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Chapter 1 Preliminaries 1.1 Introduction to Fluid Mechanics

The branch of mechanics that deals with bodies at rest as well as in motion. The

subcategory of fluid mechanics is defined as the science that deals with the behavior of

fluids at rest or in motion. Fluid mechanics itself is also divided into several categories.

The study of the motion of fluids that can be approximated as incompressible such as

liquids, especially water, and gases at low speed is usually referred to as

hydrodynamics. A subcategory of hydrodynamics is hydraulics, which deals with liquid

flows in pipes and open channels. Gas dynamics deals with the flow of fluids that

undergo significant density changes, such as the flow of gases through nozzles at high

speeds. The category aerodynamics deals with the flow of gases over bodies such as

aircraft, rockets, and automobiles at high or low speeds. Some other specialized

categories such as meteorology, oceanography and hydrology deal with naturally

occurring flows.

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Fluid mechanics is widely used both in everyday activities and in the design of modern

engineering systems from vacuum cleaners to supersonic aircraft. For example, fluid

mechanics plays a vital role in the human body. The heart is constantly pumping blood

to all parts of the human body through the arteries and veins, and the lungs are the sites

of airflow in alternating directions. All artificial hearts, breathing machines, and dialysis

systems are designed using fluid dynamics. Fluid mechanics is also used in the design

of the heating and air-conditioning system, the hydraulic brakes, the power steering, the

automatic transmission, the lubrication systems, the cooling system of the engine block

including the radiator and the water pump, and even the tires. The sleek streamlined

shape of recent model cars is the result of efforts to minimize drag by using extensive

analysis of flow over surfaces.

1.2 Types of Fluids

Fluid: Fluid is a substance that continually deforms (flows) under an applied

shear stress. Fluids are a subset of the phases of matter and include liquids, gases

and plasmas. Although the term “fluid” includes both the liquid and gas phases, in

common usage, “fluid” is often used as a synonym for “liquid”. A portion of matter

is called a liquid, if its expansion, when subjected to some external force or an

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increase in temperature, is not much that is, the liquid have definite volume which

changes slightly when subjected to external force or temperature difference. Thus

a liquid may not occupy whole of the space of the container. In gases, the volume

of a gas changes significantly when subjected to external force or temperature

difference or a change in the pressure, and so, it does not have a definite volume.

Thus a portion of matter is called gas if it has no definite volume, and occupies the

whole of the space of the container. As there is no definite volume for gases and

plasmas both are usually considered as gases. But plasma is characterized by an

ionized state of the matter, it may be present even in liquids. Thus there is a

distinction between plasmas and gases. Here again it is to be pointed out that the

division between, gases and plasmas is not a sharp one. The fluids can be

classified as;

• Ideal fluids

• Real fluids

1.2.1 Ideal fluid or Inviscid fluid

An Ideal fluid is one, which has no property other than density. No resistance is

encountered when such a fluid flows or Ideal fluids or Inviscid fluids are those fluids in

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which two contacting layers experience no tangential force (shearing stress) but act on

each other with normal force (pressure) when the fluids are in motion. This is equivalent

to stating that inviscid fluid offers no internal resistance to change in shape. The

pressure at every point of an ideal fluid is equal in all directions, whether the fluid is at

rest or in motion. Inviscid fluids are also known as effect fluids or frictionless fluids. In

true sense, no such fluid exists in nature. The assumption of ideal fluids helps in

simplifying the mathematical analysis. However fluids which have low viscosities such

as water and air can be treated as ideal fluids under certain conditions.

1.2.2 Viscous fluid or Real fluid

“Viscous fluid or real fluid are those, which have viscosity, surface tension and

compressibility in addition to the density” or viscous fluid or real fluid are those when

they are in motion the two contacting layers of those fluids experience tangential as well

as normal stresses. The property of exerting tangential of shearing stress and normal

stress in a real fluid when the fluid is in motion is known as viscosity of the fluid. In

viscous fluid internal friction plays an important role during the motion of the fluid. One

of the important characteristics of viscous fluid is that it offers internal resistance to

motion of the fluid, viscosity, being the characteristic of the real fluids, exhibits a certain

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resistance to alter the form also. Viscous or real fluids are classified into Newtonian

fluids and non Newtonian fluids.

1.2.3 Newtonian fluid

To understand the concept of Newtonian fluid, let us consider a thin layer of fluid

between two parallel plates at distance 𝑑𝑦.

Here one plate is fixed and a shearing force F is applied to the other. When conditions

are steady the force F is applied to the other and balanced by an internal force in the

fluid due to its viscosity. Newton, while discussing the properties of fluid, remarked that

in a simple rectilinear motion of a fluid two neighboring fluid layers, one moving over the

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other with some relative velocity, will experience a tangential force proportional to the

relative velocity between the two layers and inversely proportional to the distance

between the layers, that is if the two neighboring fluid layers are moving with velocities

𝑢 and 𝑢 + 𝛿𝑢 are at a distance 𝛿𝑦, then the shearing stress, is given by

𝜏 ∝ 𝜕𝑢

𝜕𝑦 𝑜𝑟 𝜏 = 𝜇

𝜕𝑢

𝜕𝑦 (1.2.1)

This is called Newtonian hypothesis and a fluid satisfying this hypothesis is called a

Newtonian fluid. It is clear from the Newton’s law that

• If 𝜏 = 0 then 𝜇 = 0, equation (1.2.1) will represent an ideal fluid.

• If 𝜕𝑢

𝜕𝑦= 0 then 𝜇 = 1, equation (1.2.1) will represent an elastic bodies.

• A fluid for which the constant of proportionality 𝜇 does not change with rate of

deformation (shear strain 𝜕𝑢

𝜕𝑦 ) is said to be an Newtonian fluid and graph 𝜏 verses

𝜕𝑢

𝜕𝑦 is a

straight line.

Where 𝜇 is known as Newtonian viscosity. It will be seen that 𝜇 is the tangential force

per unit area exerted on layers of fluid a unit distance apart and having a unit velocity

difference between them.

The diagram relating shear stress and rate of shear for Newtonian fluids represents flow

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curve of the type straight line. The example of Newtonian fluid are water, air, mercury,

benzene, ethyl alcohol, glycerin and oil etc.,

1.2.4 Non-Newtonian fluid

Non-Newtonian fluids are those fluids which do not obey Newtonian law. It can also be

stated as the non-Newtonian fluids are those for which flow curve is not linear, i.e., the

‘viscosity’ of a non-Newtonian fluid is not constant at a given temperature and pressure

but depends on other factors such as the rate of shear in the fluid. The typical examples

of these classes of fluids are paints, coal-tar, polymer solutions, condensed milk, paste,

lubricants, honey, plastics, molasses, molten rubber, printer ink, collides,

macro/molecular materials and so on.

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1.2.5 Compressible and Incompressible fluid Fluids undergo density changes when temperature and pressure variations occur in

them then they are consider compressible. For several flows situations, however density

changes are negligible and the fluid may be treated as incompressible. Therefore, flow

of liquids are treated as incompressible for small pressure and temperature variations.

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1.2.5 Ferro fluids

A magnetic colloid, also known as a ferrofluid , is a colloidal suspension of single-

domain magnetic particles, with typical dimensions of about 10 nm, dispersed in a liquid

carrier. The liquid carrier can be polar or nonpolar. Since the nineteen sixties, when

these materials were initially synthesized, their technological applications did not stop to

increase.

Ferrofluids are different from the usual magnetorheological fluids (MRF) used for

dampers, brakes and clutches, formed by micron sized particles dispersed in oil. In

MRF the application of a magnetic field causes an enormous increase of the viscosity,

so that, for strong enough fields, they may behave like a solid. On the other hand, a

Ferro fluid keeps its fluidity even if subjected to strong magnetic fields (~ 10 𝑘𝐺).

Ferrofluids are optically isotropic but, in the presence of an external magnetic field,

exhibit induced birefringence . Wetting of particular substrates can also induce

birefringence in thin Ferro fluid layers .In order to avoid agglomeration, the magnetic

particles have to be coated with a shell of an appropriate material. According to the

coating, the Ferro Fluid’s are classified into two main groups:

Surfacted ferro fluids

Ionic ferro fluids

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1.3 Some Important Types of Flow

1.3.1 Steady and Unsteady Flow

A flow in which properties and conditions of fluid motions do not change with change of

time such that the flow pattern is not affected with time iscalled steady, i.e 𝜕𝜑

𝜕𝑡= 0,

where 𝜑 may be velocity, temperature, pressure, density etc.

On the other hand, if the flow pattern depends upon time, it is called unsteady flow.

1.3.2 Laminar and Turbulent Flow

A flow in which each fluid particle traces out a definite curve and curves traced out by

any two different particle do not intersect, is said to be laminar. On the other hand, a

flow, in which each fluid particle does not trace out a definite curve and the curves

traced out by fluid particle intersect, is said to be turbulent flow. The most of the flows,

which occur in practical applications are turbulent, and this term denotes a motion in

which an irregular fluctuation (mixing, or eddying motion) is superimposed on the main

stream

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1.3.3 Rotational and Irrotational Flow

The flow in which the fluid particles rotate about their own axis is called rotational and

the flow in which the fluid particle does not rotate about their own axis is called

irrotational. Mathematically,

If ∇ × �⃗� = 0 ⇒ irrotational flow.

If ∇ × �⃗� ≠ 0 ⇒ rotational flow

1.3.4 Uniform and Non-uniform Flow

A flow in which the velocity of fluid particles are equal at each section of the channel is

called uniform flow and a flow in which the velocities of fluid particles are different at

each section of the channel is called non-uniform flow.

1.3.5 One and Two-Dimensional Flow

Considerable simplification in analysis may often be achieved, however, by selecting

the coordinate directions so that appreciable variation of the parameters occurs only two

directions, or even in only one.

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• One-dimensional flow is that in which all the flow parameters may be expressed as

functions of time and one space coordinate only.

Example: Flow in a pipe.

• In two-dimensional flow, the flow parameters are functions of time and two rectangular

space coordinates (say x and y) only, there is no variation in z-direction.

Examples: Flow between parallel planes, water flow over a weir of uniform cross

section and infinite width.

1.3.6 Magnetohydrodynamic Flow (MHD):

The flow of electrically-conducting fluid under influence of applied magnetic field is

called MHD flow. In other words, Magnetohydrodynamics is the study of the motion of

an electrically conducting fluid in the presence of external electromagnetic field. It is the

combination of two branches viz., hydrodynamics and electromagnetism. The dictionary

meaning of hydro is water but hydrodynamics includes study dynamical behaviour of

electrically conducting medium, which may be a liquid or an ionized gas in presence of

magnetic field. Both plasma and conducting fluids are related in common theory by

assuming plasma as a continuous fluid for which the kinetic theory of gases still holds

true. In MHD induced electric current produces mechanical force, which in turn modified

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the motion in the fluid. Hence, study of electrically conducting fluid flow in the presence

of transverse magnetic field assures significance. In 1937, Hartmann studied the motion

of electrically conducting fluids in presence of magnetic field. Chapman and Ferraro

developed theory of magnetic storms during 1930-1935.

Electromagnetic forces will be generated which may be of the same order of magnitude

as the hydrodynamical and inertial forces in the case when the conductor is either a

liquid or a gas. Thus, the equation of motion will have to take these electromagnetic

forces into account as well as the other forces. The science that treats these

phenomena is magnetohydrodynamics (MHD). Other variants of nomenclature are

hydromagnetics, magneto-fluid dynamics, magneto-gas dynamics etc. As we know that

MHD is relatively new but important branch of fluid dynamics. It is concerned with the

interaction of electrically conducting fluid and electro magnetic fields, such interaction

occurs both in nature and in new man-made device. The study of uniform magnetic field

on the motion of a electrically conducting fluid over a stretching sheet find its application

in various engineering disciplines such as polymer technology, where one deals with

stretching plastic sheet and metallurgy, where hydro magnetic techniques have

recently been used. The important application of MHD flow in metallurgy is the

purification process of molten metal from non-metallic inclusion using magnetic field .

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1.4 Boundary-layer theory

The concept of boundary layer, which was introduced by a German scientist Ludwig

Prandtl in the year 1904, provides a link between the ideal flow and the real flow. It

is the great achievement of Ludwig Prandtl, at the beginning of this century, set forth

the way in which these two diverging directions of fluid mechanics could be unified. In

the case of real fluids, however small their viscosity may be the fluid particles adhere

to the boundary and hence the condition of no-slip prevails. The fluid velocity at the

stationary boundary vanishes, whereas the fluid adhering to the boundary will have

the same velocity as boundary itself, if the boundary is moving. Further away from the

boundary there exists a velocity gradient 𝜕𝑢

𝜕𝑦 normal to the boundary and the fluid

exerts a shear or tangential in the direction of motion. The force caused by this shear

stress in the direction of motion is known as surface drag. The boundary in turn exists

a shear resistance to the flow i.e., a force on the fluid which is equal to magnitude and

opposite in direction to the surface drag. Due to no-slip condition, the fluid is always

retardedat or near the boundary. The retardation due to the presence of viscosity is

negligible and the velocity remains constant far away from the boundary. The

transition from zero velocity at the boundary to the full magnitude at far away from the

boundary takes place in a very thin layer. The fluid layer which has its velocity

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affected by boundary shear is called the boundary layer. The boundary layer was

developed mainly for the laminar flow of an incompressible fluid and it was extended

to the practically important turbulent compressible boundary layer flows. The

boundary layer theory can be used to describe the flow through the blade cascades in

compressors and turbines as well as through diffusers and nozzles.

1.4.1 Viscous boundary layer

The influence of boundary, due to no slip condition is confined to a very thin region in

the immediate neighborhood of the solid surface, known as viscous or momentum

boundary layer. In this thin layer there is rapid change in velocity of the fluid, from

velocity of the surface to its value that corresponds to external frictionless flow. This is

the viscosity of the fluid that gives rise to the boundary layer and for an inviscid fluid

there exists no boundary layer. For flow over a surface of finite or semi infinite length

the thickness of boundary layer increase in the down stream region. The thickness of

the boundary layer decreases with decrease in viscosity of the medium, but even for

small viscosity, the shear stress 𝜏𝑤 = 𝜇 𝜕𝑢

𝜕𝑦 is important due to large velocity gradient.

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The injection or suction across the porous surface also has a great effect on the size of

the viscous boundary layer.

1.4.2 Thermal boundary layer

A thin region in which the temperature of the fluid particles changes from its free stream

value to body surface value is called “Thermal Boundary Layer”. The thermal boundary

layer strongly depends upon the thermal conductivity of the medium i.e., higher the

conductivity of the medium, thicker would be the thermal boundary layer. Like viscous

boundary layer, injection or suction across the porous surface also has a great effect on

the size of the thermal boundary layer.

1.5 Boundary conditions

1.5.1 Slip condition:

If the boundary is rough, impermeable and moving then normal velocity is zero but the

tangential velocity is not zero (v = 0, u is nonzero) then slip condition exists.

1.5.2 No-Slip condition: When a viscous fluid flows over a solid surface, the fluid elements adjacent to the

surface attain the velocity of the surface; in other words, the relative velocity between

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the solid surface and the adjacent fluid particles is zero. This phenomenon is known as

the ‘no-slip’ condition.

For example, we know that water in a river cannot flow through large rocks, and must

go around them. That is, the water velocity normal to the rock surface must be zero, and

water approaching the surface normally comes to a complete stop at the surface.

What is not as obvious is that water approaching the rock at any angle also comes to a

complete stop at the rock surface, and thus the tangential velocity of water at the

surface is also zero.

1.6 Stretching Sheet Stretching sheet is the sheet being stretched with the stretching velocity Uw(x) along the

x-axis, keeping the origin fixed in the fluid of ambient temperature T. Thus the sheet can

be stretched linearly, non-linearly or exponentially. For illustrations, consider the flow of

a fluid past a flat sheet coinciding with the plane y = 0, the flow being confined to y > 0.

Two equal and opposite forces are applied along the x-axis, so that the wall is stretched

keeping the origin fixed. Assume that the stretching sheet has a velocity 𝑈𝑤(𝑥) as

shown

in figure 1.2.

• If the sheet stretching linearly, then the stretching velocity is 𝑈𝑤(𝑥) = 𝑈0𝑥.

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• If the sheet stretching non-linearly or exponentially, then the stretching velocity is

𝑈𝑤(𝑥) = 𝑈𝑜 𝑒𝑥/𝐿 .

Figure

Figure - 1.2: Schematic diagram of the Stretching sheet

1.7 Heat Transfer

1.7.1 Temperature

The word temperature indicates a physical property on which depends the sense

impression of hotness or coldness. Temperature has been defined as the “the state of a

substance or body with regard to sensible warmth referred to some standard of

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comparison”. Senseimpressions can give only a crude estimate and temperature is

usually measured by means of a Thermometer.

1.7.2 Heat

The conception of heat which passes from the hotter to the colder body and is thought

of as bringing about the change of temperatures. According to Max Planck, the

conception of heat, like all other physical concepts originates in the sense-perception,

but it acquires its physical significance of the events which excite the sensation. So heat

regarded physically, has no more to do with the sense of hotness than color in the

physical sense and has to do with the perception of color.

The terms Heat and Temperature in older philosophy drew little or no distinction

between them and we still use words like blood-heat and summer heat, which introduce

the term heat in connection with the idea of temperature. Joseph Black was the first to

perceive clearly the necessity of removing this confusion and he pointed out that we

must distinguish between quantity and intensity of heat, quantity corresponding to the

amount of heat and intensity to temperature.

As we know that the knowledge of heat transfer is very important for construction and

designing of power plan, which will perform in the prescribed fashion, is the objective of

the engineer. This clearly requires detailed knowledge of the principles governing heat

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transfer in the various components, which may be involved i.e., boilers, turbines,

condenser, pumps and compressors. Some of the other industrial fields of heat transfer

play an important role like heating and air conditioning, chemical reactions and process.

A detailed heat transfer analysis is essential, since the dimensions of boilers, heaters,

refrigerators and heat exchangers not only depend on the amount of heat to be

transmitted but also on the rate at which heat is to be transferred under given

conditions.

1.7.3 Types of heat transfer

Heat transfer is a science that predicts the transfer of heat energy from one body to

another by virtue of temperature difference. Heat transfer occurs as a result of three

mechanisms.

• Conduction

• Convection

• Radiation

Conduction:

In conduction heat flows due to molecular interaction, molecules not being displace or

due to the motion of free electrons. Heat conduction may be stated as the transfer of

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internal energy between the molecules. Heat flows from a region of higher temperature

to a region of lower temperature by kinetic motion or direct impact of molecules whether

the body is at rest or in motion.

Convection:

Heat transfer due to convection involves the energy exchange between a solid surface

and an adjacent fluid. Convection is a mechanism in which heat flows or transferred

between a fluid and a solid surface as a consequence of motion of fluid particles relative

to the solid surface when there exists a temperature gradient. Convection heat transfer

may be classified as “Forced Convection” and “Free or Natural Convection”.

Forced convection:

If heat transfer between a fluid and a solid surface occurs by the fluid motion induced by

by external agencies or forces then the mode of heat transfer is termed as “ forced

convection”. Heat transfer in all types of heat exchangers, nuclear reactor and air

conditioning apparatus is by forced convection.

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Natural or Free convection:

If heat transfer between a fluid and a solid surface occurs by the fluid motion due to the

density differences caused by the temperature differences between the surface and the

fluid, then the mode of heat transfer is termed as “Free Convection or Natural

Convection”. Heat flows from a heated metal plate to the atmosphere, heat flows from

hot water to the container are certain examples of free convection.

Radiation:

The phenomenon or the mode of heat transfer in the form of electromagnetic waves

without the presence of any intervening medium is called Radiation. The transfer of heat

energy from the sun to the earth is an example of Radiation.

Heat Flux:

The heat transfer per unit area is called heat flux. If q is the amount of heat transfer and

A is area normal to the direction of the heat flow, then the heat flux is

𝑄 = 𝑞/𝐴

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Heat Dissipation:

The heat generated by internal friction within the volume element of the fluid per unit

time is called heat dissipation.

Thermal Conductivity:

The concept of thermal conductivity is that “The quantity of heat passing in unit time

through each unit of area when there is a difference of temperature of one degree

between the inside and outside face of a wall of unit thickness”.

To be more specific about discussion of thermal conductivity we consider two parallel

layers of a fluid, at distance d apart are kept at different temperatures 𝑇1 𝑎𝑛𝑑 𝑇2 (One of

the layers may be a solid surface). Fourier noticed that a flow of heat is set up through

the layer such that the quantity of heat q transferred through unit area in unit time is

directly proportional to the difference of the temperature between the layers and

inversely proportional to the distance d. Thus he found 𝑞 = 𝑘𝑇1−𝑇2

𝑑

, where k is the constant of proportionality and is known as the coefficient of thermal

conductivity.

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If the distance d between the two layers of fluid is infinitesimal the above law can be

written in the differential form as = 𝑘𝑑𝑇

𝑑𝑦 , where the negative sign indicated that the heat

flows in the direction of decreasing temperature.

The dimensions of the coefficient of thermal conductivity can be determined as follows

𝑘 =Heat flux

temperature gradient

Thermal Diffusivity:

The effect of conductivity on the temperature field is determined by the ratio of k to the

product of density 𝜌 and specific heat 𝐶𝑝 rather than 𝑘 alone. This ration is known

as the thermal conductivity and it is usually denoted by 𝛼 =𝑘

𝜌𝐶𝑝

1.8. Porous Media

A porous medium is a material containing pores (voids). Examples like sponges, clothes

wicks, paper sand gravel, filters, concrete brickes, plaster walls, many naturally

occurring rocks, packed beds used for distillation, absorption etc. The skeletal portion of

the material is often called the “matrix” or “frame”. The pores are typically filled with a

fluid (liquid or gas). The skeletal material is usually a solid, but structures like foams are

often also usefully analyzed using concept of porous media. A porous medium is most

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often characterised by its porosity. Other properties of the medium (e.g., permeability,

tensile strength, electrical conductivity) can sometimes be derived from the respective

properties of its constituents (solid matrix and fluid) and the media porosity and pores

structure, but such a derivation is usually complex. Most of the studies of flow in porous

media assume the Darcy’s law is valid. However this law is known to be valid only for

relatively slow flows through porous media. In general we must consider the effect of

fluid inertia as well as of viscous diffusion at boundaries which may become significant

for material with high porosities such as fibrous and foams.

The concept of porous media is of great interest in many areas of applied science and

engineering due to their important applications in the field of agricultural engineering to

study the under-ground water resource, seepage of water in river beds, in petroleum

technology to study the movement of natural gas, oil and water through oil reservoirs, in

chemical engineering for filtration and purification processes. The petroleum industry

has been showing a lot of interest in these problems in connection with the crude oil

production from the underground reservoirs. These problems are also of much interest

in geophysics and in the study of the interaction of the geomagnetic field with the fluid in

the geothermal region. The textile technologist is interested in fluid flow through fibers,

whereas biologists are interested in water movement through plant roots of the cells of

living systems. On the other hand, it has also encountered in the field of mechanics

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(acoustics, geomechanics, soil mechanics, rock mechanics), engineering (petroleum

engineering, bio-remediation, construction engineering), geosciences (hydrogeology,

petroleum geology, geophysics), biology and biophysics, material science, etc. Fluid

flow through porous media is a subject of most common interest and has emerged a

separate field of study. The study of more general behaviour of porous media involving

deformation of the solid frame is called poromechanics.

1.8.1 Darcy’s Law

Based on the experimental research of Darcy in flow through porous medium, Navier-

Stokes equation are replaced by linear partial differential equations. Suitable

approximations are to be made to get the solution, as the governing equations of porous

media are partial differential equations. In 1856, Henri Darcy formulated the law which

governs the flow through a porous medium. Darcy’s law is given by,

𝑞 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡(∇𝑝 + 𝜌𝑔). (1.8.1)

where p is the pressure, 𝜌 is the density and 𝑔 is the acceleration due to gravity.

Equation (1.8.1) express that Darcy’s velocity 𝑞 is proportional to the sum of pressure

gradient and the gravitational force. Moreover, 𝑞 is inversely proportional to viscosity.

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This Darcy’s law is macroscopic equation of motion for Newtonian fluid in porous

median at small Reynolds numbers. Many researchers verified this law experimentally.

The constant in the equation is replaced by the permeability 𝑘 by Musket.

Now equation (1.8.1) becomes,

𝑞 = −𝑘

𝜇∇𝑝 (1.8.2)

This law is valid for the flow through isotropic porous media.

By using Darcy’s law various flows through porous media have been invistigated by

Musket, De Wiest, Bear and many other researchers. The most general form of Darcy’s

law is given by,

𝜌

𝐸=

𝑑𝑢𝑖

𝑑𝑡=

𝜕𝑝

𝜕𝑥+ 𝜌𝑥𝑖 −

𝜇

𝑘𝑢𝑖 (1.8.3)

𝑥𝑖 = The ith component of body force per unit mass,

𝑢𝑖 = The ith component of velocity,

𝐸 = The Porosity,

𝑑

𝑑𝑡= Substantial derivative.

The dimension of permeability is 𝐿2. The unit of permeability is Darcy which is used in

petroleum industry. The value of one darcy is 0.987 × 10−8𝑐𝑚2 . The hydraulic

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conductivity of the porous medium is measured in meinzers. If the porous medium has a

permeability of one Darcy, then it has the hydraulic conductivity 18.2 meinzers.

Darcy’s law is valid when the flow takes place at low speeds. But for high speed flows,

Darcy’s law is not valid. Also Darcy’s law fails to describe the flows with high speeds or

the flow near surfaces which are either permeable or rigid. In such cases, Brinkman

equation will be useful.

1.8.2 Brinkman Model

The following equation is proposed by Brinkman for the flow through porous media

where �⃗⃗� is the velocity vector

This equation is valid when the permeability k is very high. In general, the particles of

the porous media are loosely packed so that k is small. Hence there exist a two

boundary layer very near to the surface.

In 1966, Tam supplemented a theoretical proof for this equation. Katto and Masuoka

experimentally found that Brinkman equation is valid up to the magnitude of 𝑘

ℎ2 of order

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10 or so. If the porous medium is made up of spherical particles then 𝑘

ℎ2 corresponds to

considerably high values of 𝑑

ℎ where d is the diameter of the fillings and h is the

recital thickness of the porous media. Yamamoto and Yoshida made improvements on

Darcy’s law by adding corrective terms. Saffman gave the equations of motion for the

flow through porous medium by incorporating viscous stresses.

1.8.3 Non-Darcy Law

In many practical problems, the flow through porous media is curvilinear and the

curvature of the path yields the inertia effect, so that the streamlines become more

distorted and the drag increase more rapidly. Lapwood was the first person who

suggested the inclusion of convective inertial term (𝑞. ∇)𝑞 in the momentum equation.

Subsequently many research articles have appeared on the non-Darcy model. Now the

equation can be written as,

(1.8.5)

However, equation (1.8.5) does not take care of possible unsteady nature of velocity.

The flow pattern in a certain region may be unsteady and one has to consider the local

acceleration term 1

𝛿2

𝜕𝑞

𝜕𝑡 also. Adding this term equation (1.8.5) it becomes,

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(1.8.6)

This equation is known as Darcy-Lapwood-Brinkman equation. For an isotropic porous

medium equation (1.8.5) takes the form.

(1.8.7)

1.9 Dimensionless Parameters

Every physical problem involved some physical quantities, which can be measured in

different units. But the physical problem itself should not depend on the unit used for

measuring these quantities. In dimensional analysis of any problem we write down the

dimensions of each physical quantity in terms of fundamental units. Then by dividing

and rearranging the different units, we get some non-dimensional numbers.

Dimensional analysis of any problem provides information on qualitative behaviors of

the physical significance of a particular phenomenon associated with the problem.

There are usually two general methods for obtaining dimensionless parameters.

1. The inspection analysis

2. The dimensionless analysis

31

In this thesis the latter method has been used. The basic equations are made

dimensionless using certain dependent and independent characteristics values. In this

process certain dimensionless numbers appear as the co-efficient of various terms in

these equations.The some of the dimensionless parameters used in this thesis are

explained below.

1.9.1 Ekman Number

It is the ratio of viscous forces in a fluid to the fictitious forces arising from planetary

rotation.

1.9.2 Reynolds Number

The Reynolds number is the ratio of inertial forces to viscous forces and consequently it

quantifies the relative importance of these two types of forces for given flow conditions.

Thus, it is used to identify different flow regimes, such as laminar or turbulent flow. It is

one of the most important dimensionless numbers in fluid dynamics and is used, usually

along with other dimensionless numbers, to provide a criterion for determining dynamic

similitude. It is named after Osborne Reynolds (1842-1912), who first introduced this

number while discussing boundary layer theory in 1883. Typically it is given as

32

follows:

𝑅𝑒 =𝜌𝑈2/ℎ

𝜇𝑈/ℎ2=

𝜌𝑈ℎ

𝜇=

𝑈ℎ

𝑣 ,

where

U - some characteristic velocity,

h - some characteristic length,

𝑣 =𝜇

𝜌- kinematic fluid viscosity,

𝜌- fluid density.

1.9.3 Hartmann Number

Hartmann number is the ratio of electromagnetic force to the viscous force. It was first

introduced by Hartmann and is defined as:

𝐻𝑎 = 𝐵𝐿√𝜎

𝜇

where

B - the magnetic field,

L - the characteristic length scale,

𝜎- the electrical conductivity,

𝜇- the viscosity

33

1.8.4 Hall parameter

It is defined the product of cyclotron frequency of electrons and electron collision time

and is given by

𝑚 = 𝑤𝑒𝑇𝑒

where 𝑤𝑒 is cyclotron frequency of electrons and _e is electron collision time.

1.9.5 Prandtl Number

It is an important dimension parameter dealing with the properties of a fluid. It is defined

as the ratio of viscous force to thermal force of a fluid. Prandtl number physically means

or signifies the relative speed with which the momentum and heat energy are

transmitted through a fluid. It thus associates the velocity and temperature fields of a

fluid. For gases Prandtl number is of unit order and varies over a wide range in case of

liquids.

𝑃𝑟 =𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒

𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝑓𝑜𝑟𝑐𝑒=

𝜇𝐶𝑝

𝑘

where

μ - Coefficifent of viscosity,

𝐶𝑝- Specific heat at constant pressure,

34

k - Coefficient of thermal conductivity

1.9.6 Eckert Number

It is equal to the square of the fluid velocity far from the body divided by the product of

the specific heat of the fluid at constant temperature and the difference between

temperatures of the fluid and the body.

𝐸𝐶 =𝑉02

𝐶𝑝(𝑇𝑤−𝑇∞),

where

Tw - Temperature near the plate,

𝑇∞ - Temperature far away from the plate,

Cp - Specific heat at constant pressure,

V0 - Characteristic value of velocity.

1.9.7 Number Density

Number density is an intensive quantity used to describe the degree of concentration

of countable objects in the three-dimensional physical space, or Number density is the

number of specified objects per volume i.e., n = N / V.

35

1.9.8 Grashof Number

It plays a significant role in free convection heat transfer. The ratio of the product of the

inertial force and the buoyant force to the square of the viscous force in the convection

flow system is known as Grashof number. Grashof number in free convection is

analogues to Reynolds number in forced convection.

𝐺𝑟 = 𝑔𝛽 (𝑇𝑊−𝑇∞)𝑙

2

𝑣2

where

g - acceleration due to gravity,

𝛽- volumetric coefficient of thermal expansion,

l - characteristic length,

Tw - temperature of the wall,

𝑇∞ - constant temperature far away from the sheet.

1.9.9 Non-uniform heat source/sink parameter

It is defined as

𝑞′′′ = (𝑘𝑈𝑤(𝑥)

𝑥𝑣) [𝐴∗(𝑇𝑤 − 𝑇∞)𝑓

′(𝜂) + 𝐵∗(𝑇 − 𝑇∞)]

36

where 𝐴∗ and 𝐵∗ are the parameters of the space and temperature dependent internal

heat generation/absorption. It is to be noted that 𝐴∗ and 𝐵∗ are positive to internal heat

source and negative to internal heat sink, 𝑣 is the kinematic viscosity, Tw and 𝑇∞ denote

the temperature at the wall and at large distance from the wall respectively.

1.9.10 Radiation parameter

It is defined as

𝑁𝑟 =16 𝜎∗𝑇∞

3

3𝑘𝑘∗

where

𝜎∗ - Stefan-Boltzman constant

𝑘∗- mean absorption co-efficient.

1.9.11Melting parameter

𝑀 =𝐶𝑝(𝑇∞−𝑇𝑚)

𝛾+𝐶𝑠(𝑇𝑚−𝑇0),

where M is the dimensionless melting parameter, where 𝐶𝑝 is the heat capacity of the

37

fluid at constant pressure. The melting parameter is a combination of the Stefan

numbers 𝐶𝑓(𝑇∞−𝑇0)

𝛾 and

𝐶𝑠(𝑇∞−𝑇0)

𝛾 for the liquid and solid phases, respectively. We Take

Tm is the temperature of the melting surface, while the temperature in the free-stream

condition is 𝑇∞, where 𝑇𝑚 > 𝑇∞.

1.9.12 Curvature parameter

𝛾 = √𝑙𝑣

𝑏𝑎2

is the curvature parameter, 𝛾 = 0, corresponds to flat plate.

1.9.13 Shear stress/Skin Friction

Any real fluids moving along solid boundary will incur a shear stress on that boundary.

The no-slip condition dictates that the speed of the fluid at the boundary is zero, but

atsome height from the boundary the flow speed must equal that of the fluid. The region

between these two points is aptly named the boundary layer. For all Newtonian fluids in

laminar flow the shear stress is proportional to the strain rate in the fluid, where the

viscosity is the constant of proportionality. However for Non-Newtonian fluids, this is no

longer the case as for these fluids the viscosity is not constant. The shear stress is

38

imparted onto the boundary as a result of this loss of velocity. The shear stress, for a

Newtonian fluid, at a surface element parallel to a flat plate, at the point y, is given by:

𝜏(𝑦) = 𝜇𝜕𝑢

𝜕𝑦

where

μ is the dynamic viscosity of the fluid,

u is the velocity of the fluid along the boundary, and

y is the height of the boundary. Specifically, the wall shear stress is defined as:

𝜏𝑤 = 𝜏(𝑦 = 0) = 𝜇𝜕𝑢

𝜕𝑦| 𝑦 = 0

In case of wind, the shear stress at the boundary is called wind stress.

1.9.14 Nusselt Number

The convective heat transfer from the surface will depend upon the magnitude of

Ch(Tw−T), where, Ch is the heat transfer coefficient and Tw and T are the temperatures of

wall and fluid respectively. Also, if there was no flow, the heat transfer was purely due to

39

conduction, the Fouriers law states that the quantity 𝑘(𝑇𝑤−𝑇)

𝑙 would be the measure of

the heat transfer rate, where k is the thermal conductivity and l is the length. Now

Nusselt number can be written as

𝑁𝑢 =𝐶ℎ(𝑇𝑤−𝑇)

𝑘(𝑇𝑤−𝑇)/𝑙=

𝐶ℎ𝑙

𝑘

i.e., Nusselt Number is the measure of the ratio of magnitude of the convective heat

transfer rate to the magnitude of heat transfer rate that would exist when there was pure

conduction.

1.10 Laplace Transforms

Laplace Transform is an essential mathematical tool which can be used to solve several

problems in science and engineering. This technique becomes popular when Heaviside

function applied to the solution of an ODE representing a problem in electrical

engineering. Transforms are used to accomplish the solution of certain problems with

less effort and in a simple routine way. The Laplace transform method reduces the

solution of an ODE to the solution of an algebraic equation. Also, when the Laplace

transform technique is applied to a PDE, it reduces the number of independent

variables by one

40

Definition 1.10.1. Let f(t) be a continuous and single-valued function of a real variable

t defined for all t, 0 < 𝑡 < ∞, and is of exponential order. Then the Laplace transform

of f(t) is defined as a function F(s) denoted by the integral

𝐿[𝑓(𝑡)] = ∫ 𝑒−𝑠𝑡𝑓(𝑡)𝑑𝑡 ∞

0 (1.10.1)

Definition 1.9.2. Error Function:

The error function is defined as,

erf(𝑥) =2

√𝜋∫ 𝑒−𝑡

2𝜋

0𝑑𝑡 (1.10.2)

and its compliment is

erf c(𝑥) = 1 − erf(𝑥) =2

√𝜋∫ 𝑒−𝑡

2∞

𝑥𝑑𝑡 (1.10.3)

The Laplace transform of the error function is

𝐿(erf(𝑥)) =1

𝑠 𝑒

𝑠2

4 𝑒𝑟𝑓𝑐(𝑠

2)

41

1.10.1 Complex Inversion Formula/Mellin-Fourier integral

In solving partial differential equations using Laplace transform method, complex

variable theory may come in handy for finding inverse transform. Inverse Laplace

transform can be expressed as an integral which is known as inverse integral and this

integral can be evaluated by using contour integration methods.

The inverse Laplace Transforms of U, V are u, v respectively and are given by the

Integrals

42

𝑢 =1

2𝑖𝜋∫ 𝑒𝑠𝑡 𝑈 𝑑𝑡𝑟+𝑖∞

𝑟−𝑖∞ and 𝑣 =

1

2𝑖𝜋∫ 𝑒𝑠𝑡 𝑉 𝑑𝑡𝑟+𝑖∞

𝑟−𝑖∞

Which can be evaluated by means of contour integration. Since there is no branch

point, the contour chosen is the closed curve ABC formed by the line x = r and a semi

circle C with origin as center and radius R (See figure 1.1) so that

∫ 𝑒𝑠𝑡 𝑈 𝑑𝑡 = lim𝑅→∞

∫ 𝑒𝑠𝑡 𝑈 𝑑𝑡𝐵

𝐴

𝑟+𝑖∞

𝑟−𝑖∞

== lim𝑅→∞

[∮ 𝑒𝑠𝑡𝑈 𝑑𝑡 − ∫ 𝑒𝑠𝑡𝑈 𝑑𝑡𝐶𝐴𝐵𝐶

]

Using Cauchy’s theorem of residues and Jordans lemma, we have

𝑢 =1

2𝑖𝜋∫ 𝑒𝑠𝑡 𝑈 𝑑𝑡𝑟+𝑖∞

𝑟−𝑖∞= sum of residues of {𝑒𝑠𝑡𝑈} at its poles.

Similarly,

𝑣 =1

2𝑖𝜋∫ 𝑒𝑠𝑡𝑉 𝑑𝑡𝑟+𝑖∞

𝑟−𝑖∞= sum of residues of {𝑒𝑠𝑡𝑉} at its poles.

Table 1.1: Laplace transform of some important functions

43

1.11 Similarity Transformation Birkhoff (1950) first recognized that Boltzmann’s method of solving the diffusion

equation with a concentration-dependant diffusion co-efficient is based on the algebraic

symmetry of the equation and special solutions of this equation can be obtained by

solving a related ordinary differential equation. Such solutions are called “similarity

solutions” because they are geometrically similar. He also suggested that the algebraic

symmetry of the partial differential equations can be used to find similarity solutions of

other partial differential equations by solving associated ordinary differential equations.

44

Thus, the method of similarity solutions has become a very successful dealing with the

determination of a group of transformation under which a given partial differential

equation is invariant. The simplifying feature of this method is that a similarity

transformation of the form 𝑢(𝑥, 𝑡) = 𝑡𝑝 𝑣(𝜂), 𝜂 = 𝑥 𝑡−𝑞 can be found which can, then,

we used effectively to reduces the partial differential equations to an ordinary differential

equations with 𝜂 as the independent variable. The resulting ordinary differential

equations is relatively easy tosolve. In practice this method is simple and useful in

finding solutions of both linear andnonlinear partial differential equations.

1.12 Numerical Methods

Numerical methods are the way to do higher mathematics problems on a computer, a

technique widely used by scientists and engineers to solve their problems. A major

advantage for numerical analysis is that a numerical answer can be obtained even

when a problem has no “analytical” solution. It is important to realize that a numerical

analysis solution is always numerical. Analytical methods usually give a result in terms

of mathematical functions that can then be evaluated for specific instances. There is

thus advantage to the analytical results, in that the behavior and properties of the

function are often apparent. However, numerical results can be plotted to show some of

the behavior

45

of the solution.

Another important distinction is that the result from numerical method is an

approximation, but results can be made as accurate as desired. To achieve high

accuracy, many separate operations must be carried out. Here are some of the

operations that numerical methods can do:

• Solve for the roots of a nonlinear equation.

• Solve large systems of linear equations.

• Get the solutions of a set of nonlinear equations.

• Interpolate to find intermediate values within a table of data.

• Solve ODE when given initial values for the variables.

• Solve boundary-value problems and determine eigenvalues and eigenvectors.

• Obtain numerical solutions to all types of partial differential equations and so on.

In connection with numerical analysis many symbolic algebraic programmes are

available,namely Mathematica, DERIVE, Maple, MathCad, MATLAB, and MacSyma. In

this thesis the numerical solutions of the problem are solved by RKF -45 method with

the help of algebraic software MAPLE.

1.12.1 Runge Kutta Fehlberg Method

46

Runge-Kutta-Fehlberg is adaptive; that is, the method adapts the number and position

of the grid points during the course of the iteration in attempt to keep the local error

within some specified bound. Sketch of the ideas:

• Begin with two RK approximation algorithms, one with order p and with order p + 1.

• Apply the algorithms to get two approximations at a given grid point tk.

• These approximations are used to approximate the local discretization error at the grid

point. This error approximations is then used to make several decisions.

• If the error approximation exceeds some prescribed maximum bound on accuracy,

then a smaller step size is assigned, a new grid point tk is assigned, and the preceding

steps are repeated.

• If the error approximation falls below some present minimum bound on accuracy,

then the step size is increased and the next step in the iteration is performed.

• If the error approximation falls in between some user-specified minimum and

maximum values, then we may choose to leave the step size alone or we may compute

an optimal step size for the next step. The term optimal is used loosely because there

are some assumptions made and some approximations involved in getting this value.

• Typically, the approximation given to the user is reported as the more accurate p + 1

order approximation, even through, in the analysis, that approximation is used to

approximate the error in the pth order approximation

47

The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to resolve problem. It

has a procedure to determine, if the proper step size h is being used. At each step,

twomdifferent approximations for the solution are made and compared. If the two

answers are in close agreement, the approximation is accepted. If the two answers do

not agree to a specified accuracy, the step size is reduced. If the answers agree to

more significant digits than required, the step size is increased. Each step requires the

use of the following six values:

Now the approximation solution to the given I.V.P. is made using a Runge-Kutta method of order 4:

48

where the four function values f1, f3, f4 and f5 are used. Notice that f2 is not used in the

above formula. A better value for the solution is determined using a Runge-Kutta

method of order 5:

The optimal step size sh can be determined by multiplying the scalar s times the

current step size h. The scalar s is

49

Magnetite and Mn-Zn Ferrite-Water BasedNanofluidsFlow

atStagnation Point over a Stretching Surface Inspired

By Nonlinear Thermal Radiation

2.1 INTRODUTION

The thermal properties of the ordinary fluids are not sufficient to meet today’s

cooling rate requirements in industrial applications. Therefore, the nanofluids are

introduced to enhance the thermal performance of ordinary fluids. Basically, the

nanofluids are engineered colloidal suspensions of nanoparticles in a base fluid. The

nanoparticles are typically made up of metals (Al, Cu), oxides (𝐴𝑙2𝑂3, 𝑇𝑖𝑂2 and CuO),

carbides (SiC), nitrides (AlN, SiN) or nonmetals (Graphite, carbon nanotubes) and the

base fluid is usually a conductive fluid, such as water, ethylene glycol and etc., The term

of nanofluid was first introduced by Choi [1]. Buongiorno [2] proposed seven slip

mechanisms to write down conservation equations based on the Brownian diffusion and

thermophoresis effects. Recently, Gorla and Chamkha [3] have analyzed the flow of

nanofluid with natural convective boundary layer over a horizontal plate along with

porous medium. Rashidi et al. [4] have investigated entropy generation in steady MHD

flow of nanofluid due to a rotating porous disk. Gireesha et al. [5] have analyzed the

effect of suspended particles on nanofluid flow and heat transfer over a stretching sheet

50

saturated by a porous medium. The heat and mass transfer of water based nanofluid

flow over a stationary/moving vertical plate wereexplored by Mahanthesh et al. [6].

Besides, theferrofluid comprises of iron-based nanoparticles such as magnetite,

hematite, cobalt ferrite, etc.,Researchers and scientists have focused considerably on

the surface driven ferrofluid flows owing to their numerous industrial and biomedical

demands. For instance, iron-based nanoparticles can be used for efficient drug delivery

by guiding the particles via external magnets;magnetic nanoparticles are prominent in

hyperthermia. Several researchers investigated diversified characteristics of such

ferrofluid problems. For instance,Tangthieng et al. [7] addressed heat transfer

enhancement in ferrofluids subjected to steady magnetic fields.Jue [8] used semi-

implicit finite element method in orderto simulate magnetic gradient and thermal

buoyancy induced cavity ferrofluid flow.Nanjundappa et al. [9] analyzed the influenceof

magnetic field dependent viscosity on the horizontal layer of ferrofluid.Sheikholeslami

and Ganji [10] have invstigatedthe MHD flow and heattransfer of ferro nanofluid with the

effet of convective heat transfer. The stagnation point flow and heat transfer of ferrofluid

towards a stretching sheet in the presence of viscous dissipationwere investigated by

Zafar et al. [11]. They have consideredthree types of ferroparticles magnetite (𝐹𝑒3𝑂4),

cobalt ferrite (𝐶𝑜𝐹𝑒2𝑂4) and 𝑀𝑛-𝑍𝑛 ferrite (𝑀𝑛 − 𝑍𝑛𝐹𝑒2𝑂4 ) with water and kerosene as

51

conventional base fluids.Recently Sheikholeslami, Rashidi andGanji [12, 13] have

studied the effect of magnetic field with suspended ferroparticles.

The fluid flow at astagnationpoint over a stretching sheet iscrucial in theoretical and

application point of view in fluid dynamics. Chaim [14] was the first toinvestigate the

stagnation-point flow towards a stretching sheet. Mahapatra and Gupta[15] have

studied the similar problem by considering the strain-rate and the stretching rate to be

different. They obtained boundary layer adjacent to thesheet which completely depends

on the ratio between the strain-rate of stagnation-point flow and the stretching rate of

thesheet. The steady and unsteady stagnation-point flow of an incompressible viscous

fluid over a stretching surface wasstudied by Paullet and Weidman [16]. The stagnation

point flow of an electrically conducting fluid over a stretching surface under

theinfluenceof magnetic field wasinvestigated by Mahapatra et al. [17]. Pal et al. [18]

have analyzed the nanofluid flow and non-isothermal heat transfer at the stagnation-

pointover a stretching/shrinking sheet embedded in a porous medium.Gireesha et al.

[19] have presented the numerical solution for boundary layer stagnation-point flow past

a stretched surface with melting effect and aligned magnetic field. They have

incorporated the nanofluid model by considering Brownian motion with thermophoresis

mechanisms.Furthermore, the stagnation-point flow over stretching sheet under

52

different physical aspects were discussed byIshak [20], Mabood [21],Makinde[22] and

Hayat et al [23].

In view of the above discussion, the objectives of present analysis are three folds.

Firstly to model and examine the two-dimensional stagnation flow of Magneto-ferrofluids

induced by a stretched surface. Secondly to analyze the heat transfer process under the

influence of non-linear thermal radiation, heat generation and viscous dissipation

effects. Thirdly to compareflow characteristics of two ferrofluids namely 𝐹𝑒3𝑂4 andMn-

ZnFe2O4water based nanofluids. Numerical solutions are computed for the governing

nonlinear boundary value problem via similarity method.

2.2 PROBLEM FORMULATION

Consider a steady, two-dimensionalboundary layer flow of

anincompressibleferrofluid driven by stretching ofsheet at 𝑦 = 0with a fixed stagnation

point. The 𝑥-axis taken along the sheet and 𝑦-axis is normal to it. The fluid occupies the

half-plane 𝑦 > 0.Two equal and opposite forces areapplied along the sheet, so that the

sheet is stretched, keeping theposition of the origin unchanged (see figure 1). The fluid

is assumed to be electrically conducting.The magnetic field of strength 𝐵0 is applied

along 𝑦 -direction. The thermophysical properties of the base fluids (water) and the

53

ferroparticles (magnetite and Mn-Zn ferrite) are presented in Table 1.The sheet is

maintained at constant temperature 𝑇𝑤 whereas 𝑇∞ denotes the temperature outside the

thermal boundary layer.The velocity of the stretching rate and flow external to the

boundary layer are 𝑈𝑤(𝑥) = 𝑐𝑥and𝑈𝑒(𝑥) = 𝑎𝑥, where 𝑎 and 𝑐 are positive constants.

Figure1: Flow configuration and coordinate system

In terms of stream function𝜓(𝑥, 𝑦)such that𝑢 =𝜕𝜓

𝜕𝑦and𝑣 = −

𝜕𝜓

𝜕𝑥, the governing equations

are

𝜕2𝜓

𝜕𝑥𝜕𝑦−

𝜕2𝜓

𝜕𝑥𝜕𝑦= 0, (1)

𝜕𝜓

𝜕𝑦

𝜕2𝜓

𝜕𝑥𝜕𝑦−𝜕𝜓

𝜕𝑥

𝜕2𝜓

𝜕𝑦2= 𝑈𝑒

𝑑𝑈𝑒

𝑑𝑥+𝜇𝑛𝑓

𝜌𝑛𝑓(𝜕3𝜓

𝜕𝑦3) −

𝜎

𝜌𝑛𝑓𝐵02(𝜕𝜓

𝜕𝑦− 𝑈𝑒), (2)

𝜕𝜓

𝜕𝑦

𝜕𝑇

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕𝑇

𝜕𝑦= 𝛼𝑛𝑓 (

𝜕2𝑇

𝜕𝑦2) +

𝜇𝑛𝑓

(𝜌𝑐𝑝)𝑛𝑓(𝜕2𝜓

𝜕𝑦2)2

−1

(𝜌𝑐𝑝)𝑛𝑓

𝜕𝑞𝑟

𝜕𝑦+

𝑄0

(𝜌𝑐𝑝)𝑛𝑓(𝑇 − 𝑇∞). (3)

54

where u and v are the velocity components along the x - and y -axes, μnf

-dynamic

viscosity of nanofluid, σ- the electrical conductivity, B0-magnetic field, νnf the effective

kinematic viscosity of nanofluid,αnf is the effective thermal diffusivity of nanofluid, T-fluid

temperature, T∞ - ambient fluid temperature, (ρcp)nf- heat capacity of nanofluid, 𝑄0 -

dimensional heatgeneration/absorption coefficient and 𝑐𝑝-the specific heat.

The boundary conditions for the problem are given by;

𝜕𝜓

𝜕𝑦 = 𝑈𝑤(𝑥), −

𝜕𝜓

𝜕𝑥 = 0, 𝑇 = 𝑇𝑤, 𝑎𝑡 𝑦 = 0

𝜕𝜓

𝜕𝑦 → 𝑈𝑒(𝑥), 𝑇 → 𝑇∞, 𝑎𝑠 𝑦 → ∞

} (4)

The radiative heat flux expression in Eq. (3) is given by theRosseland approximation

asfollows

𝑞𝑟 = −4𝜎∗

3𝑘∗𝜕𝑇4

𝜕𝑦= −

16𝜎∗

3𝑘∗𝑇3

𝜕𝑇

𝜕𝑦, (5)

whereσ∗ and k∗ are the Stefan-Boltzman constant and the meanabsorption coefficient

respectively. In view of Eq. (5), the Eq. (3)reduces to

𝜕𝜓

𝜕𝑦

𝜕𝑇

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕𝑇

𝜕𝑦= 𝛼𝑛𝑓 (

𝜕2𝑇

𝜕𝑦2) +

𝜇𝑛𝑓

(𝜌𝑐𝑝)𝑛𝑓(𝜕2𝜓

𝜕𝑦2)2

+16𝜎∗

3(𝜌𝑐𝑝)𝑛𝑓𝑘∗[𝑇3

𝜕2𝑇

𝜕𝑦2+ 3𝑇2 (

𝜕𝑇

𝜕𝑦)2

]

+𝑄0

(𝜌𝑐𝑝)𝑛𝑓(𝑇 − 𝑇∞). (6)

The effective properties of nanofluids may be expressed in terms of the properties of

base fluid and nanoparticle and the solid volume fraction of nanofluid as follows;

55

𝜈𝑛𝑓 =𝜇𝑛𝑓

𝜌𝑛𝑓, 𝜇𝑛𝑓 =

𝜇𝑓(1 − 𝜙)2.5

,

𝜌𝑛𝑓 = (1 − 𝜙)𝜌𝑓 +𝜙𝜌𝑠,

(𝜌𝑐𝑝)𝑛𝑓 =(1 − 𝜙)(𝜌𝑐𝑝)𝑓 + 𝜙(𝜌𝑐𝑝)𝑠,

𝑘𝑛𝑓

𝑘𝑓=

(𝑘𝑠+2𝑘𝑓)−2𝜙(𝑘𝑓−𝑘𝑠)

(𝑘𝑠+2𝑘𝑓)+𝜙(𝑘𝑓−𝑘𝑠). (7)

here 𝑘𝑛𝑓 -the thermal conductivity of the nanofluid, 𝑘𝑓 -thermal conductivity of the

basefluid,

ks- thermal conductivity of solid nanoparticles,𝜎𝑓 the electrical conductivity of the base

fluid, 𝜎𝑠 the electricalconductivity of the nanoparticle, the subscripts 𝑠 and 𝑓 denotes to

the solid and fluid respectively and 𝜙-solid volume fraction of nanofluid.

We look for a similarity solution of (1)–(6) of the following form;

𝜂 = √𝑐

𝜈𝑓𝑦, 𝑓(𝜂) =

𝜓

√𝑥𝜈𝑓𝑈𝑤,

𝑇 = 𝑇∞(1 + (𝜃𝑤 − 1)𝜃), (8)

where 𝜃𝑤 =𝑇𝑤

𝑇∞, 𝜃𝑤(>1) isthe temperature ratio parameter.Followingnonlinear system of

ordinary differential equations are obtained from the Eqns. (2) and (6) by employing

similarity variables (8);

𝑓 ′′′(𝜂) + (1 − 𝜙)2.5 {(1 − 𝜙 + 𝜙

𝜌𝑠

𝜌𝑓) (𝑓(𝜂)𝑓 ′′(𝜂) − 𝑓 ′(𝜂)2 + 𝛿2)

+𝑀(𝛿 − 𝑓 ′(𝜂))} = 0, (9)

56

𝑘𝑛𝑓

𝑘𝑓𝜃 ′′(𝜂) + 𝑅 {(1 + (𝜃𝑤 − 1)𝜃)

3𝜃 ′′(𝜂) + 3(1 + (𝜃𝑤 − 1)𝜃(𝜂))2(𝜃𝑤 − 1)𝜃

′2(𝜂)}

+1

2Pr (1 − 𝜙 + 𝜙

(𝜌𝑐𝑝)𝑠(𝜌𝑐𝑝)𝑓

)𝑓(𝜂)𝜃 ′(𝜂) +𝑃𝑟𝐸𝑐

(1−𝜙)2.5𝑓 ′′2(𝜂) + 𝑃𝑟𝑄𝜃(𝜂) = 0, (10)

Subjected to the boundary condition;

𝑓(0) = 0, 𝑓 ′(0) = 1, 𝜃(0) = 1 𝑎𝑡 𝜂 = 0

𝑓 ′(𝜂) → 𝛿, 𝜃(𝜂) → 0 𝑎𝑠 𝜂 → ∞} (11)

herethe non-dimensional quantity’s are magnetic parameter, stretching

parameter,Prandtl number, radiation parameter, Eckert number, heat generation and

the temperature ratio parameters. They are respectivelydefined as follows;

𝑀 =𝜎𝐵0

2

𝜌𝑓𝑐, 𝛿 =

𝑎

𝑐,

𝑃𝑟 =(𝜇𝑐𝑝)𝑓

𝑘𝑓, 𝑅 =

16𝜎∗𝑇∞3

3𝑘∗𝑘𝑓

𝐸𝑐 =𝑈𝑤2

𝑐𝑝𝑓(𝑇𝑤−𝑇∞), 𝑄 =

𝑄0

𝑐(𝜌𝑐𝑝)𝑓, 𝜃𝑤 =

𝑇𝑤

𝑇∞}

(12)

The skin friction coefficient𝐶𝑓 and local Nusselt number 𝑁𝑢𝑥 are defined as follows;

𝐶𝑓𝑥 =𝜏𝑤

𝜌𝑓𝑢𝑤2 and 𝑁𝑢𝑥 =

𝑥𝑞𝑤

𝑘𝑓(𝑇𝑤−𝑇∞), (13)

where𝜏𝑤-surface shear stresses and 𝑞𝑤-surface heat flux given by

𝜏𝑤𝑥 = 𝜇𝑛𝑓 (𝜕𝑢

𝜕𝑦)𝑦=0

, 𝑞𝑤 = −𝑘𝑛𝑓 (𝜕𝑇

𝜕𝑦+ 𝑞𝑟)

𝑦=0 (14)

Using Equation (8) and (14) in equation (13) one can obtain;

57

√𝑅𝑒𝑥𝐶𝑓𝑥 =1

(1−𝜙)2.5𝑓 ′′(0)

𝑁𝑢𝑥

√𝑅𝑒𝑥= −(

𝑘𝑛𝑓

𝑘𝑓+ 𝑅𝜃𝑤

3) 𝜃 ′(0)}, (15)

where which 𝑅𝑒𝑥 =𝑈𝑤𝑥

𝜈𝑓 is the local Reynolds number.

Table1:Thermophysical properties of base fluid and magnetic nanoparticles [12]

𝝆(𝒌𝒈/𝒎𝟑) 𝒄𝒑(𝑱/𝒌𝒈𝑲) 𝒌(𝑾/𝒎-𝑲)

𝑾𝒂𝒕𝒆𝒓 997.1 4179 0.613

𝑭𝒆𝟑𝑶𝟒 5180 670 9.7

𝑴𝒏− 𝒁𝒏𝑭𝒆𝟐𝑶𝟒 4900 800 5

2.3 RESULT AND DISCUSSION

The coupled ordinary differential nonlinearequations (9) and (10) along with the

boundary conditions (11) are solved numerically using Runge-Kutta-Fehlberg4-5thorder

scheme coupled with shooting technique. The integration length varies with respect to

the physical parameter values and it has been suitably chosen each time such that the

boundary conditions at the outer edge of the boundary layer are satisfied. Throughout

our computation the volume fraction of nanoparticleis considered in the range 0 ≤ 𝜙 ≤

0.2 while the Prandtl number 𝑃𝑟 = 6.2 for water.For water-based ferrofluids, the

variations ofskin friction with solid volume fractionfor different values of stretching and

58

magnetic parameter areshown in Table 2. It is observed that the skin friction is smaller

in the absence of magnetic field. Also, the skin friction coefficientincreases withincrease

in volume fraction parameter. Whereasthe skin frictioncoefficientdecreases by

increasing stretching parameter. Table 3revealsthatthe Nusselt number is higher for

nonlinear radiation influence than the linear radiation. Further, the Nusselt number

increases with volume fraction parameterand it is decreasedby increasingstretching

parameter. It is interesting to note that 𝐹𝑒3𝑂4 -nanofluid has higher skin friction

coefficientthan Mn− ZnFe2O4 nanofluid.

Table2: Variation of −√𝑹𝒆𝒙𝑪𝒇𝒙 with solid volume fraction for different values of

magnetic and stretching parameterswith 𝑬𝒄 = 𝟎. 𝟐, 𝑸 = 𝟎. 𝟓, 𝑹 = 𝟏, 𝜽𝒘 = 𝟏. 𝟓.

𝝓

𝑴 = 𝟎 𝑴 = 𝟎.𝟓

𝜹 = 𝟎. 𝟏 𝜹 = 𝟎. 𝟑

𝜹 = 𝟎. 𝟓 𝜹 = 𝟎. 𝟏 𝜹 = 𝟎. 𝟑 𝜹 = 𝟎. 𝟓

𝑭𝒆𝟑𝑶𝟒

0.01 1.00207 0.87801 0.68973 1.19007 1.00948 0.77598

0.1 1.31759 1.15450 0.90692 1.50289 1.28354 0.99136

0.2 1.73762 1.52252 1.19602 1.92885 1.65533 1.28278

𝑴𝒏− 𝒁𝒏

𝑭𝒆𝟐𝑶𝟒

0.01 1.00072 0.87683 0.68880 1.18893 1.00845 0.77515

0.1 1.30450 1.14302 0.89790 1.49143 1.27323 0.98313

59

0.2 1.71089 1.49909 1.17761 1.90481 1.63382 1.26565

Table3: Variation of 𝑵𝒖𝒙

√𝑹𝒆𝒙 with solid volume fraction for different values of

magnetic and stretching parameters with𝑬𝒄 = 𝟎. 𝟐, 𝑸 = 𝟎. 𝟓,𝑴 = 𝟎. 𝟓, 𝑹 = 𝟏.

LinearRadiation Non-LinearRadiation

𝝓 𝛿 = 0.1 𝛿 = 0.3 𝛿 = 0.5 𝛿 = 0.1 𝛿 = 0.3 𝛿 = 0.5

0.01 3.951494

-

3.048904

-

7.476607 5.509048

-

1.401750

-

4.717270

𝑭𝒆𝟑𝑶𝟒 0.1 7.128524

-

1.227471

-

6.073685 8.908286

-

0.293510

-

4.001240

0.2 9.857402 0.395658

-

4.752053 12.07021 0.724722

-

3.301960

𝐌𝐧− 𝐙𝐧 0.01 2.097975

-

1.687121

-

4.091306 3.240191

-

0.863060

-

2.847900

𝐅𝐞𝟐𝐎𝟒 0.1 3.305423

-

0.926465

-

3.509962 4.423531

-

0.384430

-

2.544520

0.2 4.179113

-

0.259601

-

2.962651 5.057244 0.028913

-

2.258640

60

Figures 2-14 are plotted to bring out the salient features of different flow fields

versus different values of physical parameters.In these figures, red lines indicate the

profiles for 𝐹𝑒3𝑂4 -nanofluid and green lines indicate the profiles forMn− ZnFe2O4

nanofluid.The velocity profile for different values of stretching parameter 𝛿 is presented

in figure 2. It is observed that the flow has a boundary layer structurewhen the

stretching velocity is less than the free stream velocity (𝛿 =𝑎

𝑐> 1) .Physically, the

straining motion near the stagnation point increased to accelerate velocity of the

external stream, which leads to an increase in the thickness of the boundary layer with

𝛿. However the stretching velocity 𝑐𝑥 of the surface exceeds the free stream velocity

𝑎𝑥 (𝛿 < 1), therefore inverted boundary layer structure is formed and for 𝛿 = 1 there is

no boundary layer formation because the stretching velocity is equal to the free stream

velocity.The velocity profile for different values of magnetic parameter isdisplayed in

figure3. Because of the influence of Lorentz force the velocity field and its associated

boundary layer thickness decreases. The variation ofstretching ratio parameter δon

velocity and temperature profiles respectivelyplotted in figures 4 and 5. These figures

indicate that, the velocity field increases andtemperature field decreases with an

increasein stretching ratio parameter.

Figure 6 exhibit the effect of volume fractionparameter on temperature profiles. It

is observed that the dimensionless temperature is found to be higher and its

61

corresponding thermal boundary layer become thicker.Figures7 and 8demonstrate the

effects ofheat source and nonlinear radiation parameteron temperature profiles. It is

observed that the temperature is augmented throughout theboundary layer region as 𝑄

and 𝑅 increases.This is because theheat generation and radiation parameter provides

more heat intothe fluid, which leads to an intensification of the thermalboundary layer.

Additionally, it is noted that the non-linear thermal radiation effect is more prominent on

temperature field than that of linear thermal radiation.Also one can observe that

thetemperature profile is smaller for Mn-ZnFe2O3-nanofluid in comparisonwith Fe3O4 -

nanofluid.The variationof temperature profile θ(η) for differentvalues of temperature

ratio parameter is plotted in figure 9. It is found that the thermal boundary layer

thickness increases by increasingthe temperature ratio parameter. This is happen due

to the enhancedthermal conductivity of the flow.

The variation of skin friction coefficient for different values of

themagneticparameter andEckert numberis explained in figure10. As expected, the skin

friction decreases for strong magnetic field.The variation of skin friction for different

values of magnetic parameter andstretching parameter is presented in figure 11. It

shows that, the skin friction decreases with higher magnetic field strength of both

ferroparticles. This is due to that, the magnetic field strength restricted the fluid motion.

Here, the skin friction coefficient is higher for Mn-ZnFe2O3-nanofluid in comparison with

62

Fe3O4 -nanofluid.The Nusselt numberprofile for various values of the magnetic

parameter, Eckert number andstretching parameter is presentedin figures 12 and 13.It

is seen that the Nusselt numberis reduced when themagnetic parameter

increases.Further, we observed that the response of the Nusselt number is opposite for

larger values of the Eckert number andstretching parameter. Finally, the effect of

radiation parameter on the Nusselt number is shown in figure14.The rate of heat

transfer at the surface is higher for larger values of radiation parameter.

2.4 CONCLUSION

In this study,magnetohydrodynamic stagnation-point flow and heat transfer of two

selected ferrofluids toward a stretching sheetis investigated. We have analyzed the

effects of nonlinear thermal radiation and heat generation.The major aspects of this

work are given below;

The dimensionless velocity decreases with increase in magnetic parameter

and increases with increase in stretching parameter.

The dimensionless temperature decreases with increasing stretching

parameter.

63

The nonlinear thermal radiation plays a crucial role in cooling and

heatingprocess.

The skin friction coefficient and Nusselt number decreased for the strong

magnetic field.

The Nusselt number increases with nonlinear thermal radiation parameter.

Finally, comparison of Magnetite and Mn-Znferrite-water nanofluids, it can be

concluded that magnetite have a higher rate of skin friction and nusselt

numbers.

Fig. 2: Effect of stretching parameter on the dimensionless velocity for different 𝐹𝑒3𝑂4

and 𝑀𝑛 − 𝑍𝑛𝐹𝑒2𝑂4nanoparticles in waterbasedferrofluids.

64

Fig. 3: Effect of magneticparameter on velocity profile.

Fig. 4: Effect of stretchingparameter on velocity profile.

65

Fig. 5: Effect of stretchingparameter on temperature profile.

Fig. 6: Effect of volume fractionparameter on temperature profile.

66

Fig. 7: Effect of heat generationparameter on temperature profile.

Fig. 8: Effect of nonlinear radiationparameter on temperature profile.

67

Fig. 9: Effect of temperature ratioparameter on temperature profile.

Fig. 10: Variation of Magnetic parameter with Eckert number on skin friction

coefficient.

68

Fig. 11: Variation of Magnetic parameter with stretching parameter on skin

friction coefficient.

Fig. 12: Variation of Magnetic parameter with Eckert number on nusselt number.

69

Fig. 13: Variation of Magnetic parameter with stretching parameter on Nusselt number.

Fig. 14: the effect of nonlinear radiation with heat generation on nusselt number.

70

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74

Publications:

Published On 2017 “A Comparative Study of Magnetite and Mn–Zn Ferrite

Nanoliquids Flow Inspired by Nonlinear Thermal Radiation” In the American

Scientific Publishers, vol. 6, pp. 1-7, 2017

Research Article Entitled “MHD boundary layer slip flow and heat transfer of ferro

fluid along a stretching sheet” is Presented in two days international Conference on

Differential Geometry, Analysis and Fluid Mechanics [ICDGAFM-2016] held on 4rd-5th

February 2016, Organised by the Department of P.G Studies and Research in

Mathematics, Jnanasahyadri, Shankaraghatta-577451, Shimoga. Karnataka, India.

Presented a paper entitled “A Comparative Study of Magnetite and Mn–Zn Ferrite

Nanoliquids Flow Inspired by Nonlinear Thermal Radiation”, in UGC sponsored

two day National conference on Recent Advancements in Nano-Science and

Technology [RANST-2017] held on 21st and 22nd April-2017, organized by Department

of Chemistry , Government Science College, Chitradurga, Karnataka, India.