buoyancy-surface tension instability by the method of energy.

J. Fluid Mech., Vol. 39, pp. 347–359. Davis S.H. (1987). Thermocapillary instabilities. Ann. Rev. Fluid Mech., Vol. 19,

pp. 403–435.

111

Debler W.R. and Wolf L.W. (1970). The effects of gravity and surface tension gradients on cellular convection in fluid layers with parabolic temperature profiles. Trans. ASME : J. Heat Trans., Vol. 92, pp. 351–358.

Degan G. and Vasseur P. (2003). Influence of anisotropy on convection in porous

media with non-uniform thermal gradient. International Journal of Heat and Mass Transfer, Vol. 46, pp. 781-789.

Devakar M. and Iyengar T.K.V. (2008). Stokes’ Problems for an Incompressible

Couple Stress Fluid. Nonlinear Analysis: Modelling and Control, Vol. 1, pp. 181–190.

Devakar M. and Iyengar T.K.V. (2010). Run up flow of a couple stress fluid

between parallel plates. Nonlinear Analysis: Modelling and Control, Vol. 15, pp. 29–37.

Drazin P. and Reid W. (1981). Hydrodynamic instability. Cambridge University

Press, Cambridge. Einarsson T. (1942). The nature of springs in Iceland. (Ger) Rit. Visindafelug Isl.,

Vol. 26, p. 1. Elder J. W. (1966). Numerical experiments with free convection in a vertical slot.

J. Fluid Mech., Vol. 24, p. 823. Elder J. W. (1958). An experimental investigation of turbulent spots and

breakdown to turbulence. Ph.D. thesis. Cambridge University. Elder J. W. (1967). Steady convection in a porous medium heated from below.

J. Fluid Mech., Vol. 27, p. 29. Elsharkawy A.A. (2004). Effects of misalignment on the performance of finite

journal bearings lubricated with couple stress fluids. Int. Jl. Computer Applications in Technology, Vol. 21, pp. 137-146.

Eringen A. C. (1966). Theory of micropolar fluids. J. Math. Mech., Vol. 16,

pp. 1- 18. Ezzat M., Zakaria M., Samaan A. and Abd-El Bary A. (2006). Free convection

effects on a perfectly conducting couple stress fluid. Journal of Technical Physics, Vol. 47, pp. 5 - 30.

112

Falsaperla P., Mulone G. and Straughan B. (2010). Rotating porous convection with prescribed heat flux. International Journal of Engineering Science, pp. 685-692.

Finlayson B. A. (1972). The method of weighted residuals and variational

principles. Academic Press, New York. Frank F. C. (1965). On dilatancy in relation to seismic sources. Rev. Geophys.,

Vol. 3, pp. 484 - 503. Gaikwad S.N., Malashetty M.S. and Rama Prasad K. (2007). An analytical

study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid. International Journal of Non-Linear Mechanics, Vol. 42, pp. 903-913.

Gasser R. D. and Kazimi M. S. (1976). Onset of convection in a porous medium

with internal heat generation. J. Heat Transfer, Vol. 98, pp. 49 – 54. Gebhart B., Jaluria Y., Mahajan R.L. and Sammakia B. (1988). Buoyancy

induced flows and transport. Hemisphere Publishing Corporation, New York. Gershuni G.Z. and Zhukhovitsky E.M. (1976). Convective stability of

incompressible fluids. Keter, Jerusalem. Getling A.V. and Brausch O. (2003). Cellular flow patterns and their evolutionary

scenarios in three dimensional Rayleigh-Benard convection. Phys. Rev. E., Vol. 67, p. 46313.

Getling, A.V. (1998). Rayleigh-Bénard convection - Structures and dynamics.

World Scientific, Singapore. Gheorghitza (1961). The marginal stability in porous inhomogeneous media. Proc.

Camb. Phil. Soc., Vol. 57, p. 871. Ghosh S. K. and Anwar O. B. (2008). Theoretical Analysis of Radiative Effects

on Transient Free Convection Heat Transfer past a Hot Vertical Surface in Porous Media. Nonlinear Analysis : Modelling and Control, Vol. 13, pp. 419–432.

Gill A.E. (1966). The boundary layer regime for convection in a rectangular cavity.

J. Fluid Mech., Vol. 26, p. 515.

113

Goody R.M. (1956). The influence of radiative transfer on cellular convection. J. Fluid Mech., Vol. 1, pp. 424–435.

Gouesbet G., Maquet J., Roze C. and Darrigo R. (1990). Surface-tension and

coupled buoyancy-driven instability in a horizontal liquid layer: overstability and exchange of stability. Phys. Fluids A, Vol. 2, pp. 903–911.

Griffiths R.W. (1981). Layered double diffusive convection in porous media.

J. Fluid Mech., Vol. 102, p. 221. Guo W. and Narayanan R. (2007). Onset of Rayleigh-Marangoni convection in a

cylindrical annulus heated from below. J. Coll. Interface Science, Vol. 314, pp. 727 - 732.

Hargrove J. B., Lloyd J. R. and Radcliffe C. J. (1998). Radiation heat transfer in

electrorheological fluids: The effect of field induced chaining. 11th Int. Heat Trans. Conf., Kyongju (Korea).

Hashim I. and Siri Z. (2009). Feedback control of thermocapillary convection in a

rotating fluid layer with free-slip bottom. Sains Malaysiana, Vol. 38, pp. 119 – 124.

Hashim I. (2001). On the marginal mode for the onset of oscillatory Marangoni

convection with internal heat generation. Acta Mech., Vol. 148, pp. 157–164. Hashim I. and Arifin N.M. (2003). Oscillatory Marangoni convection in a

conducting fluid layer with a deformable free surface in the presence of a vertical magnetic field. Acta Mech., Vol. 164, pp. 199–215.

Hashim I. and Wilson S.K. (1999a). The effect of a uniform vertical magnetic

field on the linear growth rates of steady Marangoni convection in a horizontal layer of conducting fluid. Int. J. Heat Mass Trans., Vol. 42, pp. 525–533.

Hashim I. and Wilson S.K. (1999b). The effect of a uniform vertical magnetic

field on the onset of oscillatory Marangoni convection in a horizontal layer of conducting fluid. Acta Mech., Vol. 132, pp. 129–146.

Hashim I. and Wilson S.K. (1999c). The onset of Bénard-Marangoni convection

in a horizontal layer of fluid. Int. J. Engg. Sci., Vol. 37, pp. 643–662. Herron I.H. (2000). On the principle of exchange of stabilities in Rayleigh-Benard

convection. SIAM Jl. Appl. Math., Vol. 61, pp. 1362 – 1368.

114

Hill A. A. (2005). Double-diffusive convection in a porous medium with a concentration based internal heat source. Proc. Roy. Soc. A, Vol. 461, pp. 561 – 574.

Hirayama O. and Takaki R. (1993). Thermal convection of a fluid with

temperature-dependent viscosity. Fluid Dyn. Res., Vol. 12, pp. 35–47. Homsy G.M. and Sherwood A.E. (1976). Convective instabilities in porous media

with through flow. A.I.Ch.E. Journal, Vol. 22, p. 168. Horne R.N. and O’Sullivan, M.J. (1978). Convection in a porous medium heated

from below: Effect of temperature dependent viscosity and thermal expansion coefficient. Trans. ASME : J. Heat Trans., Vol. 100, pp. 448–452.

Horton C.W. and Rogers Jr. F.T. (1945). Convection currents in a porous

medium. J. Appl. Phys., Vol. 16, p. 367. Hossain M.A., Kabir S. and Rees D.A.S. (2002). Natural convection of fluid with

variable viscosity from a heated vertical wavy surface. ZAMP, Vol. 53, pp. 48–52.

Howell J. R. and Menguc M. P. (1998). Handbook of heat transfer fundamentals.

Tata McGraw Hill, New York. Howle L., Behringer R.P. and Georgiadis J. (1993). Visualization of convective

fluid flow in a porous medium. Nature, Vol. 362, pp. 230 – 232. Hubbert M.K. (1956). Darcy’s law and the field equations of the flow of

underground fluids. American Institute of Mining Engineers Transactions, Vol. 207, pp. 222 - 239.

Humera G.N., Ramana Murthy M.V., ChennaKrishna Reddy M., Rafiuddin,

Ramu A. and Rajender S. (2010). Hydromagnetic Free Convective Rivlin – Ericksen Flow through a Porous Medium with Variable Permeability. International Journal of Computational and Applied Mathematics, Vol. 5, pp. 267–275.

Indira R., Venkatachalappa M. and Siddheshwar P.G. (2008). Flow of couple-

stress fluid between two eccentric cylinders. International J. of Math. Sci. & Engg. Appls. (IJMSEA), Vol. 2, pp. 253 – 261.

Ingham D. B. and Pop I. (Editors) (1998). Transport phenomena in porous media.

Pergamon, Oxford.

115

Isa S. S. P. M., Arifin N. M., Nazar R. M. and Saad M. N. (2009). Effects of

non-uniform temperature gradient on Marangoni convection with free slip condition. American Journal of Scientific Research, Vol. 2009, pp. 37 – 44.

Ishak A. (2009). Mixed convection boundary layer flow over a horizontal plate

with thermal radiation. Heat Mass Transfer, Vol. 46, pp. 147 - 151. Jaw-Ren Lin and Chi-Ren Hung. (2007). Combined effects of non-Newtonian

couple stresses and fluid inertia on the squeeze film characteristics between a long cylinder and an infinite plate. Fluid Dynamics Research, Vol. 39, pp. 616-631.

Jaw-Ren Lin. (1998). Squeeze film characteristics of finite journal bearings: couple

stress fluid model. Tribology International, Vol. 31, pp. 201-207. Jaw-Ren Lin. (2000). Squeeze film characteristics between a sphere and a flat

plate: couple stress fluid model, Computers & Structures, Vol. 75, pp. 73-80. Jeffreys H. (1926). The stability of a layer of fluid heated below. Phil. Mag.,

Vol. 2, pp. 833–844. Joseph D.D. (1976). Stability of fluid motions I and II. Springer Verlag. Kafoussias N.G. and Williams E.W. (1995). The effect of temperature-dependent

viscosity on the free convective laminar boundary layer flow past a vertical isothermal flat plate. Acta Mech., Vol. 110, pp. 123–137.

Kafoussias N.G., Rees D.A.S. and Daskalakis J.E. (1998). Numerical study of the

combined free and forced convective laminar boundary layer flow past a vertical isothermal flat plate with temperature-dependent viscosity. Acta Mech., Vol. 127, pp. 39–50.

Katto Y. and Masuoka T. (1967). Criterion for the onset of convection flow in a

porous medium. Int. J. Heat Mass Transfer. Vol. 10, p. 297. Kays W. M. and Crawford M. E. (1980). Convective heat and mass transfer.

McGraw-Hill, New York. Khaleque T.S. and Samad M.A. (2010). Effects of radiation, heat generation and

viscous dissipation on MHD free convection flow along a stretching sheet. Research Journal of Applied Sciences, Engineering and Technology, Vol. 2, pp. 368-377.

116

Khosla P. K. and Murgai M. P. (1963). A study of the combined effect of thermal

radiative transfer and rotation on the gravitational stability of a hot fluid. J. Fluid Mech., Vol. 16, pp. 97–107.

Klarsfeld S. (1970). Champs de temperature associe’s aux mouments de convection

naturelle dams un milieu porehx limite. Rev. Gen. Thermique, Vol. 108, p. 1403. Kordylewski W., Borkowska-Pawlik B. and Slany J. (1986). Stability of three-

dimensional natural convection in a porous layer. Arch. Mech., Vol. 4, p. 383. Kourganoff V. (1952). Basic methods in heat transfer problems. Oxford University

Press, Oxford. Kozhhoukharova Zh., Kuhlmann H.C., Wanschura M. and Rath H.J. (1999).

Influence of variable viscosity on the onset of hydrothermal waves in thermocapillary liquid bridges. ZAMM, Vol. 79, pp. 535–543.

Kozhoukharova, Zh. and Roze, C. (1999). Influence of the surface deformability

and variable viscosity on buoyant-thermocapillary instability in a liquid layer. Eur. Phys. J. B, Vol. 8, pp. 125–135.

Krishnamurti R. (1968a). Finite amplitude convection with changing mean

temperature. Part I. Theory. J. Fluid Mech., Vol. 33, pp. 445–456. Krishnamurti R. (1968b). Finite amplitude convection with changing mean

temperature. Part II. Experimental test of the theory. J. Fluid Mech., Vol. 33, pp. 457–475.

Krishnamurti R. (1997). Convection induced by selective absorption of radiation:

a laboratory model of conditional instability. Dyn. Atmos. Oceans, Vol. 27, pp. 367–382.

Lam T.T. and Bayazitoglu Y. (1987). Effects of internal heat generation and

variable viscosity on Marangoni convection. Numer. Heat Trans., Vol. 11, pp. 165–182.

Lan C. H., Ezekoye O. A., Howell J. R. and Ball K. S. (2003). Stability analysis

for three-dimensional Rayleigh-Bénard convection with radiatively participating medium using spectral methods. Int. J. Heat Mass Trans., Vol. 46, pp. 1371–1383.

117

Lapwood E.R. (1948). Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc., Vol. 44, pp. 508 – 521.

Larson V. E. (2001). The effects of thermal radiation on dry convective instability.

Dyn. Atmos. Oceans, Vol. 34, pp. 45–71. Lebon, G. and Cloot, A. (1984). A nonlinear stability analysis of the Bénard-

Marangoni problem. J. Fluid Mech., Vol. 145, pp. 447–469. Lewicka M. (2007). Existence of traveling waves in the Stokes-Boussinesq system

for reactive flow. J. Differential Equations, Vol. 237, pp. 343—371. Li Y.S., Chen Z.W. and Zhan J.M . (2010). Double-diffusive Marangoni

convection in a rectangular cavity: Transition to chaos. International Journal of Heat and Mass Transfer, Vol. 53, pp. 5223-5231.

Liao W.H., Lu R.F., Chien R.D., and Lin J.W. (2005). Linear stability analysis of

long journal bearings: couple stress fluid model. Industrial Lubrication and Tribology, Vol. 57, pp. 21 – 27.

Lienhard, J. H. (1990). Thermal radiation in Rayleigh-Benard instability.

Transactions of the ASME : Journal of Heat Transfer, Vol. 112, pp. 100-109. Lin K., Dold P. and Benz K. W. (2005). Numerical study of influences of

buoyancy and solutal Marangoni convection on flow structures in a germanium-silicon floating zone. Cryst. Res. Technol., Vol. 40, pp. 550 – 556.

List E.J. (1965). The convective stability of a confined saturated porous medium

containing uniformly distributed heat sources. W.M. Keck Lab and Water Resources, Div. Engg. Appl. Sci. Calif. Inst. Tech., Memo., No. 65 – 10.

Liu P.C. (2003). Temperature Distribution in a Porous Medium Subjected to Solar

Radiative Incidence and Downward Flow: Convective Boundaries. Transactions of the ASME, Vol. 125, pp. 190 – 194.

Lord Rayleigh, O.M. (1916). On convection currents in a horizontal layer of fluid,

when the higher temperature is on the under side. Phil. Mag., Vol. 32, pp. 529–546.

Lorenz E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci., Vol. 20,

pp. 130–141.

118

Low A.R. (1929). On the criterion for stability of a layer of viscous fluid heated from below. Proc. R. Soc. Lond. A, Vol. 125, pp. 180–192.

Ma T. and Wang S. (2007). Rayleigh-Benard convection: dynamics and structure

in the physical system. Comm. Math. Sci., Vol. 5, pp. 553 – 574. Ma. T and Wang S. (2004). Dynamic bifurcation and stability in the Rayleigh-

Benard convection. Comm. Math. Sci., Vol. 2, pp. 159 – 183. Maekawa, T. and Tanasawa, I. (1988). Effect of magnetic field on the onset of

Marangoni convection. Int. J. Heat Mass Trans., Vol. 31, pp. 285–293. Magyari E. (2010). Comment on Mixed convection boundary layer flow over a

horizontal plate with thermal radiation. Heat Mass Transfer, Vol. 46, pp. 809 – 810.

Mahinder Singh and Pardeep Kumar. (2010). Magneto Thermal Convection in a

Compressible Couple-Stress Fluid, Z. Naturforsch. Vol. 65, pp. 215 – 220. Mahmud M.N., Idris R. and Hashim I. (2009). Effect of a magnetic field on the

onset of Marangoni convection in a micropolar fluid. Proc. of World Academy of Science, Engineering and Technology, Vol. 38, pp. 866 – 868.

Mahmud M.N., Idris R. and Hashim I. (2009). Effect of a magnetic field on the

onset of Marangoni convection in a micropolar fluid. Int. Jl. Mathematical and Statistical Sciences. Vol. 1, pp. 63 - 65 .

Malashetty M. S., Shivakumara I. S. and Kulkarni S. (2005). The onset of

Lapwood-Brinkman convection using a thermal non-equilibrium model. Int. Jl. Heat and Mass Transfer, Vol. 48, pp. 1155-1163.

Malashetty M. S., Swamy M. and Rajashekhar. (2008). Double diffusive

convection in a porous layer using a thermal non-equilibrium model. International Journal of Thermal Sciences, Vol. 47, pp. 1131-1147.

Malashetty M.S. and Basavaraja D. (2002). Rayleigh-Benard convection subject

to time dependent wall temperature/gravity in a fluid-saturated anisotropic porous medium. Heat and Mass Trans., Vol. 38, pp. 551 – 563.

Malashetty M.S. and Heera R. (2008). Linear and non-linear double diffusive

convection in a rotating porous layer using a thermal non-equilibrium model. International Journal of Non-Linear Mechanics, Vol. 43, pp. 600-621.

119

Malashetty M.S. and Kulkarni S. (2009). The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model. Journal of Non-Newtonian Fluid Mechanics, Vol. 162, pp. 29 - 37.

Malashetty M.S., Dulal Pal and Premila Kollur. (2010). Double-diffusive

convection in a Darcy porous medium saturated with a couple-stress fluid. Fluid Dynamics Research., Vol. 42, pp. 35502 - 35522.

Malashetty M.S., Gaikwad S.N. and Mahantesh Swamy. (2006). An analytical

study of linear and non-linear double diffusive convection with Soret effect in couple stress liquids. International Journal of Thermal Sciences, Vol. 45, pp. 897-907.

Malashetty M. S., Kulkarni and Sridhar. (2009). The onset of convection in a

couple- stress fluid-saturated porous layer using a thermal non-equilibrium model. Phys. Letters A, Vol. 373, pp. 781 – 790.

Malkus W.V.R. and Veronis G. (1958). Finite amplitude cellular convection.

J. Fluid Mech., Vol. 4, pp. 225–260. Mansour M. A. and Gorla R. S .R. (1999). Radiative effects in natural convection

flows. Int. J. Therm. Sci., Vol. 38, pp. 664–673. Mansour M.A. and El-Shaer N.A. (2001). Radiative effects on

magnetohydrodynamic natural convection flows saturated in porous media. Journal of Magnetism and Magnetic Materials, Vol. 237, pp. 327–341.

Maroto J. A., Perez-Munuzuri V. and Romero-Cano M. S. (2007). Introductory

analysis of Benard-Marangoni convection. Euro. J. Phys., Vol. 28, pp. 311 – 320.

Martin A. R., Salitel C., Chai J. and Shyy W. (1998). Convective and radiative

internal heat transfer augmentation with fiber arrays. Int. J. Heat Mass Trans., Vol. 41, pp. 3431–3440.

Maruthamanikandan S. (2003). Effect of radiation on Rayleigh-Bénard

convection in ferromagnetic fluids. Int. Jl. Appl. Mech. Engg., Vol. 8, pp. 449 – 459.

Maruthamanikandan S. (2005). Convective instabilities in Newtonian

ferromagnetic, dielectric and other complex liquids. Ph.D. Thesis.

120

Massaioli F., Benzi R. and Succi S. (1993). Exponential tails in two-dimensional Rayleigh-Benard convection. Europhys. Lett., Vol. 21, pp. 305 – 310.

Massoudi M., Tran P.X. and Wulandana R. (2009). Convection-Radiation heat

transfer in a nonlinear fluid with temperature-dependent viscosity. Mathematical Problems in Engineering, Vol. 2009, pp. 232670 – 232684.

Masuoka T. (1968). Free convection heat transfer from a vertical surface to a

porous medium. Trans. Japan Soc. Mech. Engr., Vol. 34, p. 491. Masuoka T. (1972). Heat transfer by free convection in a porous layer heated from

below. Heat Transfer, Japanese Research, Vol. 1, p. 39. Masuoka T., Yokote Y. and Katsuhara T. (1981). Heat transfer by natural

convection in a vertical porous layer. Bulletin of JSME, Vol. 24, p. 995. Mebine P. and Adigio E.M. (2009). Unsteady Free Convection Flow with Thermal

Radiation Past a Vertical Porous Plate with Newtonian Heating. Turk J Phys., Vol. 33, pp. 109 – 119.

Merkin J. H. and Chaudhary M. A. (1994). Free-convection boundary layers on

vertical surfaces driven by an exothermic surface reaction. Quart. Jl. Mech. Appl. Math., Vol. 47, pp. 405 – 428.

Modest M. F. (1993). Radiative heat transfer. Tata McGraw Hill, New York. Mohammadein A.A., Mansour M.A., Gaied S.M. and Gorla R.S.R. (1998).

Radiative Effect on Natural Convection Flows in Porous Media. Transport in Porous Media, Vol. 32, pp. 263–283.

Mohammadien A.A. and El-Amin M.F. (2000). Thermal Dispersion–Radiation

Effects on Non-Darcy Natural Convection in a Fluid Saturated Porous Medium. Transport in Porous Media, Vol. 40, pp. 153–163.

Mokhtar N. F. M., Arifin N. M., Nazar R., Ismail F. and Suleiman M. (2008).

Marangoni convection in a fluid saturated porous layer with a prescribed heat flux at its lower boundary. European Journal of Scientific Research, Vol. 24, pp. 477- 486.

Mokhtar N. F. M., Arifin N.M., Nazar R., Ismail F. and Suleiman M. (2010).

Effect of internal heat generation on Marangoni convection in a fluid saturated porous medium. International Journal of Applied Mathematics & Statistics, Vol. 19, pp. 111 – 122.

121

Mokhtar N.F.M., Arifin N.M., Nazar R., Ismail F. and Suleiman M. (2009).

Effects of Non-Uniform Temperature Gradient and Magnetic Field on Bénard Convection in Saturated Porous Medium. European Jl. Scientific Research, Vol. 34, pp. 365-371.

Mokhtar N.M., Arifin N.M., Nazar R., Ismail F. and Suleiman M.. (2010).

Effect of internal heat generation on Marangoni convection in a fluid saturated porous medium. International Journal of Applied Mathematics & Statistics, Vol. 19, pp. 111 – 122.

Morels M., Spinn C. W. and Smith J. M. (1951). Velocities and effective thermal

conductivities in packed beds. Indust. And Engg. Chem., Vol. 43, p. 225. Morrison H. L. (1947). Preliminary experiment on convection currents in a porous

medium. Phys. Rev., Vol. 71, p. 834. Morrison H. L. and Rogers Jr. F. T. (1952). Significance of flow patterns for

initial convection in porous media. J. Appl. Phys., Vol. 23, p. 1058. Morrison H. L., Rogers Jr. F. T. and Horton C. W. (1949). Convective currents

in porous media II: Observation of conditions at onset of convection. J. Appl. Phys., Vol. 20, p. 1027.

Motsa S. S. (2008). On the onset of convection in a porous layer in the presence of

Dufour and Soret effects. SJPAM, Vol. 3, pp. 58 – 65. Mukhopadhyay S. (2009). Effects of Radiation and Variable Fluid Viscosity on

Flow and Heat Transfer along a Symmetric Wedge. Journal of Applied Fluid Mechanics, Vol. 2, pp. 29-34.

Mukutmoni D. and Yang K. T. (1994). Flow transitions and pattern selection of

the Rayleigh-Benard problem in rectabgular enclosures. Sadhana, Vol. 19, pp. 649 – 670.

Murgai M. P. and Khosla P. K. (1962). A study of the combined effect of thermal

radiative transfer and a magnetic field on the gravitational convection of an ionized fluid. J. Fluid Mech., Vol. 14, pp. 433–451.

Murty Y. N. (2006). Effect of throughflow and magnetic field on Marangoni

convection in micropolar fluids. Appl. Math. Comp., Vol. 173.

122

Muskat M. (1937). Flow of homogeneous fluids through porous media. McGraw Hill, New York.

Naduvinamani N. B., Fathima S. T. and Hiremath P. S. (2003a). Hydrodynamic

lubrication of rough slider bearings with couple stress fluids. Tribol. Int., Vol. 36, pp. 949 – 959.

Naduvinamani N. B., Fathima S. T. and Hiremath P. S. (2003b). Effect of

surface roughness on characteristics of couple stress squeeze film between anisotropic porous rectangular plates. Fluid Dyn. Res., Vol. 32, pp. 217 – 231.

Naduvinamani N. B., Hiremath P. S. and Gurubasavaraj G. (2001). Squeeze

film lubrication of a short porous journal bearing with couple stress fluids. Tribol. Int., Vol. 34, pp. 739 – 747.

Naduvinamani N. B., Hiremath P. S. and Gurubasavaraj G. (2002). Surface

roughness effects in a short porous journal bearing with a couple stress fluid. Fluid Dyn. Res., Vol. 31, pp. 333 – 354.

Naduvinamani N. B., Hiremath P. S. and Gurubasavaraj G. (2005). Effects of

surface roughness on the couple stress squeeze film between a sphere and a flat plate. Tribol. Int., Vol. 38, pp. 451 – 458.

Naduvinamani N.B. and Kashinath B. (2006). Surface roughness effects on

curved pivoted slider bearings with couple stress fluid. Lubrication Science, Vol. 18, pp. 293 – 307.

Naduvinamani N.B. and Siddangouda A. (2007). A note on porous Rayleigh step

bearings lubricated with couple-stress fluids. Proc. Inst. Mech. Engineers, Part J: Journal of Engineering Tribology, Vol. 221, pp. 615-621.

Narahari M. (2010). Effects of thermal radiation and free convection currents on

the unsteady Couette flow between two vertical parallel plates with constant heat flux at one boundary, Wseas Transactions on Heat and Mass Transfer, Vol. 5, pp. 21- 30.

Nield D. A. (1964). Surface tension and buoyancy effects in cellular convection.

J. Fluid Mech., Vol. 19, pp. 341–352. Nield D. A. (1966). Surface tension and buoyancy effects in the cellular convection

of an electrically conducting liquid in a magnetic field. ZAMP, 17, pp. 131–139.

123

Nield D. A. (1968). Onset of thermohaline convection in a porous medium. Water Resources Research, Vol. 4, p. 553.

Nield D.A. and Bejan A. (1992). Convection in porous media. Springer, New

York. Normand C., Pomeau Y. and Velarde M.G. (1977). Convective instability: A

Physicists approach. Rev. Mod. Phys., Vol. 49, pp. 581–624. Onyegegbu S. (1980). Convective instability of rotating radiating fluids.

Trans. ASME : J. Heat Trans., Vol. 102, pp. 268–275. Osalusi E. (2007). Effects of thermal radiation on MHD and slip flow over a porous

rotating disk with variable properties. Rom. Journ. Phys., Vol. 52, pp. 217–229. Ostrach S. (1972). Natural convection in enclosures. Advances in Heat Transfer,

Vol. 8, Academic Press, New York. Palani G. and Abbas I. A. (2009). Free Convection MHD Flow with Thermal

Radiation from an Impulsively-Started Vertical Plate. Nonlinear Analysis: Modelling and Control, Vol. 14, pp. 73–84.

Palm E. (1960). On the tendency towards hexagonal cells in steady convection.

J. Fluid Mech., Vol. 8, pp. 183–192. Pardeep Kumar and Mahinder Singh. (2009). Rotatory Thermosolutal

Convection in a Couple-Stress Fluid, Z. Naturforsch. Vol. 64, pp. 448 – 454. Parmentier, P.M., Regnier, V.C., Lebon, G. and Legros, J.C. (1996). Nonlinear

analysis of coupled gravitational and capillary thermoconvection in thin fluid layers. Phys. Rev. E, Vol. 54, pp. 411–423.

Patil P.M. and Kulkarni P.S. (2008). Effects of chemical reaction on free

convective flow of a polar fluid through a porous medium in the presence of internal heat generation. Int. Jl. Therm. Sci., Vol. 47, pp. 1043 – 1054.

Pearson, J.R.A. (1958). On convection cells induced by surface tension.

J. Fluid Mech., Vol. 4, pp. 489–500. Pellew A. and Southwell R.V. (1940). On maintained convective motion in a fluid

heated from below. Proc. R. Soc. Lond. A, Vol. 176, pp. 312–343.

124

Perez-Garcia, C. and Carneiro, G. (1991). Linear stability analysis of Bénard-Marangoni convection in fluids with a deformable free surface. Phys. Fluids A, Vol. 3, pp. 292–298.

Platten J.K. and Legros J.C. (1984). Convection in liquids. Springer, Berlin. Pomraning G. C. (1973). The equations of radiative hydrodynamics. Dover, New

York. Postelnicu A. (2008). The onset of a Darcy–Brinkman convection using a thermal

nonequilibrium model. Part II. Int. Jl. Thermal Sciences, Vol. 47, pp. 1587-1594. Rahman M.M. and Sultana T. (2008). Radiative Heat Transfer Flow of

Micropolar Fluid with Variable Heat Flux in a Porous Medium. Nonlinear Analysis : Modelling and Control, Vol. 13, pp. 71–87.

Rajagopal K. R., Ruzica M. and Srinivasa A. R. (1996). On the Oberbeck-

Boussinesq approximation. Math. Model. Methods Appl. Sci., Vol. 6, p. 1157. Ramachandramurthy V. (1987). Marangoni and Rayleigh-Bènard convection.

Ph.D. Thesis, Bangalore University, India. Rapaka S., Chen S., Pawar R.J., Stauffer P.H. and Zhang D. (2008). Non-modal

growth of perturbations in density-driven convection in porous media. J. Fluid Mech., Vol. 609, pp. 285 – 303.

Richter A., Plettner A., Hofmann K.A. and Sandmaier H. (1991).

A micromachined electrohydrodynamic pump. Sensors and Actuators A: Physical, Vol. 29, pp. 159–168.

Richter, F.M., Nataf, H.C. and Daley, S.F. (1983). Heat transfer and horizontally

averaged temperature of convection with large viscosity variations. J. Fluid Mech., Vol. 129, pp. 173–192.

Rogers J.L and Schatz M.F. (2000). Superlattice patterns in vertically oscillated

Rayleigh-Benard convection. Phys. Rev. Lett., Vol. 85, pp. 4281 – 4284. Rogers J.L., Schatz M.F., Bougie J.L. and Swift J.B. (2000). Rayleigh-Benard

convection in a vertically oscillating fluid layer. Phys. Rev. Lett., Vol. 84, pp. 87 – 90.

Rogers Jr. F.T. and Morrison H.L. (1950). Convection currents in porous media.

J. Appl. Phys., Vol. 21, p. 1177 - 1180.

125

Rogers Jr. F.T. and Schilberg L.E. (1950). Observation of initial flow in a fluid

obeying Darcy’s law by radioactive tracer technique. J. Appl. Phys., Vol. 22, P. 233.

Rogers Jr. F.T., Schilberg L.E. and Morrison H.L. (1950). Convection currents

in porous media IV: Remarks on the theory. J. Appl. Phys., Vol. 22, 1476. Rudiger S. and Knobloch E. (2003). Mode interaction in rotating Rayleigh-Benard

convection. Fluid Dynamics Research. Vol. 33, pp. 477 – 492. Sarangi M., Majumdar B.C. and Sekhar A.S. (2005). Elastohydrodynamically

Lubricated Ball Bearings with Couple-Stress Fluids, Part I: Steady-State Analysis. Tribology Transactions, Vol. 48, pp. 401 – 414.

Sarangi M., Majumdar B.C. and Sekhar A.S. (2005). Elastohydrodynamically

Lubricated Ball Bearings with Couple-Stress Fluids, Part II: Dynamic Analysis and Application, Tribology Transactions, Vol. 48, pp. 415 – 424.

Saravanan S. (2007). Marangoni convection in an Oldroyd-B fluid layer with

throughflow. Canadian J. Phys., Vol. 85, pp. 947 – 955. Saravanan S. (2009). Centrifugal acceleration induced convection in a magnetic

fluid saturated anisotropic rotating porous medium. Trans. Porous Media, Vol. 77, pp. 79 – 86.

Sarma G.S.R. (1979). Marangoni convection in a fluid layer under the action of a

transverse magnetic field, Space Res., Vol. 19, pp. 575–578. Sarma G.S.R. (1981). Marangoni convection in a liquid layer under the

simultaneous action of a transverse magnetic field and rotation. Adv. Space Res., Vol. 1, p. 55.

Sarma G.S.R. (1985). Effects of interfacial curvature and gravity waves on the

onset of thermocapillary convection in a rotating liquid layer subjected to a transverse magnetic field. Physio-Chem. Hydrodyn., Vol. 6, pp. 283–300.

Schneider K.J. (1963). Investigation of the influence of free thermal convection on

heat transfer through globular materials. 12th Int. Congress of Refrigeration, Munich, Paper II-4.

126

Seddeek M. A. and Aboeldahab E.M. (2001). Radiation effects on unsteady mhd free convection with hall current near an infinite vertical porous plate, IJMMS, Vol. 26, pp. 249–255.

Sekhar G.N. and Jayalatha G. (2010). Rayleigh-Bénard convection in liquids with

temperature dependent viscosity. Int. Jl. Thermal Sciences, Vol. 49, pp. 67 – 75. Sekhar G.N. (1991). Convection in magnetic fluids with internal heat generation.

Trans. ASME : J. Heat Trans., Vol. 113, 122–127. Selak R. and Lebon G. (1997). Rayleigh-Marangoni thermoconvective instability

with non-Boussinesq corrections. Int. J. Heat Mass Trans., Vol. 40, pp. 785–798. Severin J. and Herwig H. (1999). Onset of convection in the Rayleigh-Bénard

flow with temperature-dependent viscosity: An asymptotic approach. ZAMP, Vol. 50, pp. 375–386.

Sharma R. C. and Thakur K. D. (2000). On couple-stress fluid heated from below

in porous medium in hydromagnetics. Czechoslovak Jl. Physics, Vol. 50, pp. 753-758.

Sharma R. C., Sunil and Pal M. (2000). On couple-stress fluid heated from below

in porous medium in the presence of rotation. Int. J. Applied Mechanics and Engineering, Vol. 5, pp. 883 – 896.

Sharma R.C. and Mehta C.B. (2005). Thermosolutal convection in compressible,

rotating, couple-stress fluid. Indian J. Phys., Vol. 79, pp. 161-165. Sharma R.C. and Sharma M. (2004a). Effect of suspended particles on couple-

stress fluid heated from below in the presence of rotation and magnetic field. Indian Jl. Pure. Appl. Math., Vol. 35, pp. 973 – 989.

Sharma R.C. and Sharma M. (2004b). On couple-stress fluid permeated with

suspended particles heated and soluted from below in porous medium. Indian J. Phys. Vol. 78 B (2), pp. 189-194.

Sharma V., Sunil and Gupta U. (2006). Stability of stratified elastico-viscous

Walters’ (Model B′) fluid in the presence of horizontal magnetic field and rotation in porous medium. Arch. Mech., Vol. 58, pp. 187–197.

Sharma R. C. and Shivani S. (2001a). On couple-stress fluid heated from below in

a porous medium. Indian Jl. Phys. B, Vol. 75, pp. 137-139.

127

Sharma R. C. and Shivani S. (2001b). On electrically conducting couple-stress fluid heated from below in porous medium in presence of uniform horizontal magnetic field. Int. J. Appl. Mech. Eng., Vol. 6, pp. 251 - 253.

Shateyi S. (2008). Thermal radiation and buoyancy effects on heat and mass

transfer over a semi-infinite stretching surface with suction and blowing. J. Appl. Math., Vol. 2008, pp. 414830 – 414841.

Shateyi S., Motsa S.S. and Sibanda P. (2010). The Effects of thermal radiation,

Hall currents, Soret, and Dufour on MHD flow by mixed convection over a vertical surface in porous media. Mathematical Problems in Engineering, Vol. 2010, pp. 627475 - 627494.

Shklyaev S. Khenner M. and Alabuzhev A.A. (2010). Oscillatory and monotonic

modes of long-wave Marangoni convection in a thin film. Physical Review E, Vol. 82, p. 025302.

Siddheshwar P. G. and Pranesh S. (2004). An analytical study of linear and non-

linear convection in Boussinesq–Stokes suspensions. International Journal of Non-Linear Mechanics, Vol. 39, pp. 165-172.

Siddheshwar P. G. and Sri Krishna C. V. (2001). Rayleigh-Benard convection in

a viscoelastic fluid-filled high-porosity medium with nonuniform basic temperature gradient. IJMMS, Vol. 25, pp. 609–619.

Siddheshwar P.G. and Abraham, A. (2003). Effect of time-periodic boundary

temperatures/body force on Rayleigh-Bénard convection in a ferromagnetic fluid. Acta Mech., Vol. 161, pp. 131–150.

Siegel R. and Howell J. R. (1992). Thermal radiation heat transfer. Taylor &

Francis, Washington. Simanovskii, Nepomnyashchy A., Shevtsova V., Colinet P. and Legros J.C.

(2006). Nonlinear Marangoni convection with the inclined temperature gradient in a multilayer systems. Phys. Rev. E., Vol. 73, pp. 66310 – 66316.

Siri Z. and Hashim I. (2008). Control of oscillatory Marangoni convection in a

rotating fluid layer. Int. Comm. in Heat and Mass Trans., Vol. 35, pp. 1130 – 1133.

Slattery J.C. (1967). Flow of viscoelastic fluids through porous media. A.I.Ch.E.

Journal, Vol. 13, p. 1066.

128

Song H. and Tong P. (2010). Scaling laws in turbulent Rayleigh-Bénard convection under different geometry, Europhysics Letters, Vol. 90. p. 44001.

Sparrow E.M., Goldstein R.J. and Jonsson V.K. (1964). Thermal instability in a

horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile. J. Fluid Mech., Vol. 18, pp. 513–528.

Spiegel E. (1960). The convective instability of a radiating fluid layer.

Astrophys. J., Vol. 132, pp. 716–728. Sprague E., Julien K., Serre E., Sanchez-Alvarez J.J. and Crespo del Arco E.

(2005). Pattern formation in Rayleigh-Benard convection in a rapidly rotating cylinder. Proceedings of the Fourth International Symposium on Turbulence and Shear Flow Phenomena.

Sri Krishna C.V. (2001). Effect of non-inertial acceleration on the onset of

convection in a second order fluid-saturated porous medium. Int. J. Engg. Sci., Vol. 39, pp. 599 – 609.

Srimani P.K. (1981). Finite amplitude cellular convection in a rotating and non-

rotating fluid saturated porous layer. Ph.D. Thesis, Bangalore University. Srinivasacharya D., Srinivasacharyulu N. and Odelu O. (2009). Flow and heat

transfer of couple stress fluid in a porous channel with expanding and contracting walls. Int. Communications in Heat and Mass Transfer, Vol. 36, pp. 180-185.

Srivastava L. M. (1986). Peristaltic transport of a couple-stress fluid, Rheologica

Acta, Vol. 25, pp. 638-641. Stengel K.C., Oliver D.S. and Booker J.R. (1982). Onset of convection in a

variable viscosity fluid. J. Fluid Mech., Vol. 120, pp. 411–431. Stokes V. K. (1966). Couple stresses in fluids. Phys. Fluids, Vol. 91,

pp. 1709 – 1715. Stokes V. K. (1984). Theories of fluids with microstructure. Springer, New York. Straughan B. (2002a). Sharp global nonlinear stability for temperature-dependent

viscosity convection. Proc. R. Soc. Lond. A, Vol. 458, pp. 1773–1782. Straughan B. (2002b). Global stability for convection induced by absorption of

radiation. Dyn. Atmos. Oceans, Vol. 35, pp. 351–361.

129

Straughan B. (2006). Global nonlinear stability in porous convection with a thermal non-equilibrium model. Proc. Roy. Soc. A, Vol. 462, pp. 409 – 418.

Straughan B. (2004). The energy method, stability and nonlinear convection.

Springer, New York. Straus J.M. (1974). Large amplitude convection in porous media. J. Fluid. Mech.,

Vol. 64, pp. 51 – 63. Suneetha S., Bhaskar Reddy N. and Ramachandra Prasad V. (2008). Thermal

radiation effects on MHD free convection flow past an impulsively started vertical plate with variable surface temperature and concentration. Journal of Naval Architecture and Marine Engineering, Vol. 2, pp. 57-70.

Sunil R. C., Sharma and Rajender Singh Chandel. (2004). Effect of Suspended

Particles on Couple-Stress Fluid Heated and Soluted from Below in Porous Medium. Jl. Porous Media, Vol. 7, pp. 9 - 18.

Sunil, R. C. Sharma and Mohinder Pal. (2002). On a couple-stress fluid heated

from below in a porous medium in the presence of a magnetic field and rotation. Jl. Porous Media, Vol. 5, pp. 149 – 158.

Sutton K.M. (1970). Onset of convection in a porous channel with net through

flow. Phys. Fluids, Vol. 13, p. 1931. Takashima, M. (1970). Nature of the neutral state in convective instability induced

by surface tension and buoyancy. J. Phys. Soc. Japan, Vol. 28, p. 810. Takashima, M. (1981a). Surface tension driven instability in a horizontal liquid

layer with a deformable free surface. I. Stationary convection. J. Phys. Soc. Japan, Vol. 50, pp. 2745–2750.

Takashima, M. (1981b). Surface tension driven instability in a horizontal liquid

layer with a deformable free surface: II. Overstability. J. Phys. Soc. Japan, 50, pp. 2751–2756.

Taniguchi H., Ohmori T., Iwata M., Arai N., Hiraga K. and Yamaguchi S.

(2002). Numerical Study of Radiation–Convection Heat Transfer. Heat Transfer—Asian Research, Vol. 31, pp. 391 – 407.

Taylor G. (1954). Diffusion and mass transport in tubes. Proc. Phys. Soc.,

Vol. 67 (B), p. 857.

130

Taylor G.I. (1971). A model for the boundary condition of a porous material. Part I. J. Fluid Mech., Vol. 49, p. 319.

Thess, A. and Nitschke, K. (1995). On Bénard-Marangoni instability in the

presence of a magnetic field. Phys. Fluids, Vol. 7, pp. 1176–1178. Thomson, J. (1882). On a changing tessellated structure in certain liquids. Proc.

Glasgow Phil. Soc., Vol. 13, pp. 469–475. Tong P. and Shen Y. (1992). Relative velocity fluctuations in turbulent Rayleigh-

Benard convection. Phys. Rev. Lett., Vol. 69, pp. 2066 – 2069. Torrance K.E. and Turcotte, D.L. (1971). Thermal convection with large

viscosity variations. J. Fluid Mech., Vol. 47, pp. 113–125. Tritton, D.J. (1979). Physical fluid dynamics. Van Nostrand Reinhold Company,

England. Turner, J.S. (1973). Buoyancy effects in fluids. Cambridge University Press,

Cambridge. Umavathi J. C., Prathap Kumar J. and Ali J. Chamkha. (2009). Convective

flow of two immiscible viscous and couple stress permeable fluids through a vertical channel, Turkish J. Eng. Env. Sci., Vol. 33, pp. 1 – 24.

Vafai K. (Editor) (2000). Handbook of porous media. Marcel Dekker, New York. Vafai K. (Editor) (2005). Handbook of porous media. Taylor & Francis, 2nd

Edition, New York. Vafai K. and Tien C.L. (1981). Boundary and inertia effects on flow and heat

transfer in porous media. Int. J. Heat Mass Transfer, Vol. 24, p. 195. Vafai K., Deasi C.P. and Chen S.C. (1993). An investigation of heat transfer

process in a chemically reacting packed bed. Numer. Heat Transfer, Vol. 24, pp. 127 – 142.

Vaidyanathan G., Ramanathan A. and Maruthamanikandan S. (2002).

The effect of magnetic field dependent viscosity on ferroconvection in sparsely distributed porous medium. Indian Jl. Pure Appl. Phys., Vol. 40, pp. 166 - 171.

131

Vanishree R. K. and Siddheshwar P.G. (2010). Effect of rotation on thermal convection in an anisotropic porous medium with temperature-dependent viscosity. Transp Porous Med., Vol. 81 pp. 73 – 87.

Veronis G. (1959). Cellular convection with finite amplitude in a rotating fluid.

J. Fluid Mech., Vol. 5, pp. 401–435. Veronis G. (1966). Motions at subcritical values of the Rayleigh number in a

rotating fluid. J. Fluid Mech., Vol. 24, pp. 545–554. Veronis G. (1968). Effect of a stabilizing gradient of solute on thermal convection.

J. Fluid Mech., Vol. 34, pp. 315–336. Vidal, A. & Acrivos, A. (1966). Nature of the neutral state in surface tension

driven convection. Phys. Fluids, Vol. 9, pp. 615–616. Walker K. and Homsy G.M. (1977). A note on convective instabilities in

Boussinesq fluids and porous media. ASME J. Heat Transfer, Vol. 99, p. 338. Wang S. and Tan W. (2009). The onset of Darcy-Brinkman thermosolutal

convection in a horizontal porous media. Phys. Lett. A., Vol. 373, pp. 776 – 780. Weber J.E. (1975). The boundary layer regime for convection in a vertical porous

layer. Int. J. Heat Mass Transfer, Vol. 18, p. 569. Westbrook O.R. (1969). Convective flow in porous media. Phys. Fluids, Vol. 12,

p. 1547. White D.B. (1988). The planforms and onset of convection with a temperature-

dependent viscosity. J. Fluid Mech., Vol. 191, pp. 247–286. Whittaker S. (1966). The equations of motion in porous media. Chem. Engg. Sci.,

Vol. 21, p. 291. Whittaker S. (1967). Diffusion and dispersion in porous media. A.I. Ch. E.

Journal, Vol. 13, p. 420. Wilson S.K. (1993a). The effect of a uniform magnetic field on the onset of steady

Bénard-Marangoni convection in a layer of conducting fluid. J. Engg. Math., Vol. 27, pp. 161–188.

132

Wilson S.K. (1993b). The effect of a uniform magnetic field on the onset of Marangoni convection in a layer of conducting fluid. Quart. J. Mech. Appl. Math., Vol. 46, pp. 211–248.

Wilson S.K. (1994). The effect of a uniform magnetic field on the onset of steady

Marangoni convection in a layer of conducting fluid with a prescribed heat flux at its lower boundary. Phys. Fluids, Vol. 6, pp. 3591–3600.

Wilson S.K. (1997). The effect of uniform internal heat generation on the onset of

steady Marangoni convection in a horizontal layer of fluid. Acta Mech., Vol. 124, pp. 63–78.

Wooding R.A. (1957). Steady state free thermal convection of liquids in a saturated

permeable medium. J. Fluid Mech., Vol. 2, p. 273. Wooding R.A. (1958). An experiment on free thermal convection of water in

saturated permeable material. J. Fluid Mech., Vol. 3, p. 582. Wooding R.A. (1959). The stability of a viscous liquid in a vertical tube containing

porous material. Proc. Roy. Soc., Vol. 252(A), p. 120. Wooding R.A. (1960). Instability of a viscous liquid of variable density in a

vertical Hele-Shaw cell. J. Fluid Mech., Vol. 7, p. 501. Wooding R.A. (1960). Rayleigh instability of a thermal boundary layer in flow

through porous medium. J. Fluid Mech., Vol. 9, p. 183. Wooding R.A. (1963). Convection in a saturated porous medium at large Reynolds

number or Peclet number. J. Fluid Mech., Vol. 15, p. 527. Xi H.W. and Gunton J.D. (1993). Spiral-pattern formation in Rayleigh-Benard

convection. Phys. Rev. E., Vol. 47, pp. 2987 – 2990. XinGang L. and MaoHua H. (2007). Coherent thermal radiation in thin films and

its application in the emissivity design of multilayer films. Chinese Science Bulletin, vol. 52, pp. 1426-1431.

Xu B., Ai X. and Li B. Q. (2007). Stabilities of combined radiation and Rayleigh–

Bénard–Marangoni convection in an open vertical cylinder. International Journal of Heat and Mass Transfer, Vol. 50, pp. 3035-3046.

133

Yan W. M. and Li H. Y. (2001). Radiation effects on mixed convection heat transfer in a vertical square duct. Int. J. Heat Mass Trans., Vol. 44, pp. 1401–1410.

Yanagisawa T. and Yamagishi Y. (2005). Rayleigh-Benard convection in

spherical shell with infinite Prandtl number at high Rayleigh number. Journal of Earth Simulator, Vol. 4, pp. 11 – 17.

Yang W. M. (1990). Thermal instability of a fluid layer induced by radiation.

Numer. Heat Trans., Vol. 17, pp 365–376. Yan-yan M. A. (2008). Performance of dynamically loaded journal bearings

lubricated with couple stress fluids considering the elasticity of the liner. Journal of Zhejiang University - Science A, Vol. 9, pp. 916 – 921.

You X.Y. (2001). Estimating the onset of natural convection in a horizontal layer of

a fluid with a temperature-dependent viscosity. Chem. Engg. J., Vol. 84, pp. 63–67.

Zakaria M. (2002). Hydromagnetic fluctuating flow of a couple stress fluid

through a porous medium. Korean Jl. Computational & Applied Mathematics, Vol. 10, pp. 175 – 191.

Zeng M., Wang Q., Ozoe H., Wang G. and Huang Z. (2009). Natural convection

of diamagnetic fluid in an enclosure filled with porous medium under magnetic field. Prog. Comp. Fluid Dyn., Vol. 9, pp. 77 – 85.

Zhao H. and Bau H.H. (2006). Limitations of linear control of thermal convection

in a porous medium. Phys. Fluids, Vol. 18, p. 74109. Zhou Q., Sun C. and Xia K.Q. (2007). Morphological evolution of thermal plumes

in turbulent Rayleigh-Benard convection. Phys. Rev. Lett., Vol. 98, p. 74501. Ziener J. and Oertel Jr. H. (1982). Convective transport and instability

phenomena. G. Braun Publishers, West Germany.