Hindawi Publishing CorporationJournal of FluidsVolume 2013 Article ID 910531 8 pageshttpdxdoiorg1011552013910531

Research ArticleHall Effect on Beacutenard Convection ofCompressible Viscoelastic Fluid through Porous Medium

Mahinder Singh1 and Chander Bhan Mehta2

1 Department of Mathematics Government Post Graduate College Seema (Rohru) Shimla District Himachal Pradesh 171207 India2Department of Mathematics Centre of Excellence Government Degree College Sanjauli Shimla DistrictHimachal Pradesh 171006 India

Correspondence should be addressed to Mahinder Singh mahinder singh91rediffmailin

Received 17 April 2013 Revised 26 July 2013 Accepted 30 July 2013

Academic Editor Amy Shen

Copyright copy 2013 M Singh and C B Mehta This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An investigation made on the effect of Hall currents on thermal instability of a compressible Walterrsquos B1015840 elasticoviscous fluidthrough porous medium is considered The analysis is carried out within the framework of linear stability theory and normalmode technique For the case of stationary convection Hall currents and compressibility have postponed the onset of convectionthrough porous medium Moreover medium permeability hasten postpone the onset of convection and magnetic field has duelcharacter on the onset of convection The critical Rayleigh numbers and the wave numbers of the associated disturbances for theonset of instability as stationary convection have been obtained and the behavior of various parameters on critical thermal Rayleighnumbers has been depicted graphically The magnetic field Hall currents found to introduce oscillatory modes in the absence ofthese effects the principle of exchange of stabilities is valid

1 Introduction

The theoretical and experimental results of the onset of ther-mal instability (Benard convection) under varying assump-tions of hydrodynamic and hydromantic stability have beendiscussed in a treatise by Chandrasekhar [1] in his celebratedmonograph If an electric field is applied at right angles tothe magnetic field the whole current will not flow along theelectric field This tendency of the electric current of flowacross an electric field in the presence of a magnetic fieldis called Hall current effect The Hall effect is likely to beimportant in many geophysical and astrophysical situationsas well as in flows of laboratory plasmaThe use of the Boussi-nesq approximation has been made throughout which statesthat the variations of density in the equations of motion cansafely be ignored everywhere except in its association withthe external force It has been shown by Sato [2] and Tani[3] that inclusion of Hall currents gave rise to a cross flowthat is a flow at right angle to the primary flow through achannel in the presence of a transverse magnetic field Inparticular Tani [3] has found that Hall effect produces a

cross-flow of double-swirl pattern in incompressible flowthrough a straight channel with arbitrary cross-section Thisbreakdown of the primary flow and formation of secondaryflow may be presumably attributed to the inherent instabilityof the primary flow in the presence of Hall current Sato [2]has pointed out that even if the distribution of the primaryflow velocity is stable to external disturbances the wholelayer may become turbulent if the distribution of the crossflow is unstable Sherman and Sutton [4] have consideredthe effect of Hall current on the efficiency of a magnetofluidgeneratorThe effect of Hall current on the thermal instabilityof a horizontal layer of electrically conducting fluid has beenstudied by Gupta [5]

Hall currents are effects whereby a conductor carrying anelectric current perpendicular to an applied magnetic fielddevelops a voltage gradient which is transverse to both thecurrent and the magnetic field It was discovered by Hall in1879 while he was working on his doctoral degree at JohnsHopkins University at Baltimore Maryland The Hall effecthas again become an active area of researchwith the discoveryof the quantizedHall effect byKlaus vonKlitzing for which he

2 Journal of Fluids

was bestowed with Nobel prize of physics in 1985 In ionizedgases (plasmas) where the magnetic field is very strong andeffects the electrical conductivity cannot be Hall currents

In the aforementioned studies the medium has beenconsidered to be nonporousThe development of geothermalpower resources has increased general interest in the proper-ties of convection in porous media The effect of a magneticfield on the stability of such a flow is of interest in geophysicsparticularly in the study of Earthrsquos core where the Earthrsquosmantle which consists of conducting fluid behaves like aporous medium which can become convectively unstable asa result of differential diffusion The other application of theresults of a magnetic field is in the study of the stability of aconvective flow in the geothermal region

When the fluids are compressible the equations govern-ing the system become quite complicated to simplify Boussi-nesq tried to justify the approximation for compressible fluidswhen the density variations arise principally from thermaleffects Spiegel and Veronis [6] have simplified the set ofequations governing the flow of compressible fluids under thefollowing assumptions

(a) The depth of the fluid layer is much less than the scaleheight as defined by them

(b) The fluctuations in temperature density and pres-sure introduced due to motion do not exceed theirtotal static variations

Under the previous approximations the flow equationsare the same as those for incompressible fluids except thatthe static temperature gradient is replaced by its excess overthe adiabatic one and 119862V is replaced by 119862

119901

Chandra [7] observed a contradiction between the theoryand experiment for the onset of convection in fluids heatedfrom below He performed the experiment in an air layerand found that the instability depended on the depth of thelayer Scanlon and Segel [8] have considered the effects ofsuspended particles on the onset of Benard convection andfound that the critical Rayleigh number is reduced becauseof the heat capacity of the particles The suspended particleswere thus found to destabilize the layer The fluids have beenconsidered to be Newtonian and the medium has beenconsidered to be nonporous in all the previous studies

One class of elastico-viscous fluids isWalters fluid (model1198611015840) which is not characterized by MaxwellrsquosOldroydrsquos con-

stitutive relation When the fluid permeates a porous mate-rial the gross effect is represented by Darcyrsquos law As a resultof this macroscopic law the usual viscous and viscoelasticterms in the equation of Waltersrsquo fluid (model 1198611015840) motionare replaced by the resistance terms [minus(1119896

1)(120583minus120583

1015840(120597120597119905)) 119902]

where 120583 and 1205831015840 are the viscosity and viscoelasticity ofWaltersrsquofluid (model 1198611015840) 119896

1is the medium permeability and 119902 is the

Darcian filter velocity of the fluidThe flow through porousmedia is of considerable interest

for petroleum engineers and geophysical fluid dynamicistsA great number of applications in geophysics may be foundin the books by Phillips [9] Ingham and Pop [10] and Nieldand Bejan [11] The scientific importance of the field has alsoincreased because hydrothermal circulation is the dominantheat transfer mechanism in young oceanic crust (Lister [12])

Generally it is accepted that comets consisting of a dustyldquosnowballrdquo of a mixture of frozen gasses which is in theprocess of their journey change from solid to gas and viceversaThe physical properties of comets that meteoroids andinterplanetary dust strongly suggest the importance of poros-ity in astrophysical context have been studied by McDonnell[13]

The stability of two superposed conducting Waltersrsquo 1198611015840elastico-viscous fluids in hydromagnetics has been studiedby Sharma and Kumar [14] and whereas the instability ofstreaming Waltersrsquo viscoelastic fluid 119861

1015840 in porous mediumhas been considered by Sharma [15] Sunil and Chand [16]Sunil et al [17 18] studied the Hall effect on thermosolutalinstability of Rivlin-Ericksen and Waltersrsquo (model 1198611015840) fluidin porous medium In one study of Singh [19] Hall currenteffect on thermosolutal instability in a viscoelastic fluidflowing through porous medium and magnetic field stablesolute gradient are found to have stabilizing effects on thesystem whereas Hall current and medium permeability havea destabilizing effect on the system The sufficient conditionsfor the nonexistence of overstability have also obtained In theone another study Singh andKumar [20] hydrodynamic andhydromagnetic stability of two stratified Walterrsquos 1198611015840 elastico-viscous superposed fluids where system is stable for stablestratification and unstable for unstable stratification and incase of horizontal magnetic field system having stabilizingeffect for unstable stratification is in contrast to the stabilityof two superposed Newtonian fluids where the system isstable for stable stratifications Gupta et al [21] have studiedthermal convection of dusty compressible Rivlin-Ericksenviscoelastic fluidwithHall currents and found that compress-ibility and magnetic field postpone the onset of convectionwhereas Hall current and suspended particles hasten theonset of convection

During the survey it has been noticed that Hall effectsare completely neglected from the studies of compressibleelastico-viscous fluid through porous medium Keeping inmind the importance of Hall currents porous medium andcompressibility in elastico-viscous fluid motivated us to goon detailed study of Walterrsquos 1198611015840 fluid heated from belowthrough porous medium We have already studied earliersome problems on Hall current effect with porous as well asnonporousmedium and suspended particles found the usefuland interesting results so compressible thermal instabilityproblemofWalterrsquos1198611015840 fluidwithHall currents effects throughporous medium studied by us here

2 Mathematical Formulation of the Problem

We have considered an infinite horizontal and compressibleelectrically conducting Walterrsquos 119861

1015840 fluid permeated withporous medium in Hall current effect bounded by the planes119911 = 0 and 119911 = 119889 as shown in Figure 1 This layer is heatedfrom below so that temperature at bottom (at 119911 = 0) and theupper layer (at 119911 = 119889) is119879

0and119879119889 respectively and a uniform

temperature gradient 120573 (=|119889119879119889119911|) is maintained A uniformvertical magnetic field intensity (0 0119867) and gravity force119892(0 0 minus119892) pervade the system

Journal of Fluids 3

Figure 1 Geometrical configuration

Let 119901 120588 119879 120572 119892 120578 120583119890 and 119902(119906 V 119908) denote respectively

the fluid pressure density temperature thermal coefficientof expansion gravitational acceleration resistivity magneticpermeability andfluid velocityUsing Spiegel andVeronisrsquo [6]assumptions the flow equations for compressible fluids arefound to be the same as those for incompressible fluids exceptthat in the equation of heat conduction the temperaturegradient 120573 is replaced by its excess over the adiabatic thatis (120573 minus 119892119888

119901) The equations expressing the conservation

of momentum mass temperature and equation of state ofWaltersrsquo (Model 1198611015840) fluid are

1

120598[120597 119902

120597119905+1

120598( 119902 sdot nabla) 119902]

= minus1

120588119898

nabla119901 + 119892 (1 +120575120588

120588119898

) minus1

1198961

(] minus ]1015840120597

120597119905) 119902

+120583119890

4120587120588119898

(nabla times ) times

nabla sdot 119902 = 0

119864120597119879

120597119905+ ( 119902 sdot nabla) 119879 = (120573 minus

119892

119888119901

)119908 + 120581nabla2119879

120588 = 120588119898[1 minus 120572 (119879 minus 119879

0)]

(1)

The magnetic permeability 120583119890 the kinematic viscosity ] the

kinematic viscoelasticity ]1015840 and the thermal diffusivity 120581 areall assumed to be constants Maxwellrsquos equations relevant tothe problems are

120598119889

119889119905= ( sdot nabla) 119902 + 120598120578nabla

2 minus

119888120598

4120587119873119890nabla times [(nabla times times )]

nabla sdot = 0

(2)

where 119889119889119905 = 120597120597119905+120598minus1 119902 sdotnabla stands for convective derivative

Here 119864 = 120598 + (1 minus 120598)(120588119904119888119904120588119898119888119891) is a constant and 1198641015840 is

a constant analogous to 119864 but corresponding to solute ratherthan heat 120588

119904 119888119904and 120588119898 119888119891stand for density and heat capacity

of solid (porous matrix) material and fluid respectively Thesteady state solution is

119902 = (0 0 0) 119879 = minus120573119911 + 1198790

120588 = 120588119898(1 + 120572120573119911 minus 120572

10158401205731015840119911)

(3)

Let 120575120588 120575119901 120579 ℎ(ℎ119909 ℎ119910 ℎ119911) and 119902(119906 V 119908) denote respec-

tively the perturbations in density 120588 pressure 119901 temperature119879 magnetic field (0 0119867) and filter velocity (zero initially)Then the linearized hydromagnetic perturbation equationsthrough porousmedium (Joseph [22]Walterrsquos [23] Shermanand Sutton [4] and Spiegel and Veronis [6]) relevant to theproblem are

(1

120598)120597 119902

120597119905= minus nabla(

120575119901

120588119898

) + 119892120575120588

120588119898

minus1

1198961

(120592 minus 1205921015840 120597

120597119905) 119902

+120583119890

4120587120588119898

(nabla times ℎ) times

nabla sdot 119902 = 0

120598120597ℎ

120597119905= ( sdot nabla) 119902 + 120598120578nabla

2ℎ minus

119888120598

4120587119873119890120578nabla times [(nabla times ℎ) times ]

nabla sdot ℎ = 0

119864120597120579

120597119905= (120573 minus

119892

119888119901

)119908 + 120581nabla2120579

(4)

And change in density 120575120588 caused by perturbation 120579 in tem-perature is given by

120575120588 = minus120572120588119898120579 (5)

where 120572 is the coefficient of thermal expansionWriting (4) in scalar form using (5) and eliminating 119906 V

ℎ119909 ℎ119910 and 120575119901 between them we obtain

1

120598

120597

120597119905nabla2119908 = 119892120572(

1205972

1205971199092+1205972

1205971199102)120579 minus

1

1198961

(120592 minus 1205921015840 120597

120597119905)nabla2119908

+120583119890119867

4120587120588119898

nabla2120597ℎ119911

120597119911

120598120597ℎ119911

120597119905= 119867

120597119908

120597119911+ 120598120578nabla

2ℎ119911minus

119888119867120598

4120587119873119890

120597120585

120597119911

120598120597120585

120597119905= 119867

120597120589

120597119911+ 120598120578nabla

2120585 +

119888119867120598

4120587119873119890nabla2120597ℎ119911

120597119911

1

120598

120597120589

120597119905= minus

1

1198961

(120592 minus 1205921015840 120597

120597119905) 120589 +

120583119890119867

4120587120588119898

120597120585

120597119911

119864120597120579

120597119905= (120573 minus

119892

119888119901

)119908 + 120581nabla2120579

(6)

4 Journal of Fluids

where nabla2 = 12059721205971199092 +12059721205971199102 +12059721205971199112 120589 = 120597V120597119909minus120597119906120597119910 and120585 = (120597120597119909)ℎ

119910minus(120597120597119910)ℎ

119909denote the 119911-components of vorticity

and current density respectivelyConsider the case in which both the boundaries are

free the medium adjoining the fluid is perfectly conductingand the temperatures at the boundaries are kept fixed Thecase of two free boundaries is a little artificial except instellar atmospheres (Spiegel [24]) and in certain geophysicalsituations where it is most appropriate but it allows us tohave an analytical solution It has been shown by Spiegel thatthe assumption of free boundary conditions is not a seriousone so in free boundary conditions the vertical velocity tem-perature fluctuation horizontal stress and all vanish on theboundaries The boundary conditions appropriate to theproblem are (Chandrasekhar [1])

119908 = 01205972119908

1205971199112= 0 120579 = 0

120597120589

120597119911= 0

ℎ119911= 0 at 119911 = 0 119911 = 119889

(7)

3 The Dispersion Relation

Analyzing the disturbances into normalmodes we seek solu-tions whose dependence on 119909 119910 and 119905 is given by

[119908 120579 ℎ119911 120577 120585] = [119882 (119911) Θ (119911) 119870 (119911) 119885 (119911) 119883 (119911)]

times exp (119894119896119909119909 + 119894119896

119910119910 + 119899119905)

(8)

where 119896119909 119896119910are horizontal wave numbers 119896 (=radic1198962

119909+ 1198962119910) is

the resultantwave number and 119899 is a complex constant Usingthe dimensionless variables 119886 = 119896119889 120590 = 119899119889

2] 1199011= 120592120581

1199012= 120592120578 119875

119897= 11989611198892 119865 = 120592

10158401198892 119909lowast = 119909119889 119910lowast = 119910119889 119911lowast =

119911119889 and119863 = 119889119889119911lowast and removing the stars for convenience

(5)-(6) with the help of (8) become

[120590

120598+1 minus 120590119865

119875119897

] (1198632minus 1198862)119882 +

11989212057211988921198862

]Θ

minus120583119890119867119889

4120587120588119898](1198632minus 1198862)119863119870 = 0

(120590

120598+1 minus 120590119865

119875119897

)119885 =120583119890119867119889

4120587120588119898]119863119883

(1198632minus 1198862minus 1199012120590)119870 = minus(

119867119889

120598120578)119863119882 +

119888119867119889

4120587119873119890120578119863119883

(1198632minus 1198862minus 1199012120590)119883

= minus(119867119889

120598120578)119863119885 minus

119888119867

4120587119873119890120578119889(1198632minus 1198862)119863119870

(1198632minus 1198862minus 1198641199011120590)Θ = minus(

119866 minus 1

119866)1205731198892

120581119882

(9)

From boundary conditions (7) using expression (8) we have

119882 = 1198632119882 = 0 Θ = 0 119863119885 = 0 119870 = 0

at 119911 = 0 119911 = 1(10)

Eliminating Θ 119870 119885 and119883 from (9) we obtain

(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862) (1198632minus 1198862minus 1198641199011120590)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590)2

+119876

120598(1198632minus 1198862minus 1199012120590)1198632

minus 119872(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862)1198632]119882

minus 1198771198862(119866 minus 1

119866)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590)2

+119876

120598(1198632minus 1198862minus 1199012120590)1198632

minus 119872(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862)1198632]119882

+119876

120598(1198632minus 1198862) (1198632minus 1198862minus 1198641199011120590)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590) +

119876

1205981198632]1198632119882 = 0

(11)

Here 119877 = 1198921205721205731198894120592120581 is thermal Rayleigh number 119876 =

120583119890119867211988924120587120588119898]120578 is Chandrasekhar number and 119872 = (119888119867

4120587119873119890120578)2 is nondimensional number according to Hall cur-

rentsIt can be shown with the help of (9) and boundary

conditions (10) that all the even order derivatives of 119882 vanishat the boundaries and hence the proper solution of (10) char-acterizing the lowest mode is

119882 = 119882119900sin120587119911 (12)

where119882119900is a constant Substituting (12) in (11) and letting119909 =

11988621205872 1198771= 119877120587

4 1198761= 119876120587

2 1198941205901= 120590120587

2 and 119875 = 1205872119875119897

we obtain the dispersion relation

Journal of Fluids 5

1198771119909 = (

119866

119866 minus 1)

times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909) (1 + 119909 + 119864119894119901

11205901)

times [1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909 + 119894119901

21205901)2

+1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909)]

+119876

120598(1 + 119909) (1 + 119909 + 119894119864119901

11205901) [(1 + 119909 + 119894119901

21205901) times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598]

(1 + 119909 + 11989411990121205901)2

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872 (1 + 119909)

1

119875+ 1198941205901(1

120598minus1205872119865

119875)

(13)

4 The Stationary Convection

For the case of stationary convection 120590 = 0 and (13) reducesto

1198771= (

119866

119866 minus 1)(

1 + 119909

119909)

times((1 + 119909) 119875 + 119876

1120598)2

minus119872(1 + 119909) 1198752

(1 + 119909) 119875 + 1198761120598 minus 119872119875

(14)

In order to investigate the effects of Hall current mediumpermeability and magnetic field we examine the behaviour

of 1198891198771119889119872 119889119877

1119889119875 and 119889119877

11198891198761analytically Equation

(14) yields

1198891198771

119889119872=

119866

119866 minus 1

1198761(1 + 119909)

120598119909119875

((1 + 119909) 119875 + 1198761120598)

((1 + 119909) 119875 + 1198761120598 minus119872119875)

2

(15)

which is positive The Hall current therefore had postponethe onset of thermal convection through porous medium for119866 gt 1 It is evident from (14) that

1198891198771

119889119875= minus

119866

119866 minus 1

(1 + 119909)

1199091198752

((1 + 119909) 1198752) (1 + 119909 minus119872)

2+ 21198761(1 + 119909) (1 + 119909 +119872) 120598119875 + (119876

1120598)2

(1 + 119909 +119872)

((1 + 119909 minus119872) 119875 + 1198761120598)2

1198891198771

1198891198761

=119866

119866 minus 1

(1 + 119909)

120598119909

((1 + 119909) 119875 + 1198761120598) ((1 + 119909) 119875 + 119876

1120598 minus 2119872119875) + (119872119875

2) (1 + 119909)

((1 + 119909 minus119872) 119875 + 1198761120598)2

(16)

which imply that for 119866 gt 1 medium permeability hastenpostpone the onset of convection where as magnetic fieldhas postponed the onset of convection inWaltersrsquo 1198611015840 elastico-viscous fluid through porous medium for 119876

1gt (120598119875)[2119872 minus

(1 + 119909)] and hasten postpone the onset of convection if1198761lt (120598119875)[2119872 minus (1 + 119909)] Therefore magnetic field has

duel character in presence of Hall currents through porousmedium For fixed 119875 119876

1 and 119872 let 119866 (accounting for the

compressibility effects) also be kept fixed in (14) Then wefind that

119877119888= (

119866

119866 minus 1)119877119888 (17)

where 119877119888and 119877

119888denote respectively the critical Rayleigh

numbers in the presence and absence of compressibilityThus the effect of compressibility is to postpone the onset ofthermal instability The cases 119866 lt 1 and 119866 = 1 correspondto negative and infinite values of Rayleigh number which arenot relevant in the present study 119866 gt 1 is relevant here

The compressibility therefore has postponed the onset ofconvection

5 Graphical Results and Discussion

The dispersion relation (14) in case of stationary convectionhas been computed by concerning mathematical softwareThe results have been displayed graphically for variousparameters of interest The effects of these parameters espe-cially Hall parameter medium permeability magnetic fieldRayleigh number with wave number have been studied InFigure 2 Rayleigh number 119877

1is plotted against wave num-

ber 119909 (=10ndash80) for different values of Hall parameter 119872 (=

10ndash40) and fixed values of medium permeability parameter119875 = 3 119866 = 10 magnetic field parameter 119876

1= 100 and 120598 =

05 Here we find that with the increase in the value of Hallcurrent parameter value of Rayleigh number is increasedshowing that theHall currents parameter has stabilizing effecton the system

6 Journal of Fluids

240

250

260

270

280

10 20 30 40 50 60 70 80

M = 10

M = 20

M = 30

M = 40

Wave number (x)

Rayl

eigh

num

ber (R1)

Figure 2 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119866 = 10 1198761= 100 and 120598 = 05

In Figure 3 Rayleigh number 1198771is plotted against wave

number 119909 (=1ndash5) and for different medium permeabil-ity parameter 119875 (=1 2 3 7) for fixed magnetic fieldparameter 119876

1= 100 Hall current parameter119872 = 10 119866 =

10 and 120598 = 05 are considered We find that as mediumpermeability 119875 increases value of Rayleigh number 119877

1

decreases which indicates the destabilizing effect of mediumpermeability

In Figure 4 Rayleigh number 1198771is plotted against wave

number 119909 (=10ndash80) and for different values of magneticfield parameter 119876

1(=10ndash40) for fixed values of medium

permeability 119875 = 3 Hall current parameter119872 = 10 119866 = 10

and 120598 = 05 are considered It is clear from the graph that withthe increase in the value of magnetic field parameter thereis decrease as well as increase in the Rayleigh number 119877

1

implying the destabilizing as well as stabilizing effect on thesystem

6 The Case of Overstability

In the present section we discuss the possibility as to whetherinstability may occur as overstability Since for overstabilitywe wish to determine the critical Rayleigh number for theonset of instability via a state of pure oscillations it willsuffice to find conditions forwhich (13) will admit of solutionswith 120590

1real Equating real and imaginary parts of (13) and

eliminating 1198771between them we obtain

11986031198883

1+ 11986021198882

1+ 11986011198881+ 119860119900= 0 (18)

250

300

350

400

450

500

1 2 3 4 5

Rayl

eigh

num

ber (R1)

Wave number (x)

P = 1

P = 2

P = 3

P = 7

Figure 3 Variation of Rayleigh number 1198771against wave number 119909

for 1198761= 100 119872 = 10 119866 = 10 and 120598 = 05

where

1198881= 1205902

1 119887 = 1 + 119909 (19)

1198603= 1199014

2(1

120598minus1205872119865

119875)

2

[1198641199011

119875+ 119887(

1

120598minus1205872119865

119875)] (20)

119860119900=1

119875(1

120598minus1205872119865

119875)1198875

+ [1198641199011

119875+2

119875(1198761

120598minus119872

119875)(

1

120598minus1205872119865

119875)] 1198874

+ [

[

(1198761

120598minus119872

119875)

2

(1

120598minus1205872119865

119875) +

21198641199011

1198752(1198761

120598minus119872

119875)

+1198761

1205981198752(1198641199011minus 1199012) ]

]

1198873

+ [1198721198761

1205981198752(31198641199011+ 1199012) + (

1198761

120598)

2

times 2

119875(1198641199011minus 1199012) + 119864119901

1minus119872(

1

120598minus1205872119865

119875)

+11986411990111198722

1198753] 1198872

+ (1198761

120598)

2

[1198641199011119872

119875+1198761

120598(1198641199011minus 1199012)] 119887

(21)

The three values of 1198881 1205901being real are positiveThe product

of the roots of (18) is minus11986001198603 and if this is to be positive then

Journal of Fluids 7

10

30

50

70

90

110

130

10 20 30 40 50 60 70 80

Q1 = 10

Q1 = 20

Q1 = 30

Q1 = 40

Rayl

eigh

num

ber (R1)

Wave number (x)

Figure 4 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119872 = 10 119866 = 100 and 120598 = 05

1198600lt 0 since from (20) 119860

3gt 0 if 1120598 gt 120587

2119865119875 Equation

(17) shows that this is clearly impossible if

1

120598gt1205872119865

119875 119864119901

1gt 1199012 119864119901

1gt 119872(

1

120598minus1205872119865

119875) (22)

which imply that

1205921015840lt1198961

120598 119864

120592

120581gt max[120592

120578 (

119888119867

4120587119873119890120578)

21198961minus 1205921015840120598

1198961120598

]

(23)

Thus 1205921015840 lt 1198961 120598 and 119864(120592120581) gt max[120592120578 (1198881198674120587119873119890120578)2((119896

1minus

1205921015840120598)1198961120598)] are sufficient conditions for the nonexistence of

overstability the violation of which does not necessarilyimply the occurrence of overstability

7 Concluding Remarks

Combined effect of various parameters that is magneticfield compressibility mediumpermeability and hall currentseffect has been investigated on thermal instability of aWalterrsquos 1198611015840 fluid The principle concluding remarks are as thefollowing

(i) For the stationary convection Walterrsquos 1198611015840 fluid be-haves like an ordinary Newtonian fluid due to thevanishing of the viscoelastic parameter

(ii) The presence of magnetic field (and therefore Hallcurrents) and medium permeability effects introduceoscillatory modes in the system in the absence ofthese effects the principle of exchange of stabilities isvalid

(iii) The sufficient conditions for the occurrence of over-stability are 1205921015840 lt 119896

1120598 and 119864(120592120581) gt max[120592120578 (119888119867

4120587119873119890120578)2((1198961minus1205921015840120598)1198961120598)] violation of which does not

necessarily imply the occurrence of overstability(iv) From (17) it is clear that effect of compressibility has

postponed the onset of convection

(v) To investigate the effects of medium permeabilitymagnetic permeability and Hall currents in com-pressible Walterrsquos 1198611015840 viscoelastic fluid we examinedthe expressions 119889119877

1119889119872 119889119877

1119889119875 and 119889119877

11198891198761ana-

lytically Hall current effect has postponed the onsetof convection andmedium permeability hastened theonset of convection where magnetic field has post-poned the onset of convection as well as hastened theonset of convection

Nomenclature

119892 Acceleration due to gravity (msminus2)119870 Stokersquos drag coefficient (kg sminus1)119896 Wave number (mminus1)119896119909 119896119910 Horizontal wave-numbers (mminus1)

1198961 Medium permeability (m2)

119898 Mass of single particle (g)119873 Suspended particle number

density (mminus3)119899 Growth rate (sminus1)119901 Fluid pressure (Pa)119905 Time (s) Fluid velocity (msminus1)V Suspended particle velocity (msminus1) Magnetic field intensity vector

having component (0 0119867) (G)120573(= |119889119879119889119911|) Steady adverse temperature

gradient (Kmminus1)1198731199011

Thermal Prandtl number (minus)1198731199012

Magnetic Prandtl number (minus)119877 = 119892120572120573119889

4120592120581 thermal Rayleigh number

119876 = 120583119890119867211988924120587120588119898]120578 Chandrasekhar number

119872 = (1198881198674120587119873119890120578)2 Nondimensional number

according to Hall currents119891 The mass fraction120577 119885 Component of vorticity120585 119885 Component of current density119873119862

119877and 119873119862

119877 Critical Rayleigh numbers in the

absence and presence ofcompressibility

Greek Letters

120598 Medium porosity (m0 s0 k0)120583 Dynamic viscosity (kmminus1 sminus1)1205831015840 Fluid viscoelasticity (kmminus1 sminus1)

] Kinematic viscosity (m2 sminus1)]1015840 Kinematic viscoelasticity (m2 sminus1)120588 Density (kgmminus3)

Acknowledgment

The authors are grateful to the referees for their technicalcomments and valuable suggestions resulting in a significantimprovement of the paper

8 Journal of Fluids

References

[1] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publications New York NY USA 1981

[2] H Sato ldquoThe Hall effect in the viscous flow of ionized gasbetween parallel plates under transversemagnetic fieldrdquo Journalof the Physical Society of Japan vol 16 no 7 pp 1427ndash1433 1961

[3] I Tani ldquoSteady flow of conducting fluid in channels undertransverse magnetic field with consideration of Hall EffectrdquoJournal of Aerospace Science vol 29 pp 297ndash305 1962

[4] A Sherman and G W Sutton Magnetohydrodynamics North-western University Press Evanston Ill USA 1962

[5] A S Gupta ldquoHall effects on thermal instabilityrdquo Revue Rou-maine de Mathematique Pures et Appliquees pp 665ndash677 1967

[6] E A Spiegel andGVeronisrsquo ldquoOn the Boussinesq approximationfor a compressible fluidrdquoThe Astrophysical Journal vol 131 pp442ndash447 1960

[7] K Chandra ldquoInstability of fluids heated from belowrdquo Proceed-ings of the Royal Society A vol 164 pp 231ndash242 1938

[8] J W Scanlon and L A Segel ldquoSome effects of suspended par-ticles on the onset of Benard convectionrdquo Physics of Fluids vol16 no 10 pp 1573ndash1578 1973

[9] O M Phillips Flow and Reaction in Permeable Rocks Cam-bridge University Press Cambridge UK 1991

[10] D B Ingham and I Pop Transport Phenomena in PorousMedium Pergamon Press Oxford UK 1998

[11] D A Nield and A Bejan Convection in Porous MediumSpringer New York NY USA 2nd edition 1999

[12] C R B Lister ldquoOn the thermal balance of a mid-ocean ridgerdquoGeophysics Journal of the Royal Astronomical Society Continuesvol 26 pp 515ndash535 1972

[13] J A M McDonnell Cosmic Dust John Wiley amp Sons TorontoCanada 1978

[14] R C Sharma and P Kumar ldquoRayleigh-Taylor instability oftwo superposed conducting Walterrsquos B1015840 elastico-viscous fluidsin hydromagneticsrdquo Proceedings of the National Academy ofSciences A vol 68 no 2 pp 151ndash161 1998

[15] R C Sharma ldquoMHD instability of rotating superposed fluidsthrough porous mediumrdquo Acta Physica Academiae ScientiarumHungaricae vol 42 no 1 pp 21ndash28 1977

[16] S Sunil and T Chand ldquoRayleigh-Taylor instability of plasma inpresence of a variable magnetic field and suspended particlesin porous mediumrdquo Indian Journal of Physics vol 71 no 1 pp95ndash105 1997

[17] S Sunil R C Sharma and V Sharma ldquoStability of stratifiedWalterrsquos B1015840 visco-elastic fluid in stratified porous mediumrdquoStudia Geotechnica etMechenica vol 261 no 2 pp 35ndash52 2004

[18] S Sunil R C Sharma and S Chand ldquoHall effect on thermalinstability of Rivlin-Ericksen fluidrdquo Indian Journal of Pure andApplied Mathematics vol 31 no 1 pp 49ndash59 2000

[19] M Singh ldquoHall Current effect on thermosolutal instability ina visco-elastic fluid flowing in a porous mediumrdquo InternationalJournal of Applied Mechanics and Engineering vol 16 no 1 pp69ndash82 2011

[20] M Singh and P Kumar ldquoHydrodynamic and hydromagneticstability of two stratifiedWalterrsquosB1015840 elastico-viscous superposedfluidsrdquo International Journal of AppliedMechanics and Engineer-ing vol 16 no 1 p 233 2011

[21] U Gupta P Aggarwal and R K Wanchoo ldquoThermal con-vection of dusty compressible Rivlin-Ericksen viscoelastic fluid

with Hall currentsrdquo Thermal Science vol 16 no 1 pp 177ndash1912012

[22] D D Joseph Stability of Fluid Motion II Springer New YorkNY USA 1976

[23] K Walterrsquos ldquoThe solution of flow problems in case of materialswith memoryrdquo Journal of Mecanique vol 1 pp 469ndash479 1962

[24] E A Spiegel ldquoConveive instability in a compressible atmo-sphererdquo Journal of Astrophysics vol 141 pp 1068ndash1090 1965

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

2 Journal of Fluids

was bestowed with Nobel prize of physics in 1985 In ionizedgases (plasmas) where the magnetic field is very strong andeffects the electrical conductivity cannot be Hall currents

In the aforementioned studies the medium has beenconsidered to be nonporousThe development of geothermalpower resources has increased general interest in the proper-ties of convection in porous media The effect of a magneticfield on the stability of such a flow is of interest in geophysicsparticularly in the study of Earthrsquos core where the Earthrsquosmantle which consists of conducting fluid behaves like aporous medium which can become convectively unstable asa result of differential diffusion The other application of theresults of a magnetic field is in the study of the stability of aconvective flow in the geothermal region

When the fluids are compressible the equations govern-ing the system become quite complicated to simplify Boussi-nesq tried to justify the approximation for compressible fluidswhen the density variations arise principally from thermaleffects Spiegel and Veronis [6] have simplified the set ofequations governing the flow of compressible fluids under thefollowing assumptions

(a) The depth of the fluid layer is much less than the scaleheight as defined by them

(b) The fluctuations in temperature density and pres-sure introduced due to motion do not exceed theirtotal static variations

Under the previous approximations the flow equationsare the same as those for incompressible fluids except thatthe static temperature gradient is replaced by its excess overthe adiabatic one and 119862V is replaced by 119862

119901

Chandra [7] observed a contradiction between the theoryand experiment for the onset of convection in fluids heatedfrom below He performed the experiment in an air layerand found that the instability depended on the depth of thelayer Scanlon and Segel [8] have considered the effects ofsuspended particles on the onset of Benard convection andfound that the critical Rayleigh number is reduced becauseof the heat capacity of the particles The suspended particleswere thus found to destabilize the layer The fluids have beenconsidered to be Newtonian and the medium has beenconsidered to be nonporous in all the previous studies

One class of elastico-viscous fluids isWalters fluid (model1198611015840) which is not characterized by MaxwellrsquosOldroydrsquos con-

stitutive relation When the fluid permeates a porous mate-rial the gross effect is represented by Darcyrsquos law As a resultof this macroscopic law the usual viscous and viscoelasticterms in the equation of Waltersrsquo fluid (model 1198611015840) motionare replaced by the resistance terms [minus(1119896

1)(120583minus120583

1015840(120597120597119905)) 119902]

where 120583 and 1205831015840 are the viscosity and viscoelasticity ofWaltersrsquofluid (model 1198611015840) 119896

1is the medium permeability and 119902 is the

Darcian filter velocity of the fluidThe flow through porousmedia is of considerable interest

for petroleum engineers and geophysical fluid dynamicistsA great number of applications in geophysics may be foundin the books by Phillips [9] Ingham and Pop [10] and Nieldand Bejan [11] The scientific importance of the field has alsoincreased because hydrothermal circulation is the dominantheat transfer mechanism in young oceanic crust (Lister [12])

Generally it is accepted that comets consisting of a dustyldquosnowballrdquo of a mixture of frozen gasses which is in theprocess of their journey change from solid to gas and viceversaThe physical properties of comets that meteoroids andinterplanetary dust strongly suggest the importance of poros-ity in astrophysical context have been studied by McDonnell[13]

The stability of two superposed conducting Waltersrsquo 1198611015840elastico-viscous fluids in hydromagnetics has been studiedby Sharma and Kumar [14] and whereas the instability ofstreaming Waltersrsquo viscoelastic fluid 119861

1015840 in porous mediumhas been considered by Sharma [15] Sunil and Chand [16]Sunil et al [17 18] studied the Hall effect on thermosolutalinstability of Rivlin-Ericksen and Waltersrsquo (model 1198611015840) fluidin porous medium In one study of Singh [19] Hall currenteffect on thermosolutal instability in a viscoelastic fluidflowing through porous medium and magnetic field stablesolute gradient are found to have stabilizing effects on thesystem whereas Hall current and medium permeability havea destabilizing effect on the system The sufficient conditionsfor the nonexistence of overstability have also obtained In theone another study Singh andKumar [20] hydrodynamic andhydromagnetic stability of two stratified Walterrsquos 1198611015840 elastico-viscous superposed fluids where system is stable for stablestratification and unstable for unstable stratification and incase of horizontal magnetic field system having stabilizingeffect for unstable stratification is in contrast to the stabilityof two superposed Newtonian fluids where the system isstable for stable stratifications Gupta et al [21] have studiedthermal convection of dusty compressible Rivlin-Ericksenviscoelastic fluidwithHall currents and found that compress-ibility and magnetic field postpone the onset of convectionwhereas Hall current and suspended particles hasten theonset of convection

During the survey it has been noticed that Hall effectsare completely neglected from the studies of compressibleelastico-viscous fluid through porous medium Keeping inmind the importance of Hall currents porous medium andcompressibility in elastico-viscous fluid motivated us to goon detailed study of Walterrsquos 1198611015840 fluid heated from belowthrough porous medium We have already studied earliersome problems on Hall current effect with porous as well asnonporousmedium and suspended particles found the usefuland interesting results so compressible thermal instabilityproblemofWalterrsquos1198611015840 fluidwithHall currents effects throughporous medium studied by us here

2 Mathematical Formulation of the Problem

We have considered an infinite horizontal and compressibleelectrically conducting Walterrsquos 119861

1015840 fluid permeated withporous medium in Hall current effect bounded by the planes119911 = 0 and 119911 = 119889 as shown in Figure 1 This layer is heatedfrom below so that temperature at bottom (at 119911 = 0) and theupper layer (at 119911 = 119889) is119879

0and119879119889 respectively and a uniform

temperature gradient 120573 (=|119889119879119889119911|) is maintained A uniformvertical magnetic field intensity (0 0119867) and gravity force119892(0 0 minus119892) pervade the system

Journal of Fluids 3

Figure 1 Geometrical configuration

Let 119901 120588 119879 120572 119892 120578 120583119890 and 119902(119906 V 119908) denote respectively

the fluid pressure density temperature thermal coefficientof expansion gravitational acceleration resistivity magneticpermeability andfluid velocityUsing Spiegel andVeronisrsquo [6]assumptions the flow equations for compressible fluids arefound to be the same as those for incompressible fluids exceptthat in the equation of heat conduction the temperaturegradient 120573 is replaced by its excess over the adiabatic thatis (120573 minus 119892119888

119901) The equations expressing the conservation

of momentum mass temperature and equation of state ofWaltersrsquo (Model 1198611015840) fluid are

1

120598[120597 119902

120597119905+1

120598( 119902 sdot nabla) 119902]

= minus1

120588119898

nabla119901 + 119892 (1 +120575120588

120588119898

) minus1

1198961

(] minus ]1015840120597

120597119905) 119902

+120583119890

4120587120588119898

(nabla times ) times

nabla sdot 119902 = 0

119864120597119879

120597119905+ ( 119902 sdot nabla) 119879 = (120573 minus

119892

119888119901

)119908 + 120581nabla2119879

120588 = 120588119898[1 minus 120572 (119879 minus 119879

0)]

(1)

The magnetic permeability 120583119890 the kinematic viscosity ] the

kinematic viscoelasticity ]1015840 and the thermal diffusivity 120581 areall assumed to be constants Maxwellrsquos equations relevant tothe problems are

120598119889

119889119905= ( sdot nabla) 119902 + 120598120578nabla

2 minus

119888120598

4120587119873119890nabla times [(nabla times times )]

nabla sdot = 0

(2)

where 119889119889119905 = 120597120597119905+120598minus1 119902 sdotnabla stands for convective derivative

Here 119864 = 120598 + (1 minus 120598)(120588119904119888119904120588119898119888119891) is a constant and 1198641015840 is

a constant analogous to 119864 but corresponding to solute ratherthan heat 120588

119904 119888119904and 120588119898 119888119891stand for density and heat capacity

of solid (porous matrix) material and fluid respectively Thesteady state solution is

119902 = (0 0 0) 119879 = minus120573119911 + 1198790

120588 = 120588119898(1 + 120572120573119911 minus 120572

10158401205731015840119911)

(3)

Let 120575120588 120575119901 120579 ℎ(ℎ119909 ℎ119910 ℎ119911) and 119902(119906 V 119908) denote respec-

tively the perturbations in density 120588 pressure 119901 temperature119879 magnetic field (0 0119867) and filter velocity (zero initially)Then the linearized hydromagnetic perturbation equationsthrough porousmedium (Joseph [22]Walterrsquos [23] Shermanand Sutton [4] and Spiegel and Veronis [6]) relevant to theproblem are

(1

120598)120597 119902

120597119905= minus nabla(

120575119901

120588119898

) + 119892120575120588

120588119898

minus1

1198961

(120592 minus 1205921015840 120597

120597119905) 119902

+120583119890

4120587120588119898

(nabla times ℎ) times

nabla sdot 119902 = 0

120598120597ℎ

120597119905= ( sdot nabla) 119902 + 120598120578nabla

2ℎ minus

119888120598

4120587119873119890120578nabla times [(nabla times ℎ) times ]

nabla sdot ℎ = 0

119864120597120579

120597119905= (120573 minus

119892

119888119901

)119908 + 120581nabla2120579

(4)

And change in density 120575120588 caused by perturbation 120579 in tem-perature is given by

120575120588 = minus120572120588119898120579 (5)

where 120572 is the coefficient of thermal expansionWriting (4) in scalar form using (5) and eliminating 119906 V

ℎ119909 ℎ119910 and 120575119901 between them we obtain

1

120598

120597

120597119905nabla2119908 = 119892120572(

1205972

1205971199092+1205972

1205971199102)120579 minus

1

1198961

(120592 minus 1205921015840 120597

120597119905)nabla2119908

+120583119890119867

4120587120588119898

nabla2120597ℎ119911

120597119911

120598120597ℎ119911

120597119905= 119867

120597119908

120597119911+ 120598120578nabla

2ℎ119911minus

119888119867120598

4120587119873119890

120597120585

120597119911

120598120597120585

120597119905= 119867

120597120589

120597119911+ 120598120578nabla

2120585 +

119888119867120598

4120587119873119890nabla2120597ℎ119911

120597119911

1

120598

120597120589

120597119905= minus

1

1198961

(120592 minus 1205921015840 120597

120597119905) 120589 +

120583119890119867

4120587120588119898

120597120585

120597119911

119864120597120579

120597119905= (120573 minus

119892

119888119901

)119908 + 120581nabla2120579

(6)

4 Journal of Fluids

where nabla2 = 12059721205971199092 +12059721205971199102 +12059721205971199112 120589 = 120597V120597119909minus120597119906120597119910 and120585 = (120597120597119909)ℎ

119910minus(120597120597119910)ℎ

119909denote the 119911-components of vorticity

and current density respectivelyConsider the case in which both the boundaries are

free the medium adjoining the fluid is perfectly conductingand the temperatures at the boundaries are kept fixed Thecase of two free boundaries is a little artificial except instellar atmospheres (Spiegel [24]) and in certain geophysicalsituations where it is most appropriate but it allows us tohave an analytical solution It has been shown by Spiegel thatthe assumption of free boundary conditions is not a seriousone so in free boundary conditions the vertical velocity tem-perature fluctuation horizontal stress and all vanish on theboundaries The boundary conditions appropriate to theproblem are (Chandrasekhar [1])

119908 = 01205972119908

1205971199112= 0 120579 = 0

120597120589

120597119911= 0

ℎ119911= 0 at 119911 = 0 119911 = 119889

(7)

3 The Dispersion Relation

Analyzing the disturbances into normalmodes we seek solu-tions whose dependence on 119909 119910 and 119905 is given by

[119908 120579 ℎ119911 120577 120585] = [119882 (119911) Θ (119911) 119870 (119911) 119885 (119911) 119883 (119911)]

times exp (119894119896119909119909 + 119894119896

119910119910 + 119899119905)

(8)

where 119896119909 119896119910are horizontal wave numbers 119896 (=radic1198962

119909+ 1198962119910) is

the resultantwave number and 119899 is a complex constant Usingthe dimensionless variables 119886 = 119896119889 120590 = 119899119889

2] 1199011= 120592120581

1199012= 120592120578 119875

119897= 11989611198892 119865 = 120592

10158401198892 119909lowast = 119909119889 119910lowast = 119910119889 119911lowast =

119911119889 and119863 = 119889119889119911lowast and removing the stars for convenience

(5)-(6) with the help of (8) become

[120590

120598+1 minus 120590119865

119875119897

] (1198632minus 1198862)119882 +

11989212057211988921198862

]Θ

minus120583119890119867119889

4120587120588119898](1198632minus 1198862)119863119870 = 0

(120590

120598+1 minus 120590119865

119875119897

)119885 =120583119890119867119889

4120587120588119898]119863119883

(1198632minus 1198862minus 1199012120590)119870 = minus(

119867119889

120598120578)119863119882 +

119888119867119889

4120587119873119890120578119863119883

(1198632minus 1198862minus 1199012120590)119883

= minus(119867119889

120598120578)119863119885 minus

119888119867

4120587119873119890120578119889(1198632minus 1198862)119863119870

(1198632minus 1198862minus 1198641199011120590)Θ = minus(

119866 minus 1

119866)1205731198892

120581119882

(9)

From boundary conditions (7) using expression (8) we have

119882 = 1198632119882 = 0 Θ = 0 119863119885 = 0 119870 = 0

at 119911 = 0 119911 = 1(10)

Eliminating Θ 119870 119885 and119883 from (9) we obtain

(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862) (1198632minus 1198862minus 1198641199011120590)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590)2

+119876

120598(1198632minus 1198862minus 1199012120590)1198632

minus 119872(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862)1198632]119882

minus 1198771198862(119866 minus 1

119866)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590)2

+119876

120598(1198632minus 1198862minus 1199012120590)1198632

minus 119872(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862)1198632]119882

+119876

120598(1198632minus 1198862) (1198632minus 1198862minus 1198641199011120590)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590) +

119876

1205981198632]1198632119882 = 0

(11)

Here 119877 = 1198921205721205731198894120592120581 is thermal Rayleigh number 119876 =

120583119890119867211988924120587120588119898]120578 is Chandrasekhar number and 119872 = (119888119867

4120587119873119890120578)2 is nondimensional number according to Hall cur-

rentsIt can be shown with the help of (9) and boundary

conditions (10) that all the even order derivatives of 119882 vanishat the boundaries and hence the proper solution of (10) char-acterizing the lowest mode is

119882 = 119882119900sin120587119911 (12)

where119882119900is a constant Substituting (12) in (11) and letting119909 =

11988621205872 1198771= 119877120587

4 1198761= 119876120587

2 1198941205901= 120590120587

2 and 119875 = 1205872119875119897

we obtain the dispersion relation

Journal of Fluids 5

1198771119909 = (

119866

119866 minus 1)

times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909) (1 + 119909 + 119864119894119901

11205901)

times [1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909 + 119894119901

21205901)2

+1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909)]

+119876

120598(1 + 119909) (1 + 119909 + 119894119864119901

11205901) [(1 + 119909 + 119894119901

21205901) times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598]

(1 + 119909 + 11989411990121205901)2

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872 (1 + 119909)

1

119875+ 1198941205901(1

120598minus1205872119865

119875)

(13)

4 The Stationary Convection

For the case of stationary convection 120590 = 0 and (13) reducesto

1198771= (

119866

119866 minus 1)(

1 + 119909

119909)

times((1 + 119909) 119875 + 119876

1120598)2

minus119872(1 + 119909) 1198752

(1 + 119909) 119875 + 1198761120598 minus 119872119875

(14)

In order to investigate the effects of Hall current mediumpermeability and magnetic field we examine the behaviour

of 1198891198771119889119872 119889119877

1119889119875 and 119889119877

11198891198761analytically Equation

(14) yields

1198891198771

119889119872=

119866

119866 minus 1

1198761(1 + 119909)

120598119909119875

((1 + 119909) 119875 + 1198761120598)

((1 + 119909) 119875 + 1198761120598 minus119872119875)

2

(15)

which is positive The Hall current therefore had postponethe onset of thermal convection through porous medium for119866 gt 1 It is evident from (14) that

1198891198771

119889119875= minus

119866

119866 minus 1

(1 + 119909)

1199091198752

((1 + 119909) 1198752) (1 + 119909 minus119872)

2+ 21198761(1 + 119909) (1 + 119909 +119872) 120598119875 + (119876

1120598)2

(1 + 119909 +119872)

((1 + 119909 minus119872) 119875 + 1198761120598)2

1198891198771

1198891198761

=119866

119866 minus 1

(1 + 119909)

120598119909

((1 + 119909) 119875 + 1198761120598) ((1 + 119909) 119875 + 119876

1120598 minus 2119872119875) + (119872119875

2) (1 + 119909)

((1 + 119909 minus119872) 119875 + 1198761120598)2

(16)

which imply that for 119866 gt 1 medium permeability hastenpostpone the onset of convection where as magnetic fieldhas postponed the onset of convection inWaltersrsquo 1198611015840 elastico-viscous fluid through porous medium for 119876

1gt (120598119875)[2119872 minus

(1 + 119909)] and hasten postpone the onset of convection if1198761lt (120598119875)[2119872 minus (1 + 119909)] Therefore magnetic field has

duel character in presence of Hall currents through porousmedium For fixed 119875 119876

1 and 119872 let 119866 (accounting for the

compressibility effects) also be kept fixed in (14) Then wefind that

119877119888= (

119866

119866 minus 1)119877119888 (17)

where 119877119888and 119877

119888denote respectively the critical Rayleigh

numbers in the presence and absence of compressibilityThus the effect of compressibility is to postpone the onset ofthermal instability The cases 119866 lt 1 and 119866 = 1 correspondto negative and infinite values of Rayleigh number which arenot relevant in the present study 119866 gt 1 is relevant here

The compressibility therefore has postponed the onset ofconvection

5 Graphical Results and Discussion

The dispersion relation (14) in case of stationary convectionhas been computed by concerning mathematical softwareThe results have been displayed graphically for variousparameters of interest The effects of these parameters espe-cially Hall parameter medium permeability magnetic fieldRayleigh number with wave number have been studied InFigure 2 Rayleigh number 119877

1is plotted against wave num-

ber 119909 (=10ndash80) for different values of Hall parameter 119872 (=

10ndash40) and fixed values of medium permeability parameter119875 = 3 119866 = 10 magnetic field parameter 119876

1= 100 and 120598 =

05 Here we find that with the increase in the value of Hallcurrent parameter value of Rayleigh number is increasedshowing that theHall currents parameter has stabilizing effecton the system

6 Journal of Fluids

240

250

260

270

280

10 20 30 40 50 60 70 80

M = 10

M = 20

M = 30

M = 40

Wave number (x)

Rayl

eigh

num

ber (R1)

Figure 2 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119866 = 10 1198761= 100 and 120598 = 05

In Figure 3 Rayleigh number 1198771is plotted against wave

number 119909 (=1ndash5) and for different medium permeabil-ity parameter 119875 (=1 2 3 7) for fixed magnetic fieldparameter 119876

1= 100 Hall current parameter119872 = 10 119866 =

10 and 120598 = 05 are considered We find that as mediumpermeability 119875 increases value of Rayleigh number 119877

1

decreases which indicates the destabilizing effect of mediumpermeability

In Figure 4 Rayleigh number 1198771is plotted against wave

number 119909 (=10ndash80) and for different values of magneticfield parameter 119876

1(=10ndash40) for fixed values of medium

permeability 119875 = 3 Hall current parameter119872 = 10 119866 = 10

and 120598 = 05 are considered It is clear from the graph that withthe increase in the value of magnetic field parameter thereis decrease as well as increase in the Rayleigh number 119877

1

implying the destabilizing as well as stabilizing effect on thesystem

6 The Case of Overstability

In the present section we discuss the possibility as to whetherinstability may occur as overstability Since for overstabilitywe wish to determine the critical Rayleigh number for theonset of instability via a state of pure oscillations it willsuffice to find conditions forwhich (13) will admit of solutionswith 120590

1real Equating real and imaginary parts of (13) and

eliminating 1198771between them we obtain

11986031198883

1+ 11986021198882

1+ 11986011198881+ 119860119900= 0 (18)

250

300

350

400

450

500

1 2 3 4 5

Rayl

eigh

num

ber (R1)

Wave number (x)

P = 1

P = 2

P = 3

P = 7

Figure 3 Variation of Rayleigh number 1198771against wave number 119909

for 1198761= 100 119872 = 10 119866 = 10 and 120598 = 05

where

1198881= 1205902

1 119887 = 1 + 119909 (19)

1198603= 1199014

2(1

120598minus1205872119865

119875)

2

[1198641199011

119875+ 119887(

1

120598minus1205872119865

119875)] (20)

119860119900=1

119875(1

120598minus1205872119865

119875)1198875

+ [1198641199011

119875+2

119875(1198761

120598minus119872

119875)(

1

120598minus1205872119865

119875)] 1198874

+ [

[

(1198761

120598minus119872

119875)

2

(1

120598minus1205872119865

119875) +

21198641199011

1198752(1198761

120598minus119872

119875)

+1198761

1205981198752(1198641199011minus 1199012) ]

]

1198873

+ [1198721198761

1205981198752(31198641199011+ 1199012) + (

1198761

120598)

2

times 2

119875(1198641199011minus 1199012) + 119864119901

1minus119872(

1

120598minus1205872119865

119875)

+11986411990111198722

1198753] 1198872

+ (1198761

120598)

2

[1198641199011119872

119875+1198761

120598(1198641199011minus 1199012)] 119887

(21)

The three values of 1198881 1205901being real are positiveThe product

of the roots of (18) is minus11986001198603 and if this is to be positive then

Journal of Fluids 7

10

30

50

70

90

110

130

10 20 30 40 50 60 70 80

Q1 = 10

Q1 = 20

Q1 = 30

Q1 = 40

Rayl

eigh

num

ber (R1)

Wave number (x)

Figure 4 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119872 = 10 119866 = 100 and 120598 = 05

1198600lt 0 since from (20) 119860

3gt 0 if 1120598 gt 120587

2119865119875 Equation

(17) shows that this is clearly impossible if

1

120598gt1205872119865

119875 119864119901

1gt 1199012 119864119901

1gt 119872(

1

120598minus1205872119865

119875) (22)

which imply that

1205921015840lt1198961

120598 119864

120592

120581gt max[120592

120578 (

119888119867

4120587119873119890120578)

21198961minus 1205921015840120598

1198961120598

]

(23)

Thus 1205921015840 lt 1198961 120598 and 119864(120592120581) gt max[120592120578 (1198881198674120587119873119890120578)2((119896

1minus

1205921015840120598)1198961120598)] are sufficient conditions for the nonexistence of

overstability the violation of which does not necessarilyimply the occurrence of overstability

7 Concluding Remarks

Combined effect of various parameters that is magneticfield compressibility mediumpermeability and hall currentseffect has been investigated on thermal instability of aWalterrsquos 1198611015840 fluid The principle concluding remarks are as thefollowing

(i) For the stationary convection Walterrsquos 1198611015840 fluid be-haves like an ordinary Newtonian fluid due to thevanishing of the viscoelastic parameter

(ii) The presence of magnetic field (and therefore Hallcurrents) and medium permeability effects introduceoscillatory modes in the system in the absence ofthese effects the principle of exchange of stabilities isvalid

(iii) The sufficient conditions for the occurrence of over-stability are 1205921015840 lt 119896

1120598 and 119864(120592120581) gt max[120592120578 (119888119867

4120587119873119890120578)2((1198961minus1205921015840120598)1198961120598)] violation of which does not

necessarily imply the occurrence of overstability(iv) From (17) it is clear that effect of compressibility has

postponed the onset of convection

(v) To investigate the effects of medium permeabilitymagnetic permeability and Hall currents in com-pressible Walterrsquos 1198611015840 viscoelastic fluid we examinedthe expressions 119889119877

1119889119872 119889119877

1119889119875 and 119889119877

11198891198761ana-

lytically Hall current effect has postponed the onsetof convection andmedium permeability hastened theonset of convection where magnetic field has post-poned the onset of convection as well as hastened theonset of convection

Nomenclature

119892 Acceleration due to gravity (msminus2)119870 Stokersquos drag coefficient (kg sminus1)119896 Wave number (mminus1)119896119909 119896119910 Horizontal wave-numbers (mminus1)

1198961 Medium permeability (m2)

119898 Mass of single particle (g)119873 Suspended particle number

density (mminus3)119899 Growth rate (sminus1)119901 Fluid pressure (Pa)119905 Time (s) Fluid velocity (msminus1)V Suspended particle velocity (msminus1) Magnetic field intensity vector

having component (0 0119867) (G)120573(= |119889119879119889119911|) Steady adverse temperature

gradient (Kmminus1)1198731199011

Thermal Prandtl number (minus)1198731199012

Magnetic Prandtl number (minus)119877 = 119892120572120573119889

4120592120581 thermal Rayleigh number

119876 = 120583119890119867211988924120587120588119898]120578 Chandrasekhar number

119872 = (1198881198674120587119873119890120578)2 Nondimensional number

according to Hall currents119891 The mass fraction120577 119885 Component of vorticity120585 119885 Component of current density119873119862

119877and 119873119862

119877 Critical Rayleigh numbers in the

absence and presence ofcompressibility

Greek Letters

120598 Medium porosity (m0 s0 k0)120583 Dynamic viscosity (kmminus1 sminus1)1205831015840 Fluid viscoelasticity (kmminus1 sminus1)

] Kinematic viscosity (m2 sminus1)]1015840 Kinematic viscoelasticity (m2 sminus1)120588 Density (kgmminus3)

Acknowledgment

The authors are grateful to the referees for their technicalcomments and valuable suggestions resulting in a significantimprovement of the paper

8 Journal of Fluids

References

[1] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publications New York NY USA 1981

[2] H Sato ldquoThe Hall effect in the viscous flow of ionized gasbetween parallel plates under transversemagnetic fieldrdquo Journalof the Physical Society of Japan vol 16 no 7 pp 1427ndash1433 1961

[3] I Tani ldquoSteady flow of conducting fluid in channels undertransverse magnetic field with consideration of Hall EffectrdquoJournal of Aerospace Science vol 29 pp 297ndash305 1962

[4] A Sherman and G W Sutton Magnetohydrodynamics North-western University Press Evanston Ill USA 1962

[5] A S Gupta ldquoHall effects on thermal instabilityrdquo Revue Rou-maine de Mathematique Pures et Appliquees pp 665ndash677 1967

[6] E A Spiegel andGVeronisrsquo ldquoOn the Boussinesq approximationfor a compressible fluidrdquoThe Astrophysical Journal vol 131 pp442ndash447 1960

[7] K Chandra ldquoInstability of fluids heated from belowrdquo Proceed-ings of the Royal Society A vol 164 pp 231ndash242 1938

[8] J W Scanlon and L A Segel ldquoSome effects of suspended par-ticles on the onset of Benard convectionrdquo Physics of Fluids vol16 no 10 pp 1573ndash1578 1973

[9] O M Phillips Flow and Reaction in Permeable Rocks Cam-bridge University Press Cambridge UK 1991

[10] D B Ingham and I Pop Transport Phenomena in PorousMedium Pergamon Press Oxford UK 1998

[11] D A Nield and A Bejan Convection in Porous MediumSpringer New York NY USA 2nd edition 1999

[12] C R B Lister ldquoOn the thermal balance of a mid-ocean ridgerdquoGeophysics Journal of the Royal Astronomical Society Continuesvol 26 pp 515ndash535 1972

[13] J A M McDonnell Cosmic Dust John Wiley amp Sons TorontoCanada 1978

[14] R C Sharma and P Kumar ldquoRayleigh-Taylor instability oftwo superposed conducting Walterrsquos B1015840 elastico-viscous fluidsin hydromagneticsrdquo Proceedings of the National Academy ofSciences A vol 68 no 2 pp 151ndash161 1998

[15] R C Sharma ldquoMHD instability of rotating superposed fluidsthrough porous mediumrdquo Acta Physica Academiae ScientiarumHungaricae vol 42 no 1 pp 21ndash28 1977

[16] S Sunil and T Chand ldquoRayleigh-Taylor instability of plasma inpresence of a variable magnetic field and suspended particlesin porous mediumrdquo Indian Journal of Physics vol 71 no 1 pp95ndash105 1997

[17] S Sunil R C Sharma and V Sharma ldquoStability of stratifiedWalterrsquos B1015840 visco-elastic fluid in stratified porous mediumrdquoStudia Geotechnica etMechenica vol 261 no 2 pp 35ndash52 2004

[18] S Sunil R C Sharma and S Chand ldquoHall effect on thermalinstability of Rivlin-Ericksen fluidrdquo Indian Journal of Pure andApplied Mathematics vol 31 no 1 pp 49ndash59 2000

[19] M Singh ldquoHall Current effect on thermosolutal instability ina visco-elastic fluid flowing in a porous mediumrdquo InternationalJournal of Applied Mechanics and Engineering vol 16 no 1 pp69ndash82 2011

[20] M Singh and P Kumar ldquoHydrodynamic and hydromagneticstability of two stratifiedWalterrsquosB1015840 elastico-viscous superposedfluidsrdquo International Journal of AppliedMechanics and Engineer-ing vol 16 no 1 p 233 2011

[21] U Gupta P Aggarwal and R K Wanchoo ldquoThermal con-vection of dusty compressible Rivlin-Ericksen viscoelastic fluid

with Hall currentsrdquo Thermal Science vol 16 no 1 pp 177ndash1912012

[22] D D Joseph Stability of Fluid Motion II Springer New YorkNY USA 1976

[23] K Walterrsquos ldquoThe solution of flow problems in case of materialswith memoryrdquo Journal of Mecanique vol 1 pp 469ndash479 1962

[24] E A Spiegel ldquoConveive instability in a compressible atmo-sphererdquo Journal of Astrophysics vol 141 pp 1068ndash1090 1965

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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ThermodynamicsJournal of

Journal of Fluids 3

Figure 1 Geometrical configuration

Let 119901 120588 119879 120572 119892 120578 120583119890 and 119902(119906 V 119908) denote respectively

the fluid pressure density temperature thermal coefficientof expansion gravitational acceleration resistivity magneticpermeability andfluid velocityUsing Spiegel andVeronisrsquo [6]assumptions the flow equations for compressible fluids arefound to be the same as those for incompressible fluids exceptthat in the equation of heat conduction the temperaturegradient 120573 is replaced by its excess over the adiabatic thatis (120573 minus 119892119888

119901) The equations expressing the conservation

of momentum mass temperature and equation of state ofWaltersrsquo (Model 1198611015840) fluid are

1

120598[120597 119902

120597119905+1

120598( 119902 sdot nabla) 119902]

= minus1

120588119898

nabla119901 + 119892 (1 +120575120588

120588119898

) minus1

1198961

(] minus ]1015840120597

120597119905) 119902

+120583119890

4120587120588119898

(nabla times ) times

nabla sdot 119902 = 0

119864120597119879

120597119905+ ( 119902 sdot nabla) 119879 = (120573 minus

119892

119888119901

)119908 + 120581nabla2119879

120588 = 120588119898[1 minus 120572 (119879 minus 119879

0)]

(1)

The magnetic permeability 120583119890 the kinematic viscosity ] the

kinematic viscoelasticity ]1015840 and the thermal diffusivity 120581 areall assumed to be constants Maxwellrsquos equations relevant tothe problems are

120598119889

119889119905= ( sdot nabla) 119902 + 120598120578nabla

2 minus

119888120598

4120587119873119890nabla times [(nabla times times )]

nabla sdot = 0

(2)

where 119889119889119905 = 120597120597119905+120598minus1 119902 sdotnabla stands for convective derivative

Here 119864 = 120598 + (1 minus 120598)(120588119904119888119904120588119898119888119891) is a constant and 1198641015840 is

a constant analogous to 119864 but corresponding to solute ratherthan heat 120588

119904 119888119904and 120588119898 119888119891stand for density and heat capacity

of solid (porous matrix) material and fluid respectively Thesteady state solution is

119902 = (0 0 0) 119879 = minus120573119911 + 1198790

120588 = 120588119898(1 + 120572120573119911 minus 120572

10158401205731015840119911)

(3)

Let 120575120588 120575119901 120579 ℎ(ℎ119909 ℎ119910 ℎ119911) and 119902(119906 V 119908) denote respec-

tively the perturbations in density 120588 pressure 119901 temperature119879 magnetic field (0 0119867) and filter velocity (zero initially)Then the linearized hydromagnetic perturbation equationsthrough porousmedium (Joseph [22]Walterrsquos [23] Shermanand Sutton [4] and Spiegel and Veronis [6]) relevant to theproblem are

(1

120598)120597 119902

120597119905= minus nabla(

120575119901

120588119898

) + 119892120575120588

120588119898

minus1

1198961

(120592 minus 1205921015840 120597

120597119905) 119902

+120583119890

4120587120588119898

(nabla times ℎ) times

nabla sdot 119902 = 0

120598120597ℎ

120597119905= ( sdot nabla) 119902 + 120598120578nabla

2ℎ minus

119888120598

4120587119873119890120578nabla times [(nabla times ℎ) times ]

nabla sdot ℎ = 0

119864120597120579

120597119905= (120573 minus

119892

119888119901

)119908 + 120581nabla2120579

(4)

And change in density 120575120588 caused by perturbation 120579 in tem-perature is given by

120575120588 = minus120572120588119898120579 (5)

where 120572 is the coefficient of thermal expansionWriting (4) in scalar form using (5) and eliminating 119906 V

ℎ119909 ℎ119910 and 120575119901 between them we obtain

1

120598

120597

120597119905nabla2119908 = 119892120572(

1205972

1205971199092+1205972

1205971199102)120579 minus

1

1198961

(120592 minus 1205921015840 120597

120597119905)nabla2119908

+120583119890119867

4120587120588119898

nabla2120597ℎ119911

120597119911

120598120597ℎ119911

120597119905= 119867

120597119908

120597119911+ 120598120578nabla

2ℎ119911minus

119888119867120598

4120587119873119890

120597120585

120597119911

120598120597120585

120597119905= 119867

120597120589

120597119911+ 120598120578nabla

2120585 +

119888119867120598

4120587119873119890nabla2120597ℎ119911

120597119911

1

120598

120597120589

120597119905= minus

1

1198961

(120592 minus 1205921015840 120597

120597119905) 120589 +

120583119890119867

4120587120588119898

120597120585

120597119911

119864120597120579

120597119905= (120573 minus

119892

119888119901

)119908 + 120581nabla2120579

(6)

4 Journal of Fluids

where nabla2 = 12059721205971199092 +12059721205971199102 +12059721205971199112 120589 = 120597V120597119909minus120597119906120597119910 and120585 = (120597120597119909)ℎ

119910minus(120597120597119910)ℎ

119909denote the 119911-components of vorticity

and current density respectivelyConsider the case in which both the boundaries are

free the medium adjoining the fluid is perfectly conductingand the temperatures at the boundaries are kept fixed Thecase of two free boundaries is a little artificial except instellar atmospheres (Spiegel [24]) and in certain geophysicalsituations where it is most appropriate but it allows us tohave an analytical solution It has been shown by Spiegel thatthe assumption of free boundary conditions is not a seriousone so in free boundary conditions the vertical velocity tem-perature fluctuation horizontal stress and all vanish on theboundaries The boundary conditions appropriate to theproblem are (Chandrasekhar [1])

119908 = 01205972119908

1205971199112= 0 120579 = 0

120597120589

120597119911= 0

ℎ119911= 0 at 119911 = 0 119911 = 119889

(7)

3 The Dispersion Relation

Analyzing the disturbances into normalmodes we seek solu-tions whose dependence on 119909 119910 and 119905 is given by

[119908 120579 ℎ119911 120577 120585] = [119882 (119911) Θ (119911) 119870 (119911) 119885 (119911) 119883 (119911)]

times exp (119894119896119909119909 + 119894119896

119910119910 + 119899119905)

(8)

where 119896119909 119896119910are horizontal wave numbers 119896 (=radic1198962

119909+ 1198962119910) is

the resultantwave number and 119899 is a complex constant Usingthe dimensionless variables 119886 = 119896119889 120590 = 119899119889

2] 1199011= 120592120581

1199012= 120592120578 119875

119897= 11989611198892 119865 = 120592

10158401198892 119909lowast = 119909119889 119910lowast = 119910119889 119911lowast =

119911119889 and119863 = 119889119889119911lowast and removing the stars for convenience

(5)-(6) with the help of (8) become

[120590

120598+1 minus 120590119865

119875119897

] (1198632minus 1198862)119882 +

11989212057211988921198862

]Θ

minus120583119890119867119889

4120587120588119898](1198632minus 1198862)119863119870 = 0

(120590

120598+1 minus 120590119865

119875119897

)119885 =120583119890119867119889

4120587120588119898]119863119883

(1198632minus 1198862minus 1199012120590)119870 = minus(

119867119889

120598120578)119863119882 +

119888119867119889

4120587119873119890120578119863119883

(1198632minus 1198862minus 1199012120590)119883

= minus(119867119889

120598120578)119863119885 minus

119888119867

4120587119873119890120578119889(1198632minus 1198862)119863119870

(1198632minus 1198862minus 1198641199011120590)Θ = minus(

119866 minus 1

119866)1205731198892

120581119882

(9)

From boundary conditions (7) using expression (8) we have

119882 = 1198632119882 = 0 Θ = 0 119863119885 = 0 119870 = 0

at 119911 = 0 119911 = 1(10)

Eliminating Θ 119870 119885 and119883 from (9) we obtain

(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862) (1198632minus 1198862minus 1198641199011120590)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590)2

+119876

120598(1198632minus 1198862minus 1199012120590)1198632

minus 119872(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862)1198632]119882

minus 1198771198862(119866 minus 1

119866)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590)2

+119876

120598(1198632minus 1198862minus 1199012120590)1198632

minus 119872(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862)1198632]119882

+119876

120598(1198632minus 1198862) (1198632minus 1198862minus 1198641199011120590)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590) +

119876

1205981198632]1198632119882 = 0

(11)

Here 119877 = 1198921205721205731198894120592120581 is thermal Rayleigh number 119876 =

120583119890119867211988924120587120588119898]120578 is Chandrasekhar number and 119872 = (119888119867

4120587119873119890120578)2 is nondimensional number according to Hall cur-

rentsIt can be shown with the help of (9) and boundary

conditions (10) that all the even order derivatives of 119882 vanishat the boundaries and hence the proper solution of (10) char-acterizing the lowest mode is

119882 = 119882119900sin120587119911 (12)

where119882119900is a constant Substituting (12) in (11) and letting119909 =

11988621205872 1198771= 119877120587

4 1198761= 119876120587

2 1198941205901= 120590120587

2 and 119875 = 1205872119875119897

we obtain the dispersion relation

Journal of Fluids 5

1198771119909 = (

119866

119866 minus 1)

times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909) (1 + 119909 + 119864119894119901

11205901)

times [1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909 + 119894119901

21205901)2

+1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909)]

+119876

120598(1 + 119909) (1 + 119909 + 119894119864119901

11205901) [(1 + 119909 + 119894119901

21205901) times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598]

(1 + 119909 + 11989411990121205901)2

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872 (1 + 119909)

1

119875+ 1198941205901(1

120598minus1205872119865

119875)

(13)

4 The Stationary Convection

For the case of stationary convection 120590 = 0 and (13) reducesto

1198771= (

119866

119866 minus 1)(

1 + 119909

119909)

times((1 + 119909) 119875 + 119876

1120598)2

minus119872(1 + 119909) 1198752

(1 + 119909) 119875 + 1198761120598 minus 119872119875

(14)

In order to investigate the effects of Hall current mediumpermeability and magnetic field we examine the behaviour

of 1198891198771119889119872 119889119877

1119889119875 and 119889119877

11198891198761analytically Equation

(14) yields

1198891198771

119889119872=

119866

119866 minus 1

1198761(1 + 119909)

120598119909119875

((1 + 119909) 119875 + 1198761120598)

((1 + 119909) 119875 + 1198761120598 minus119872119875)

2

(15)

which is positive The Hall current therefore had postponethe onset of thermal convection through porous medium for119866 gt 1 It is evident from (14) that

1198891198771

119889119875= minus

119866

119866 minus 1

(1 + 119909)

1199091198752

((1 + 119909) 1198752) (1 + 119909 minus119872)

2+ 21198761(1 + 119909) (1 + 119909 +119872) 120598119875 + (119876

1120598)2

(1 + 119909 +119872)

((1 + 119909 minus119872) 119875 + 1198761120598)2

1198891198771

1198891198761

=119866

119866 minus 1

(1 + 119909)

120598119909

((1 + 119909) 119875 + 1198761120598) ((1 + 119909) 119875 + 119876

1120598 minus 2119872119875) + (119872119875

2) (1 + 119909)

((1 + 119909 minus119872) 119875 + 1198761120598)2

(16)

which imply that for 119866 gt 1 medium permeability hastenpostpone the onset of convection where as magnetic fieldhas postponed the onset of convection inWaltersrsquo 1198611015840 elastico-viscous fluid through porous medium for 119876

1gt (120598119875)[2119872 minus

(1 + 119909)] and hasten postpone the onset of convection if1198761lt (120598119875)[2119872 minus (1 + 119909)] Therefore magnetic field has

duel character in presence of Hall currents through porousmedium For fixed 119875 119876

1 and 119872 let 119866 (accounting for the

compressibility effects) also be kept fixed in (14) Then wefind that

119877119888= (

119866

119866 minus 1)119877119888 (17)

where 119877119888and 119877

119888denote respectively the critical Rayleigh

numbers in the presence and absence of compressibilityThus the effect of compressibility is to postpone the onset ofthermal instability The cases 119866 lt 1 and 119866 = 1 correspondto negative and infinite values of Rayleigh number which arenot relevant in the present study 119866 gt 1 is relevant here

The compressibility therefore has postponed the onset ofconvection

5 Graphical Results and Discussion

The dispersion relation (14) in case of stationary convectionhas been computed by concerning mathematical softwareThe results have been displayed graphically for variousparameters of interest The effects of these parameters espe-cially Hall parameter medium permeability magnetic fieldRayleigh number with wave number have been studied InFigure 2 Rayleigh number 119877

1is plotted against wave num-

ber 119909 (=10ndash80) for different values of Hall parameter 119872 (=

10ndash40) and fixed values of medium permeability parameter119875 = 3 119866 = 10 magnetic field parameter 119876

1= 100 and 120598 =

05 Here we find that with the increase in the value of Hallcurrent parameter value of Rayleigh number is increasedshowing that theHall currents parameter has stabilizing effecton the system

6 Journal of Fluids

240

250

260

270

280

10 20 30 40 50 60 70 80

M = 10

M = 20

M = 30

M = 40

Wave number (x)

Rayl

eigh

num

ber (R1)

Figure 2 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119866 = 10 1198761= 100 and 120598 = 05

In Figure 3 Rayleigh number 1198771is plotted against wave

number 119909 (=1ndash5) and for different medium permeabil-ity parameter 119875 (=1 2 3 7) for fixed magnetic fieldparameter 119876

1= 100 Hall current parameter119872 = 10 119866 =

10 and 120598 = 05 are considered We find that as mediumpermeability 119875 increases value of Rayleigh number 119877

1

decreases which indicates the destabilizing effect of mediumpermeability

In Figure 4 Rayleigh number 1198771is plotted against wave

number 119909 (=10ndash80) and for different values of magneticfield parameter 119876

1(=10ndash40) for fixed values of medium

permeability 119875 = 3 Hall current parameter119872 = 10 119866 = 10

and 120598 = 05 are considered It is clear from the graph that withthe increase in the value of magnetic field parameter thereis decrease as well as increase in the Rayleigh number 119877

1

implying the destabilizing as well as stabilizing effect on thesystem

6 The Case of Overstability

In the present section we discuss the possibility as to whetherinstability may occur as overstability Since for overstabilitywe wish to determine the critical Rayleigh number for theonset of instability via a state of pure oscillations it willsuffice to find conditions forwhich (13) will admit of solutionswith 120590

1real Equating real and imaginary parts of (13) and

eliminating 1198771between them we obtain

11986031198883

1+ 11986021198882

1+ 11986011198881+ 119860119900= 0 (18)

250

300

350

400

450

500

1 2 3 4 5

Rayl

eigh

num

ber (R1)

Wave number (x)

P = 1

P = 2

P = 3

P = 7

Figure 3 Variation of Rayleigh number 1198771against wave number 119909

for 1198761= 100 119872 = 10 119866 = 10 and 120598 = 05

where

1198881= 1205902

1 119887 = 1 + 119909 (19)

1198603= 1199014

2(1

120598minus1205872119865

119875)

2

[1198641199011

119875+ 119887(

1

120598minus1205872119865

119875)] (20)

119860119900=1

119875(1

120598minus1205872119865

119875)1198875

+ [1198641199011

119875+2

119875(1198761

120598minus119872

119875)(

1

120598minus1205872119865

119875)] 1198874

+ [

[

(1198761

120598minus119872

119875)

2

(1

120598minus1205872119865

119875) +

21198641199011

1198752(1198761

120598minus119872

119875)

+1198761

1205981198752(1198641199011minus 1199012) ]

]

1198873

+ [1198721198761

1205981198752(31198641199011+ 1199012) + (

1198761

120598)

2

times 2

119875(1198641199011minus 1199012) + 119864119901

1minus119872(

1

120598minus1205872119865

119875)

+11986411990111198722

1198753] 1198872

+ (1198761

120598)

2

[1198641199011119872

119875+1198761

120598(1198641199011minus 1199012)] 119887

(21)

The three values of 1198881 1205901being real are positiveThe product

of the roots of (18) is minus11986001198603 and if this is to be positive then

Journal of Fluids 7

10

30

50

70

90

110

130

10 20 30 40 50 60 70 80

Q1 = 10

Q1 = 20

Q1 = 30

Q1 = 40

Rayl

eigh

num

ber (R1)

Wave number (x)

Figure 4 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119872 = 10 119866 = 100 and 120598 = 05

1198600lt 0 since from (20) 119860

3gt 0 if 1120598 gt 120587

2119865119875 Equation

(17) shows that this is clearly impossible if

1

120598gt1205872119865

119875 119864119901

1gt 1199012 119864119901

1gt 119872(

1

120598minus1205872119865

119875) (22)

which imply that

1205921015840lt1198961

120598 119864

120592

120581gt max[120592

120578 (

119888119867

4120587119873119890120578)

21198961minus 1205921015840120598

1198961120598

]

(23)

Thus 1205921015840 lt 1198961 120598 and 119864(120592120581) gt max[120592120578 (1198881198674120587119873119890120578)2((119896

1minus

1205921015840120598)1198961120598)] are sufficient conditions for the nonexistence of

overstability the violation of which does not necessarilyimply the occurrence of overstability

7 Concluding Remarks

Combined effect of various parameters that is magneticfield compressibility mediumpermeability and hall currentseffect has been investigated on thermal instability of aWalterrsquos 1198611015840 fluid The principle concluding remarks are as thefollowing

(i) For the stationary convection Walterrsquos 1198611015840 fluid be-haves like an ordinary Newtonian fluid due to thevanishing of the viscoelastic parameter

(ii) The presence of magnetic field (and therefore Hallcurrents) and medium permeability effects introduceoscillatory modes in the system in the absence ofthese effects the principle of exchange of stabilities isvalid

(iii) The sufficient conditions for the occurrence of over-stability are 1205921015840 lt 119896

1120598 and 119864(120592120581) gt max[120592120578 (119888119867

4120587119873119890120578)2((1198961minus1205921015840120598)1198961120598)] violation of which does not

necessarily imply the occurrence of overstability(iv) From (17) it is clear that effect of compressibility has

postponed the onset of convection

(v) To investigate the effects of medium permeabilitymagnetic permeability and Hall currents in com-pressible Walterrsquos 1198611015840 viscoelastic fluid we examinedthe expressions 119889119877

1119889119872 119889119877

1119889119875 and 119889119877

11198891198761ana-

lytically Hall current effect has postponed the onsetof convection andmedium permeability hastened theonset of convection where magnetic field has post-poned the onset of convection as well as hastened theonset of convection

Nomenclature

119892 Acceleration due to gravity (msminus2)119870 Stokersquos drag coefficient (kg sminus1)119896 Wave number (mminus1)119896119909 119896119910 Horizontal wave-numbers (mminus1)

1198961 Medium permeability (m2)

119898 Mass of single particle (g)119873 Suspended particle number

density (mminus3)119899 Growth rate (sminus1)119901 Fluid pressure (Pa)119905 Time (s) Fluid velocity (msminus1)V Suspended particle velocity (msminus1) Magnetic field intensity vector

having component (0 0119867) (G)120573(= |119889119879119889119911|) Steady adverse temperature

gradient (Kmminus1)1198731199011

Thermal Prandtl number (minus)1198731199012

Magnetic Prandtl number (minus)119877 = 119892120572120573119889

4120592120581 thermal Rayleigh number

119876 = 120583119890119867211988924120587120588119898]120578 Chandrasekhar number

119872 = (1198881198674120587119873119890120578)2 Nondimensional number

according to Hall currents119891 The mass fraction120577 119885 Component of vorticity120585 119885 Component of current density119873119862

119877and 119873119862

119877 Critical Rayleigh numbers in the

absence and presence ofcompressibility

Greek Letters

120598 Medium porosity (m0 s0 k0)120583 Dynamic viscosity (kmminus1 sminus1)1205831015840 Fluid viscoelasticity (kmminus1 sminus1)

] Kinematic viscosity (m2 sminus1)]1015840 Kinematic viscoelasticity (m2 sminus1)120588 Density (kgmminus3)

Acknowledgment

The authors are grateful to the referees for their technicalcomments and valuable suggestions resulting in a significantimprovement of the paper

8 Journal of Fluids

References

[1] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publications New York NY USA 1981

[2] H Sato ldquoThe Hall effect in the viscous flow of ionized gasbetween parallel plates under transversemagnetic fieldrdquo Journalof the Physical Society of Japan vol 16 no 7 pp 1427ndash1433 1961

[3] I Tani ldquoSteady flow of conducting fluid in channels undertransverse magnetic field with consideration of Hall EffectrdquoJournal of Aerospace Science vol 29 pp 297ndash305 1962

[4] A Sherman and G W Sutton Magnetohydrodynamics North-western University Press Evanston Ill USA 1962

[5] A S Gupta ldquoHall effects on thermal instabilityrdquo Revue Rou-maine de Mathematique Pures et Appliquees pp 665ndash677 1967

[6] E A Spiegel andGVeronisrsquo ldquoOn the Boussinesq approximationfor a compressible fluidrdquoThe Astrophysical Journal vol 131 pp442ndash447 1960

[7] K Chandra ldquoInstability of fluids heated from belowrdquo Proceed-ings of the Royal Society A vol 164 pp 231ndash242 1938

[8] J W Scanlon and L A Segel ldquoSome effects of suspended par-ticles on the onset of Benard convectionrdquo Physics of Fluids vol16 no 10 pp 1573ndash1578 1973

[9] O M Phillips Flow and Reaction in Permeable Rocks Cam-bridge University Press Cambridge UK 1991

[10] D B Ingham and I Pop Transport Phenomena in PorousMedium Pergamon Press Oxford UK 1998

[11] D A Nield and A Bejan Convection in Porous MediumSpringer New York NY USA 2nd edition 1999

[12] C R B Lister ldquoOn the thermal balance of a mid-ocean ridgerdquoGeophysics Journal of the Royal Astronomical Society Continuesvol 26 pp 515ndash535 1972

[13] J A M McDonnell Cosmic Dust John Wiley amp Sons TorontoCanada 1978

[14] R C Sharma and P Kumar ldquoRayleigh-Taylor instability oftwo superposed conducting Walterrsquos B1015840 elastico-viscous fluidsin hydromagneticsrdquo Proceedings of the National Academy ofSciences A vol 68 no 2 pp 151ndash161 1998

[15] R C Sharma ldquoMHD instability of rotating superposed fluidsthrough porous mediumrdquo Acta Physica Academiae ScientiarumHungaricae vol 42 no 1 pp 21ndash28 1977

[16] S Sunil and T Chand ldquoRayleigh-Taylor instability of plasma inpresence of a variable magnetic field and suspended particlesin porous mediumrdquo Indian Journal of Physics vol 71 no 1 pp95ndash105 1997

[17] S Sunil R C Sharma and V Sharma ldquoStability of stratifiedWalterrsquos B1015840 visco-elastic fluid in stratified porous mediumrdquoStudia Geotechnica etMechenica vol 261 no 2 pp 35ndash52 2004

[18] S Sunil R C Sharma and S Chand ldquoHall effect on thermalinstability of Rivlin-Ericksen fluidrdquo Indian Journal of Pure andApplied Mathematics vol 31 no 1 pp 49ndash59 2000

[19] M Singh ldquoHall Current effect on thermosolutal instability ina visco-elastic fluid flowing in a porous mediumrdquo InternationalJournal of Applied Mechanics and Engineering vol 16 no 1 pp69ndash82 2011

[20] M Singh and P Kumar ldquoHydrodynamic and hydromagneticstability of two stratifiedWalterrsquosB1015840 elastico-viscous superposedfluidsrdquo International Journal of AppliedMechanics and Engineer-ing vol 16 no 1 p 233 2011

[21] U Gupta P Aggarwal and R K Wanchoo ldquoThermal con-vection of dusty compressible Rivlin-Ericksen viscoelastic fluid

with Hall currentsrdquo Thermal Science vol 16 no 1 pp 177ndash1912012

[22] D D Joseph Stability of Fluid Motion II Springer New YorkNY USA 1976

[23] K Walterrsquos ldquoThe solution of flow problems in case of materialswith memoryrdquo Journal of Mecanique vol 1 pp 469ndash479 1962

[24] E A Spiegel ldquoConveive instability in a compressible atmo-sphererdquo Journal of Astrophysics vol 141 pp 1068ndash1090 1965

Submit your manuscripts athttpwwwhindawicom

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 Computational  Methods in Physics

Journal of

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ThermodynamicsJournal of

4 Journal of Fluids

where nabla2 = 12059721205971199092 +12059721205971199102 +12059721205971199112 120589 = 120597V120597119909minus120597119906120597119910 and120585 = (120597120597119909)ℎ

119910minus(120597120597119910)ℎ

119909denote the 119911-components of vorticity

and current density respectivelyConsider the case in which both the boundaries are

free the medium adjoining the fluid is perfectly conductingand the temperatures at the boundaries are kept fixed Thecase of two free boundaries is a little artificial except instellar atmospheres (Spiegel [24]) and in certain geophysicalsituations where it is most appropriate but it allows us tohave an analytical solution It has been shown by Spiegel thatthe assumption of free boundary conditions is not a seriousone so in free boundary conditions the vertical velocity tem-perature fluctuation horizontal stress and all vanish on theboundaries The boundary conditions appropriate to theproblem are (Chandrasekhar [1])

119908 = 01205972119908

1205971199112= 0 120579 = 0

120597120589

120597119911= 0

ℎ119911= 0 at 119911 = 0 119911 = 119889

(7)

3 The Dispersion Relation

Analyzing the disturbances into normalmodes we seek solu-tions whose dependence on 119909 119910 and 119905 is given by

[119908 120579 ℎ119911 120577 120585] = [119882 (119911) Θ (119911) 119870 (119911) 119885 (119911) 119883 (119911)]

times exp (119894119896119909119909 + 119894119896

119910119910 + 119899119905)

(8)

where 119896119909 119896119910are horizontal wave numbers 119896 (=radic1198962

119909+ 1198962119910) is

the resultantwave number and 119899 is a complex constant Usingthe dimensionless variables 119886 = 119896119889 120590 = 119899119889

2] 1199011= 120592120581

1199012= 120592120578 119875

119897= 11989611198892 119865 = 120592

10158401198892 119909lowast = 119909119889 119910lowast = 119910119889 119911lowast =

119911119889 and119863 = 119889119889119911lowast and removing the stars for convenience

(5)-(6) with the help of (8) become

[120590

120598+1 minus 120590119865

119875119897

] (1198632minus 1198862)119882 +

11989212057211988921198862

]Θ

minus120583119890119867119889

4120587120588119898](1198632minus 1198862)119863119870 = 0

(120590

120598+1 minus 120590119865

119875119897

)119885 =120583119890119867119889

4120587120588119898]119863119883

(1198632minus 1198862minus 1199012120590)119870 = minus(

119867119889

120598120578)119863119882 +

119888119867119889

4120587119873119890120578119863119883

(1198632minus 1198862minus 1199012120590)119883

= minus(119867119889

120598120578)119863119885 minus

119888119867

4120587119873119890120578119889(1198632minus 1198862)119863119870

(1198632minus 1198862minus 1198641199011120590)Θ = minus(

119866 minus 1

119866)1205731198892

120581119882

(9)

From boundary conditions (7) using expression (8) we have

119882 = 1198632119882 = 0 Θ = 0 119863119885 = 0 119870 = 0

at 119911 = 0 119911 = 1(10)

Eliminating Θ 119870 119885 and119883 from (9) we obtain

(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862) (1198632minus 1198862minus 1198641199011120590)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590)2

+119876

120598(1198632minus 1198862minus 1199012120590)1198632

minus 119872(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862)1198632]119882

minus 1198771198862(119866 minus 1

119866)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590)2

+119876

120598(1198632minus 1198862minus 1199012120590)1198632

minus 119872(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862)1198632]119882

+119876

120598(1198632minus 1198862) (1198632minus 1198862minus 1198641199011120590)

times [(120590

120598+1 minus 120590119865

119875119897

) (1198632minus 1198862minus 1199012120590) +

119876

1205981198632]1198632119882 = 0

(11)

Here 119877 = 1198921205721205731198894120592120581 is thermal Rayleigh number 119876 =

120583119890119867211988924120587120588119898]120578 is Chandrasekhar number and 119872 = (119888119867

4120587119873119890120578)2 is nondimensional number according to Hall cur-

rentsIt can be shown with the help of (9) and boundary

conditions (10) that all the even order derivatives of 119882 vanishat the boundaries and hence the proper solution of (10) char-acterizing the lowest mode is

119882 = 119882119900sin120587119911 (12)

where119882119900is a constant Substituting (12) in (11) and letting119909 =

11988621205872 1198771= 119877120587

4 1198761= 119876120587

2 1198941205901= 120590120587

2 and 119875 = 1205872119875119897

we obtain the dispersion relation

Journal of Fluids 5

1198771119909 = (

119866

119866 minus 1)

times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909) (1 + 119909 + 119864119894119901

11205901)

times [1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909 + 119894119901

21205901)2

+1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909)]

+119876

120598(1 + 119909) (1 + 119909 + 119894119864119901

11205901) [(1 + 119909 + 119894119901

21205901) times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598]

(1 + 119909 + 11989411990121205901)2

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872 (1 + 119909)

1

119875+ 1198941205901(1

120598minus1205872119865

119875)

(13)

4 The Stationary Convection

For the case of stationary convection 120590 = 0 and (13) reducesto

1198771= (

119866

119866 minus 1)(

1 + 119909

119909)

times((1 + 119909) 119875 + 119876

1120598)2

minus119872(1 + 119909) 1198752

(1 + 119909) 119875 + 1198761120598 minus 119872119875

(14)

In order to investigate the effects of Hall current mediumpermeability and magnetic field we examine the behaviour

of 1198891198771119889119872 119889119877

1119889119875 and 119889119877

11198891198761analytically Equation

(14) yields

1198891198771

119889119872=

119866

119866 minus 1

1198761(1 + 119909)

120598119909119875

((1 + 119909) 119875 + 1198761120598)

((1 + 119909) 119875 + 1198761120598 minus119872119875)

2

(15)

which is positive The Hall current therefore had postponethe onset of thermal convection through porous medium for119866 gt 1 It is evident from (14) that

1198891198771

119889119875= minus

119866

119866 minus 1

(1 + 119909)

1199091198752

((1 + 119909) 1198752) (1 + 119909 minus119872)

2+ 21198761(1 + 119909) (1 + 119909 +119872) 120598119875 + (119876

1120598)2

(1 + 119909 +119872)

((1 + 119909 minus119872) 119875 + 1198761120598)2

1198891198771

1198891198761

=119866

119866 minus 1

(1 + 119909)

120598119909

((1 + 119909) 119875 + 1198761120598) ((1 + 119909) 119875 + 119876

1120598 minus 2119872119875) + (119872119875

2) (1 + 119909)

((1 + 119909 minus119872) 119875 + 1198761120598)2

(16)

which imply that for 119866 gt 1 medium permeability hastenpostpone the onset of convection where as magnetic fieldhas postponed the onset of convection inWaltersrsquo 1198611015840 elastico-viscous fluid through porous medium for 119876

1gt (120598119875)[2119872 minus

(1 + 119909)] and hasten postpone the onset of convection if1198761lt (120598119875)[2119872 minus (1 + 119909)] Therefore magnetic field has

duel character in presence of Hall currents through porousmedium For fixed 119875 119876

1 and 119872 let 119866 (accounting for the

compressibility effects) also be kept fixed in (14) Then wefind that

119877119888= (

119866

119866 minus 1)119877119888 (17)

where 119877119888and 119877

119888denote respectively the critical Rayleigh

numbers in the presence and absence of compressibilityThus the effect of compressibility is to postpone the onset ofthermal instability The cases 119866 lt 1 and 119866 = 1 correspondto negative and infinite values of Rayleigh number which arenot relevant in the present study 119866 gt 1 is relevant here

The compressibility therefore has postponed the onset ofconvection

5 Graphical Results and Discussion

The dispersion relation (14) in case of stationary convectionhas been computed by concerning mathematical softwareThe results have been displayed graphically for variousparameters of interest The effects of these parameters espe-cially Hall parameter medium permeability magnetic fieldRayleigh number with wave number have been studied InFigure 2 Rayleigh number 119877

1is plotted against wave num-

ber 119909 (=10ndash80) for different values of Hall parameter 119872 (=

10ndash40) and fixed values of medium permeability parameter119875 = 3 119866 = 10 magnetic field parameter 119876

1= 100 and 120598 =

05 Here we find that with the increase in the value of Hallcurrent parameter value of Rayleigh number is increasedshowing that theHall currents parameter has stabilizing effecton the system

6 Journal of Fluids

240

250

260

270

280

10 20 30 40 50 60 70 80

M = 10

M = 20

M = 30

M = 40

Wave number (x)

Rayl

eigh

num

ber (R1)

Figure 2 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119866 = 10 1198761= 100 and 120598 = 05

In Figure 3 Rayleigh number 1198771is plotted against wave

number 119909 (=1ndash5) and for different medium permeabil-ity parameter 119875 (=1 2 3 7) for fixed magnetic fieldparameter 119876

1= 100 Hall current parameter119872 = 10 119866 =

10 and 120598 = 05 are considered We find that as mediumpermeability 119875 increases value of Rayleigh number 119877

1

decreases which indicates the destabilizing effect of mediumpermeability

In Figure 4 Rayleigh number 1198771is plotted against wave

number 119909 (=10ndash80) and for different values of magneticfield parameter 119876

1(=10ndash40) for fixed values of medium

permeability 119875 = 3 Hall current parameter119872 = 10 119866 = 10

and 120598 = 05 are considered It is clear from the graph that withthe increase in the value of magnetic field parameter thereis decrease as well as increase in the Rayleigh number 119877

1

implying the destabilizing as well as stabilizing effect on thesystem

6 The Case of Overstability

In the present section we discuss the possibility as to whetherinstability may occur as overstability Since for overstabilitywe wish to determine the critical Rayleigh number for theonset of instability via a state of pure oscillations it willsuffice to find conditions forwhich (13) will admit of solutionswith 120590

1real Equating real and imaginary parts of (13) and

eliminating 1198771between them we obtain

11986031198883

1+ 11986021198882

1+ 11986011198881+ 119860119900= 0 (18)

250

300

350

400

450

500

1 2 3 4 5

Rayl

eigh

num

ber (R1)

Wave number (x)

P = 1

P = 2

P = 3

P = 7

Figure 3 Variation of Rayleigh number 1198771against wave number 119909

for 1198761= 100 119872 = 10 119866 = 10 and 120598 = 05

where

1198881= 1205902

1 119887 = 1 + 119909 (19)

1198603= 1199014

2(1

120598minus1205872119865

119875)

2

[1198641199011

119875+ 119887(

1

120598minus1205872119865

119875)] (20)

119860119900=1

119875(1

120598minus1205872119865

119875)1198875

+ [1198641199011

119875+2

119875(1198761

120598minus119872

119875)(

1

120598minus1205872119865

119875)] 1198874

+ [

[

(1198761

120598minus119872

119875)

2

(1

120598minus1205872119865

119875) +

21198641199011

1198752(1198761

120598minus119872

119875)

+1198761

1205981198752(1198641199011minus 1199012) ]

]

1198873

+ [1198721198761

1205981198752(31198641199011+ 1199012) + (

1198761

120598)

2

times 2

119875(1198641199011minus 1199012) + 119864119901

1minus119872(

1

120598minus1205872119865

119875)

+11986411990111198722

1198753] 1198872

+ (1198761

120598)

2

[1198641199011119872

119875+1198761

120598(1198641199011minus 1199012)] 119887

(21)

The three values of 1198881 1205901being real are positiveThe product

of the roots of (18) is minus11986001198603 and if this is to be positive then

Journal of Fluids 7

10

30

50

70

90

110

130

10 20 30 40 50 60 70 80

Q1 = 10

Q1 = 20

Q1 = 30

Q1 = 40

Rayl

eigh

num

ber (R1)

Wave number (x)

Figure 4 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119872 = 10 119866 = 100 and 120598 = 05

1198600lt 0 since from (20) 119860

3gt 0 if 1120598 gt 120587

2119865119875 Equation

(17) shows that this is clearly impossible if

1

120598gt1205872119865

119875 119864119901

1gt 1199012 119864119901

1gt 119872(

1

120598minus1205872119865

119875) (22)

which imply that

1205921015840lt1198961

120598 119864

120592

120581gt max[120592

120578 (

119888119867

4120587119873119890120578)

21198961minus 1205921015840120598

1198961120598

]

(23)

Thus 1205921015840 lt 1198961 120598 and 119864(120592120581) gt max[120592120578 (1198881198674120587119873119890120578)2((119896

1minus

1205921015840120598)1198961120598)] are sufficient conditions for the nonexistence of

overstability the violation of which does not necessarilyimply the occurrence of overstability

7 Concluding Remarks

Combined effect of various parameters that is magneticfield compressibility mediumpermeability and hall currentseffect has been investigated on thermal instability of aWalterrsquos 1198611015840 fluid The principle concluding remarks are as thefollowing

(i) For the stationary convection Walterrsquos 1198611015840 fluid be-haves like an ordinary Newtonian fluid due to thevanishing of the viscoelastic parameter

(ii) The presence of magnetic field (and therefore Hallcurrents) and medium permeability effects introduceoscillatory modes in the system in the absence ofthese effects the principle of exchange of stabilities isvalid

(iii) The sufficient conditions for the occurrence of over-stability are 1205921015840 lt 119896

1120598 and 119864(120592120581) gt max[120592120578 (119888119867

4120587119873119890120578)2((1198961minus1205921015840120598)1198961120598)] violation of which does not

necessarily imply the occurrence of overstability(iv) From (17) it is clear that effect of compressibility has

postponed the onset of convection

(v) To investigate the effects of medium permeabilitymagnetic permeability and Hall currents in com-pressible Walterrsquos 1198611015840 viscoelastic fluid we examinedthe expressions 119889119877

1119889119872 119889119877

1119889119875 and 119889119877

11198891198761ana-

lytically Hall current effect has postponed the onsetof convection andmedium permeability hastened theonset of convection where magnetic field has post-poned the onset of convection as well as hastened theonset of convection

Nomenclature

119892 Acceleration due to gravity (msminus2)119870 Stokersquos drag coefficient (kg sminus1)119896 Wave number (mminus1)119896119909 119896119910 Horizontal wave-numbers (mminus1)

1198961 Medium permeability (m2)

119898 Mass of single particle (g)119873 Suspended particle number

density (mminus3)119899 Growth rate (sminus1)119901 Fluid pressure (Pa)119905 Time (s) Fluid velocity (msminus1)V Suspended particle velocity (msminus1) Magnetic field intensity vector

having component (0 0119867) (G)120573(= |119889119879119889119911|) Steady adverse temperature

gradient (Kmminus1)1198731199011

Thermal Prandtl number (minus)1198731199012

Magnetic Prandtl number (minus)119877 = 119892120572120573119889

4120592120581 thermal Rayleigh number

119876 = 120583119890119867211988924120587120588119898]120578 Chandrasekhar number

119872 = (1198881198674120587119873119890120578)2 Nondimensional number

according to Hall currents119891 The mass fraction120577 119885 Component of vorticity120585 119885 Component of current density119873119862

119877and 119873119862

119877 Critical Rayleigh numbers in the

absence and presence ofcompressibility

Greek Letters

120598 Medium porosity (m0 s0 k0)120583 Dynamic viscosity (kmminus1 sminus1)1205831015840 Fluid viscoelasticity (kmminus1 sminus1)

] Kinematic viscosity (m2 sminus1)]1015840 Kinematic viscoelasticity (m2 sminus1)120588 Density (kgmminus3)

Acknowledgment

The authors are grateful to the referees for their technicalcomments and valuable suggestions resulting in a significantimprovement of the paper

8 Journal of Fluids

References

[1] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publications New York NY USA 1981

[2] H Sato ldquoThe Hall effect in the viscous flow of ionized gasbetween parallel plates under transversemagnetic fieldrdquo Journalof the Physical Society of Japan vol 16 no 7 pp 1427ndash1433 1961

[3] I Tani ldquoSteady flow of conducting fluid in channels undertransverse magnetic field with consideration of Hall EffectrdquoJournal of Aerospace Science vol 29 pp 297ndash305 1962

[4] A Sherman and G W Sutton Magnetohydrodynamics North-western University Press Evanston Ill USA 1962

[5] A S Gupta ldquoHall effects on thermal instabilityrdquo Revue Rou-maine de Mathematique Pures et Appliquees pp 665ndash677 1967

[6] E A Spiegel andGVeronisrsquo ldquoOn the Boussinesq approximationfor a compressible fluidrdquoThe Astrophysical Journal vol 131 pp442ndash447 1960

[7] K Chandra ldquoInstability of fluids heated from belowrdquo Proceed-ings of the Royal Society A vol 164 pp 231ndash242 1938

[8] J W Scanlon and L A Segel ldquoSome effects of suspended par-ticles on the onset of Benard convectionrdquo Physics of Fluids vol16 no 10 pp 1573ndash1578 1973

[9] O M Phillips Flow and Reaction in Permeable Rocks Cam-bridge University Press Cambridge UK 1991

[10] D B Ingham and I Pop Transport Phenomena in PorousMedium Pergamon Press Oxford UK 1998

[11] D A Nield and A Bejan Convection in Porous MediumSpringer New York NY USA 2nd edition 1999

[12] C R B Lister ldquoOn the thermal balance of a mid-ocean ridgerdquoGeophysics Journal of the Royal Astronomical Society Continuesvol 26 pp 515ndash535 1972

[13] J A M McDonnell Cosmic Dust John Wiley amp Sons TorontoCanada 1978

[14] R C Sharma and P Kumar ldquoRayleigh-Taylor instability oftwo superposed conducting Walterrsquos B1015840 elastico-viscous fluidsin hydromagneticsrdquo Proceedings of the National Academy ofSciences A vol 68 no 2 pp 151ndash161 1998

[15] R C Sharma ldquoMHD instability of rotating superposed fluidsthrough porous mediumrdquo Acta Physica Academiae ScientiarumHungaricae vol 42 no 1 pp 21ndash28 1977

[16] S Sunil and T Chand ldquoRayleigh-Taylor instability of plasma inpresence of a variable magnetic field and suspended particlesin porous mediumrdquo Indian Journal of Physics vol 71 no 1 pp95ndash105 1997

[17] S Sunil R C Sharma and V Sharma ldquoStability of stratifiedWalterrsquos B1015840 visco-elastic fluid in stratified porous mediumrdquoStudia Geotechnica etMechenica vol 261 no 2 pp 35ndash52 2004

[18] S Sunil R C Sharma and S Chand ldquoHall effect on thermalinstability of Rivlin-Ericksen fluidrdquo Indian Journal of Pure andApplied Mathematics vol 31 no 1 pp 49ndash59 2000

[19] M Singh ldquoHall Current effect on thermosolutal instability ina visco-elastic fluid flowing in a porous mediumrdquo InternationalJournal of Applied Mechanics and Engineering vol 16 no 1 pp69ndash82 2011

[20] M Singh and P Kumar ldquoHydrodynamic and hydromagneticstability of two stratifiedWalterrsquosB1015840 elastico-viscous superposedfluidsrdquo International Journal of AppliedMechanics and Engineer-ing vol 16 no 1 p 233 2011

[21] U Gupta P Aggarwal and R K Wanchoo ldquoThermal con-vection of dusty compressible Rivlin-Ericksen viscoelastic fluid

with Hall currentsrdquo Thermal Science vol 16 no 1 pp 177ndash1912012

[22] D D Joseph Stability of Fluid Motion II Springer New YorkNY USA 1976

[23] K Walterrsquos ldquoThe solution of flow problems in case of materialswith memoryrdquo Journal of Mecanique vol 1 pp 469ndash479 1962

[24] E A Spiegel ldquoConveive instability in a compressible atmo-sphererdquo Journal of Astrophysics vol 141 pp 1068ndash1090 1965

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Superconductivity

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 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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Volume 2014

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Biophysics

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ThermodynamicsJournal of

Journal of Fluids 5

1198771119909 = (

119866

119866 minus 1)

times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909) (1 + 119909 + 119864119894119901

11205901)

times [1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909 + 119894119901

21205901)2

+1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872

1

119875+ 1198941205901(1

120598minus1205872119865

119875) (1 + 119909)]

+119876

120598(1 + 119909) (1 + 119909 + 119894119864119901

11205901) [(1 + 119909 + 119894119901

21205901) times

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598]

(1 + 119909 + 11989411990121205901)2

1

119875+ 1198941205901(1

120598minus1205872119865

119875) +

1198761

120598(1 + 119909 + 119894119901

21205901) minus 119872 (1 + 119909)

1

119875+ 1198941205901(1

120598minus1205872119865

119875)

(13)

4 The Stationary Convection

For the case of stationary convection 120590 = 0 and (13) reducesto

1198771= (

119866

119866 minus 1)(

1 + 119909

119909)

times((1 + 119909) 119875 + 119876

1120598)2

minus119872(1 + 119909) 1198752

(1 + 119909) 119875 + 1198761120598 minus 119872119875

(14)

In order to investigate the effects of Hall current mediumpermeability and magnetic field we examine the behaviour

of 1198891198771119889119872 119889119877

1119889119875 and 119889119877

11198891198761analytically Equation

(14) yields

1198891198771

119889119872=

119866

119866 minus 1

1198761(1 + 119909)

120598119909119875

((1 + 119909) 119875 + 1198761120598)

((1 + 119909) 119875 + 1198761120598 minus119872119875)

2

(15)

which is positive The Hall current therefore had postponethe onset of thermal convection through porous medium for119866 gt 1 It is evident from (14) that

1198891198771

119889119875= minus

119866

119866 minus 1

(1 + 119909)

1199091198752

((1 + 119909) 1198752) (1 + 119909 minus119872)

2+ 21198761(1 + 119909) (1 + 119909 +119872) 120598119875 + (119876

1120598)2

(1 + 119909 +119872)

((1 + 119909 minus119872) 119875 + 1198761120598)2

1198891198771

1198891198761

=119866

119866 minus 1

(1 + 119909)

120598119909

((1 + 119909) 119875 + 1198761120598) ((1 + 119909) 119875 + 119876

1120598 minus 2119872119875) + (119872119875

2) (1 + 119909)

((1 + 119909 minus119872) 119875 + 1198761120598)2

(16)

which imply that for 119866 gt 1 medium permeability hastenpostpone the onset of convection where as magnetic fieldhas postponed the onset of convection inWaltersrsquo 1198611015840 elastico-viscous fluid through porous medium for 119876

1gt (120598119875)[2119872 minus

(1 + 119909)] and hasten postpone the onset of convection if1198761lt (120598119875)[2119872 minus (1 + 119909)] Therefore magnetic field has

duel character in presence of Hall currents through porousmedium For fixed 119875 119876

1 and 119872 let 119866 (accounting for the

compressibility effects) also be kept fixed in (14) Then wefind that

119877119888= (

119866

119866 minus 1)119877119888 (17)

where 119877119888and 119877

119888denote respectively the critical Rayleigh

numbers in the presence and absence of compressibilityThus the effect of compressibility is to postpone the onset ofthermal instability The cases 119866 lt 1 and 119866 = 1 correspondto negative and infinite values of Rayleigh number which arenot relevant in the present study 119866 gt 1 is relevant here

The compressibility therefore has postponed the onset ofconvection

5 Graphical Results and Discussion

The dispersion relation (14) in case of stationary convectionhas been computed by concerning mathematical softwareThe results have been displayed graphically for variousparameters of interest The effects of these parameters espe-cially Hall parameter medium permeability magnetic fieldRayleigh number with wave number have been studied InFigure 2 Rayleigh number 119877

1is plotted against wave num-

ber 119909 (=10ndash80) for different values of Hall parameter 119872 (=

10ndash40) and fixed values of medium permeability parameter119875 = 3 119866 = 10 magnetic field parameter 119876

1= 100 and 120598 =

05 Here we find that with the increase in the value of Hallcurrent parameter value of Rayleigh number is increasedshowing that theHall currents parameter has stabilizing effecton the system

6 Journal of Fluids

240

250

260

270

280

10 20 30 40 50 60 70 80

M = 10

M = 20

M = 30

M = 40

Wave number (x)

Rayl

eigh

num

ber (R1)

Figure 2 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119866 = 10 1198761= 100 and 120598 = 05

In Figure 3 Rayleigh number 1198771is plotted against wave

number 119909 (=1ndash5) and for different medium permeabil-ity parameter 119875 (=1 2 3 7) for fixed magnetic fieldparameter 119876

1= 100 Hall current parameter119872 = 10 119866 =

10 and 120598 = 05 are considered We find that as mediumpermeability 119875 increases value of Rayleigh number 119877

1

decreases which indicates the destabilizing effect of mediumpermeability

In Figure 4 Rayleigh number 1198771is plotted against wave

number 119909 (=10ndash80) and for different values of magneticfield parameter 119876

1(=10ndash40) for fixed values of medium

permeability 119875 = 3 Hall current parameter119872 = 10 119866 = 10

and 120598 = 05 are considered It is clear from the graph that withthe increase in the value of magnetic field parameter thereis decrease as well as increase in the Rayleigh number 119877

1

implying the destabilizing as well as stabilizing effect on thesystem

6 The Case of Overstability

In the present section we discuss the possibility as to whetherinstability may occur as overstability Since for overstabilitywe wish to determine the critical Rayleigh number for theonset of instability via a state of pure oscillations it willsuffice to find conditions forwhich (13) will admit of solutionswith 120590

1real Equating real and imaginary parts of (13) and

eliminating 1198771between them we obtain

11986031198883

1+ 11986021198882

1+ 11986011198881+ 119860119900= 0 (18)

250

300

350

400

450

500

1 2 3 4 5

Rayl

eigh

num

ber (R1)

Wave number (x)

P = 1

P = 2

P = 3

P = 7

Figure 3 Variation of Rayleigh number 1198771against wave number 119909

for 1198761= 100 119872 = 10 119866 = 10 and 120598 = 05

where

1198881= 1205902

1 119887 = 1 + 119909 (19)

1198603= 1199014

2(1

120598minus1205872119865

119875)

2

[1198641199011

119875+ 119887(

1

120598minus1205872119865

119875)] (20)

119860119900=1

119875(1

120598minus1205872119865

119875)1198875

+ [1198641199011

119875+2

119875(1198761

120598minus119872

119875)(

1

120598minus1205872119865

119875)] 1198874

+ [

[

(1198761

120598minus119872

119875)

2

(1

120598minus1205872119865

119875) +

21198641199011

1198752(1198761

120598minus119872

119875)

+1198761

1205981198752(1198641199011minus 1199012) ]

]

1198873

+ [1198721198761

1205981198752(31198641199011+ 1199012) + (

1198761

120598)

2

times 2

119875(1198641199011minus 1199012) + 119864119901

1minus119872(

1

120598minus1205872119865

119875)

+11986411990111198722

1198753] 1198872

+ (1198761

120598)

2

[1198641199011119872

119875+1198761

120598(1198641199011minus 1199012)] 119887

(21)

The three values of 1198881 1205901being real are positiveThe product

of the roots of (18) is minus11986001198603 and if this is to be positive then

Journal of Fluids 7

10

30

50

70

90

110

130

10 20 30 40 50 60 70 80

Q1 = 10

Q1 = 20

Q1 = 30

Q1 = 40

Rayl

eigh

num

ber (R1)

Wave number (x)

Figure 4 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119872 = 10 119866 = 100 and 120598 = 05

1198600lt 0 since from (20) 119860

3gt 0 if 1120598 gt 120587

2119865119875 Equation

(17) shows that this is clearly impossible if

1

120598gt1205872119865

119875 119864119901

1gt 1199012 119864119901

1gt 119872(

1

120598minus1205872119865

119875) (22)

which imply that

1205921015840lt1198961

120598 119864

120592

120581gt max[120592

120578 (

119888119867

4120587119873119890120578)

21198961minus 1205921015840120598

1198961120598

]

(23)

Thus 1205921015840 lt 1198961 120598 and 119864(120592120581) gt max[120592120578 (1198881198674120587119873119890120578)2((119896

1minus

1205921015840120598)1198961120598)] are sufficient conditions for the nonexistence of

overstability the violation of which does not necessarilyimply the occurrence of overstability

7 Concluding Remarks

Combined effect of various parameters that is magneticfield compressibility mediumpermeability and hall currentseffect has been investigated on thermal instability of aWalterrsquos 1198611015840 fluid The principle concluding remarks are as thefollowing

(i) For the stationary convection Walterrsquos 1198611015840 fluid be-haves like an ordinary Newtonian fluid due to thevanishing of the viscoelastic parameter

(ii) The presence of magnetic field (and therefore Hallcurrents) and medium permeability effects introduceoscillatory modes in the system in the absence ofthese effects the principle of exchange of stabilities isvalid

(iii) The sufficient conditions for the occurrence of over-stability are 1205921015840 lt 119896

1120598 and 119864(120592120581) gt max[120592120578 (119888119867

4120587119873119890120578)2((1198961minus1205921015840120598)1198961120598)] violation of which does not

necessarily imply the occurrence of overstability(iv) From (17) it is clear that effect of compressibility has

postponed the onset of convection

(v) To investigate the effects of medium permeabilitymagnetic permeability and Hall currents in com-pressible Walterrsquos 1198611015840 viscoelastic fluid we examinedthe expressions 119889119877

1119889119872 119889119877

1119889119875 and 119889119877

11198891198761ana-

lytically Hall current effect has postponed the onsetof convection andmedium permeability hastened theonset of convection where magnetic field has post-poned the onset of convection as well as hastened theonset of convection

Nomenclature

119892 Acceleration due to gravity (msminus2)119870 Stokersquos drag coefficient (kg sminus1)119896 Wave number (mminus1)119896119909 119896119910 Horizontal wave-numbers (mminus1)

1198961 Medium permeability (m2)

119898 Mass of single particle (g)119873 Suspended particle number

density (mminus3)119899 Growth rate (sminus1)119901 Fluid pressure (Pa)119905 Time (s) Fluid velocity (msminus1)V Suspended particle velocity (msminus1) Magnetic field intensity vector

having component (0 0119867) (G)120573(= |119889119879119889119911|) Steady adverse temperature

gradient (Kmminus1)1198731199011

Thermal Prandtl number (minus)1198731199012

Magnetic Prandtl number (minus)119877 = 119892120572120573119889

4120592120581 thermal Rayleigh number

119876 = 120583119890119867211988924120587120588119898]120578 Chandrasekhar number

119872 = (1198881198674120587119873119890120578)2 Nondimensional number

according to Hall currents119891 The mass fraction120577 119885 Component of vorticity120585 119885 Component of current density119873119862

119877and 119873119862

119877 Critical Rayleigh numbers in the

absence and presence ofcompressibility

Greek Letters

120598 Medium porosity (m0 s0 k0)120583 Dynamic viscosity (kmminus1 sminus1)1205831015840 Fluid viscoelasticity (kmminus1 sminus1)

] Kinematic viscosity (m2 sminus1)]1015840 Kinematic viscoelasticity (m2 sminus1)120588 Density (kgmminus3)

Acknowledgment

The authors are grateful to the referees for their technicalcomments and valuable suggestions resulting in a significantimprovement of the paper

8 Journal of Fluids

References

[1] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publications New York NY USA 1981

[2] H Sato ldquoThe Hall effect in the viscous flow of ionized gasbetween parallel plates under transversemagnetic fieldrdquo Journalof the Physical Society of Japan vol 16 no 7 pp 1427ndash1433 1961

[3] I Tani ldquoSteady flow of conducting fluid in channels undertransverse magnetic field with consideration of Hall EffectrdquoJournal of Aerospace Science vol 29 pp 297ndash305 1962

[4] A Sherman and G W Sutton Magnetohydrodynamics North-western University Press Evanston Ill USA 1962

[5] A S Gupta ldquoHall effects on thermal instabilityrdquo Revue Rou-maine de Mathematique Pures et Appliquees pp 665ndash677 1967

[6] E A Spiegel andGVeronisrsquo ldquoOn the Boussinesq approximationfor a compressible fluidrdquoThe Astrophysical Journal vol 131 pp442ndash447 1960

[7] K Chandra ldquoInstability of fluids heated from belowrdquo Proceed-ings of the Royal Society A vol 164 pp 231ndash242 1938

[8] J W Scanlon and L A Segel ldquoSome effects of suspended par-ticles on the onset of Benard convectionrdquo Physics of Fluids vol16 no 10 pp 1573ndash1578 1973

[9] O M Phillips Flow and Reaction in Permeable Rocks Cam-bridge University Press Cambridge UK 1991

[10] D B Ingham and I Pop Transport Phenomena in PorousMedium Pergamon Press Oxford UK 1998

[11] D A Nield and A Bejan Convection in Porous MediumSpringer New York NY USA 2nd edition 1999

[12] C R B Lister ldquoOn the thermal balance of a mid-ocean ridgerdquoGeophysics Journal of the Royal Astronomical Society Continuesvol 26 pp 515ndash535 1972

[13] J A M McDonnell Cosmic Dust John Wiley amp Sons TorontoCanada 1978

[14] R C Sharma and P Kumar ldquoRayleigh-Taylor instability oftwo superposed conducting Walterrsquos B1015840 elastico-viscous fluidsin hydromagneticsrdquo Proceedings of the National Academy ofSciences A vol 68 no 2 pp 151ndash161 1998

[15] R C Sharma ldquoMHD instability of rotating superposed fluidsthrough porous mediumrdquo Acta Physica Academiae ScientiarumHungaricae vol 42 no 1 pp 21ndash28 1977

[16] S Sunil and T Chand ldquoRayleigh-Taylor instability of plasma inpresence of a variable magnetic field and suspended particlesin porous mediumrdquo Indian Journal of Physics vol 71 no 1 pp95ndash105 1997

[17] S Sunil R C Sharma and V Sharma ldquoStability of stratifiedWalterrsquos B1015840 visco-elastic fluid in stratified porous mediumrdquoStudia Geotechnica etMechenica vol 261 no 2 pp 35ndash52 2004

[18] S Sunil R C Sharma and S Chand ldquoHall effect on thermalinstability of Rivlin-Ericksen fluidrdquo Indian Journal of Pure andApplied Mathematics vol 31 no 1 pp 49ndash59 2000

[19] M Singh ldquoHall Current effect on thermosolutal instability ina visco-elastic fluid flowing in a porous mediumrdquo InternationalJournal of Applied Mechanics and Engineering vol 16 no 1 pp69ndash82 2011

[20] M Singh and P Kumar ldquoHydrodynamic and hydromagneticstability of two stratifiedWalterrsquosB1015840 elastico-viscous superposedfluidsrdquo International Journal of AppliedMechanics and Engineer-ing vol 16 no 1 p 233 2011

[21] U Gupta P Aggarwal and R K Wanchoo ldquoThermal con-vection of dusty compressible Rivlin-Ericksen viscoelastic fluid

with Hall currentsrdquo Thermal Science vol 16 no 1 pp 177ndash1912012

[22] D D Joseph Stability of Fluid Motion II Springer New YorkNY USA 1976

[23] K Walterrsquos ldquoThe solution of flow problems in case of materialswith memoryrdquo Journal of Mecanique vol 1 pp 469ndash479 1962

[24] E A Spiegel ldquoConveive instability in a compressible atmo-sphererdquo Journal of Astrophysics vol 141 pp 1068ndash1090 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Journal of Fluids

240

250

260

270

280

10 20 30 40 50 60 70 80

M = 10

M = 20

M = 30

M = 40

Wave number (x)

Rayl

eigh

num

ber (R1)

Figure 2 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119866 = 10 1198761= 100 and 120598 = 05

In Figure 3 Rayleigh number 1198771is plotted against wave

number 119909 (=1ndash5) and for different medium permeabil-ity parameter 119875 (=1 2 3 7) for fixed magnetic fieldparameter 119876

1= 100 Hall current parameter119872 = 10 119866 =

10 and 120598 = 05 are considered We find that as mediumpermeability 119875 increases value of Rayleigh number 119877

1

decreases which indicates the destabilizing effect of mediumpermeability

In Figure 4 Rayleigh number 1198771is plotted against wave

number 119909 (=10ndash80) and for different values of magneticfield parameter 119876

1(=10ndash40) for fixed values of medium

permeability 119875 = 3 Hall current parameter119872 = 10 119866 = 10

and 120598 = 05 are considered It is clear from the graph that withthe increase in the value of magnetic field parameter thereis decrease as well as increase in the Rayleigh number 119877

1

implying the destabilizing as well as stabilizing effect on thesystem

6 The Case of Overstability

In the present section we discuss the possibility as to whetherinstability may occur as overstability Since for overstabilitywe wish to determine the critical Rayleigh number for theonset of instability via a state of pure oscillations it willsuffice to find conditions forwhich (13) will admit of solutionswith 120590

1real Equating real and imaginary parts of (13) and

eliminating 1198771between them we obtain

11986031198883

1+ 11986021198882

1+ 11986011198881+ 119860119900= 0 (18)

250

300

350

400

450

500

1 2 3 4 5

Rayl

eigh

num

ber (R1)

Wave number (x)

P = 1

P = 2

P = 3

P = 7

Figure 3 Variation of Rayleigh number 1198771against wave number 119909

for 1198761= 100 119872 = 10 119866 = 10 and 120598 = 05

where

1198881= 1205902

1 119887 = 1 + 119909 (19)

1198603= 1199014

2(1

120598minus1205872119865

119875)

2

[1198641199011

119875+ 119887(

1

120598minus1205872119865

119875)] (20)

119860119900=1

119875(1

120598minus1205872119865

119875)1198875

+ [1198641199011

119875+2

119875(1198761

120598minus119872

119875)(

1

120598minus1205872119865

119875)] 1198874

+ [

[

(1198761

120598minus119872

119875)

2

(1

120598minus1205872119865

119875) +

21198641199011

1198752(1198761

120598minus119872

119875)

+1198761

1205981198752(1198641199011minus 1199012) ]

]

1198873

+ [1198721198761

1205981198752(31198641199011+ 1199012) + (

1198761

120598)

2

times 2

119875(1198641199011minus 1199012) + 119864119901

1minus119872(

1

120598minus1205872119865

119875)

+11986411990111198722

1198753] 1198872

+ (1198761

120598)

2

[1198641199011119872

119875+1198761

120598(1198641199011minus 1199012)] 119887

(21)

The three values of 1198881 1205901being real are positiveThe product

of the roots of (18) is minus11986001198603 and if this is to be positive then

Journal of Fluids 7

10

30

50

70

90

110

130

10 20 30 40 50 60 70 80

Q1 = 10

Q1 = 20

Q1 = 30

Q1 = 40

Rayl

eigh

num

ber (R1)

Wave number (x)

Figure 4 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119872 = 10 119866 = 100 and 120598 = 05

1198600lt 0 since from (20) 119860

3gt 0 if 1120598 gt 120587

2119865119875 Equation

(17) shows that this is clearly impossible if

1

120598gt1205872119865

119875 119864119901

1gt 1199012 119864119901

1gt 119872(

1

120598minus1205872119865

119875) (22)

which imply that

1205921015840lt1198961

120598 119864

120592

120581gt max[120592

120578 (

119888119867

4120587119873119890120578)

21198961minus 1205921015840120598

1198961120598

]

(23)

Thus 1205921015840 lt 1198961 120598 and 119864(120592120581) gt max[120592120578 (1198881198674120587119873119890120578)2((119896

1minus

1205921015840120598)1198961120598)] are sufficient conditions for the nonexistence of

overstability the violation of which does not necessarilyimply the occurrence of overstability

7 Concluding Remarks

Combined effect of various parameters that is magneticfield compressibility mediumpermeability and hall currentseffect has been investigated on thermal instability of aWalterrsquos 1198611015840 fluid The principle concluding remarks are as thefollowing

(i) For the stationary convection Walterrsquos 1198611015840 fluid be-haves like an ordinary Newtonian fluid due to thevanishing of the viscoelastic parameter

(ii) The presence of magnetic field (and therefore Hallcurrents) and medium permeability effects introduceoscillatory modes in the system in the absence ofthese effects the principle of exchange of stabilities isvalid

(iii) The sufficient conditions for the occurrence of over-stability are 1205921015840 lt 119896

1120598 and 119864(120592120581) gt max[120592120578 (119888119867

4120587119873119890120578)2((1198961minus1205921015840120598)1198961120598)] violation of which does not

necessarily imply the occurrence of overstability(iv) From (17) it is clear that effect of compressibility has

postponed the onset of convection

(v) To investigate the effects of medium permeabilitymagnetic permeability and Hall currents in com-pressible Walterrsquos 1198611015840 viscoelastic fluid we examinedthe expressions 119889119877

1119889119872 119889119877

1119889119875 and 119889119877

11198891198761ana-

lytically Hall current effect has postponed the onsetof convection andmedium permeability hastened theonset of convection where magnetic field has post-poned the onset of convection as well as hastened theonset of convection

Nomenclature

119892 Acceleration due to gravity (msminus2)119870 Stokersquos drag coefficient (kg sminus1)119896 Wave number (mminus1)119896119909 119896119910 Horizontal wave-numbers (mminus1)

1198961 Medium permeability (m2)

119898 Mass of single particle (g)119873 Suspended particle number

density (mminus3)119899 Growth rate (sminus1)119901 Fluid pressure (Pa)119905 Time (s) Fluid velocity (msminus1)V Suspended particle velocity (msminus1) Magnetic field intensity vector

having component (0 0119867) (G)120573(= |119889119879119889119911|) Steady adverse temperature

gradient (Kmminus1)1198731199011

Thermal Prandtl number (minus)1198731199012

Magnetic Prandtl number (minus)119877 = 119892120572120573119889

4120592120581 thermal Rayleigh number

119876 = 120583119890119867211988924120587120588119898]120578 Chandrasekhar number

119872 = (1198881198674120587119873119890120578)2 Nondimensional number

according to Hall currents119891 The mass fraction120577 119885 Component of vorticity120585 119885 Component of current density119873119862

119877and 119873119862

119877 Critical Rayleigh numbers in the

absence and presence ofcompressibility

Greek Letters

120598 Medium porosity (m0 s0 k0)120583 Dynamic viscosity (kmminus1 sminus1)1205831015840 Fluid viscoelasticity (kmminus1 sminus1)

] Kinematic viscosity (m2 sminus1)]1015840 Kinematic viscoelasticity (m2 sminus1)120588 Density (kgmminus3)

Acknowledgment

The authors are grateful to the referees for their technicalcomments and valuable suggestions resulting in a significantimprovement of the paper

8 Journal of Fluids

References

[1] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publications New York NY USA 1981

[2] H Sato ldquoThe Hall effect in the viscous flow of ionized gasbetween parallel plates under transversemagnetic fieldrdquo Journalof the Physical Society of Japan vol 16 no 7 pp 1427ndash1433 1961

[3] I Tani ldquoSteady flow of conducting fluid in channels undertransverse magnetic field with consideration of Hall EffectrdquoJournal of Aerospace Science vol 29 pp 297ndash305 1962

[4] A Sherman and G W Sutton Magnetohydrodynamics North-western University Press Evanston Ill USA 1962

[5] A S Gupta ldquoHall effects on thermal instabilityrdquo Revue Rou-maine de Mathematique Pures et Appliquees pp 665ndash677 1967

[6] E A Spiegel andGVeronisrsquo ldquoOn the Boussinesq approximationfor a compressible fluidrdquoThe Astrophysical Journal vol 131 pp442ndash447 1960

[7] K Chandra ldquoInstability of fluids heated from belowrdquo Proceed-ings of the Royal Society A vol 164 pp 231ndash242 1938

[8] J W Scanlon and L A Segel ldquoSome effects of suspended par-ticles on the onset of Benard convectionrdquo Physics of Fluids vol16 no 10 pp 1573ndash1578 1973

[9] O M Phillips Flow and Reaction in Permeable Rocks Cam-bridge University Press Cambridge UK 1991

[10] D B Ingham and I Pop Transport Phenomena in PorousMedium Pergamon Press Oxford UK 1998

[11] D A Nield and A Bejan Convection in Porous MediumSpringer New York NY USA 2nd edition 1999

[12] C R B Lister ldquoOn the thermal balance of a mid-ocean ridgerdquoGeophysics Journal of the Royal Astronomical Society Continuesvol 26 pp 515ndash535 1972

[13] J A M McDonnell Cosmic Dust John Wiley amp Sons TorontoCanada 1978

[14] R C Sharma and P Kumar ldquoRayleigh-Taylor instability oftwo superposed conducting Walterrsquos B1015840 elastico-viscous fluidsin hydromagneticsrdquo Proceedings of the National Academy ofSciences A vol 68 no 2 pp 151ndash161 1998

[15] R C Sharma ldquoMHD instability of rotating superposed fluidsthrough porous mediumrdquo Acta Physica Academiae ScientiarumHungaricae vol 42 no 1 pp 21ndash28 1977

[16] S Sunil and T Chand ldquoRayleigh-Taylor instability of plasma inpresence of a variable magnetic field and suspended particlesin porous mediumrdquo Indian Journal of Physics vol 71 no 1 pp95ndash105 1997

[17] S Sunil R C Sharma and V Sharma ldquoStability of stratifiedWalterrsquos B1015840 visco-elastic fluid in stratified porous mediumrdquoStudia Geotechnica etMechenica vol 261 no 2 pp 35ndash52 2004

[18] S Sunil R C Sharma and S Chand ldquoHall effect on thermalinstability of Rivlin-Ericksen fluidrdquo Indian Journal of Pure andApplied Mathematics vol 31 no 1 pp 49ndash59 2000

[19] M Singh ldquoHall Current effect on thermosolutal instability ina visco-elastic fluid flowing in a porous mediumrdquo InternationalJournal of Applied Mechanics and Engineering vol 16 no 1 pp69ndash82 2011

[20] M Singh and P Kumar ldquoHydrodynamic and hydromagneticstability of two stratifiedWalterrsquosB1015840 elastico-viscous superposedfluidsrdquo International Journal of AppliedMechanics and Engineer-ing vol 16 no 1 p 233 2011

[21] U Gupta P Aggarwal and R K Wanchoo ldquoThermal con-vection of dusty compressible Rivlin-Ericksen viscoelastic fluid

with Hall currentsrdquo Thermal Science vol 16 no 1 pp 177ndash1912012

[22] D D Joseph Stability of Fluid Motion II Springer New YorkNY USA 1976

[23] K Walterrsquos ldquoThe solution of flow problems in case of materialswith memoryrdquo Journal of Mecanique vol 1 pp 469ndash479 1962

[24] E A Spiegel ldquoConveive instability in a compressible atmo-sphererdquo Journal of Astrophysics vol 141 pp 1068ndash1090 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 7

10

30

50

70

90

110

130

10 20 30 40 50 60 70 80

Q1 = 10

Q1 = 20

Q1 = 30

Q1 = 40

Rayl

eigh

num

ber (R1)

Wave number (x)

Figure 4 Variation of Rayleigh number 1198771against wave number 119909

for 119875 = 3 119872 = 10 119866 = 100 and 120598 = 05

1198600lt 0 since from (20) 119860

3gt 0 if 1120598 gt 120587

2119865119875 Equation

(17) shows that this is clearly impossible if

1

120598gt1205872119865

119875 119864119901

1gt 1199012 119864119901

1gt 119872(

1

120598minus1205872119865

119875) (22)

which imply that

1205921015840lt1198961

120598 119864

120592

120581gt max[120592

120578 (

119888119867

4120587119873119890120578)

21198961minus 1205921015840120598

1198961120598

]

(23)

Thus 1205921015840 lt 1198961 120598 and 119864(120592120581) gt max[120592120578 (1198881198674120587119873119890120578)2((119896

1minus

1205921015840120598)1198961120598)] are sufficient conditions for the nonexistence of

overstability the violation of which does not necessarilyimply the occurrence of overstability

7 Concluding Remarks

Combined effect of various parameters that is magneticfield compressibility mediumpermeability and hall currentseffect has been investigated on thermal instability of aWalterrsquos 1198611015840 fluid The principle concluding remarks are as thefollowing

(i) For the stationary convection Walterrsquos 1198611015840 fluid be-haves like an ordinary Newtonian fluid due to thevanishing of the viscoelastic parameter

(ii) The presence of magnetic field (and therefore Hallcurrents) and medium permeability effects introduceoscillatory modes in the system in the absence ofthese effects the principle of exchange of stabilities isvalid

(iii) The sufficient conditions for the occurrence of over-stability are 1205921015840 lt 119896

1120598 and 119864(120592120581) gt max[120592120578 (119888119867

4120587119873119890120578)2((1198961minus1205921015840120598)1198961120598)] violation of which does not

necessarily imply the occurrence of overstability(iv) From (17) it is clear that effect of compressibility has

postponed the onset of convection

(v) To investigate the effects of medium permeabilitymagnetic permeability and Hall currents in com-pressible Walterrsquos 1198611015840 viscoelastic fluid we examinedthe expressions 119889119877

1119889119872 119889119877

1119889119875 and 119889119877

11198891198761ana-

lytically Hall current effect has postponed the onsetof convection andmedium permeability hastened theonset of convection where magnetic field has post-poned the onset of convection as well as hastened theonset of convection

Nomenclature

119892 Acceleration due to gravity (msminus2)119870 Stokersquos drag coefficient (kg sminus1)119896 Wave number (mminus1)119896119909 119896119910 Horizontal wave-numbers (mminus1)

1198961 Medium permeability (m2)

119898 Mass of single particle (g)119873 Suspended particle number

density (mminus3)119899 Growth rate (sminus1)119901 Fluid pressure (Pa)119905 Time (s) Fluid velocity (msminus1)V Suspended particle velocity (msminus1) Magnetic field intensity vector

having component (0 0119867) (G)120573(= |119889119879119889119911|) Steady adverse temperature

gradient (Kmminus1)1198731199011

Thermal Prandtl number (minus)1198731199012

Magnetic Prandtl number (minus)119877 = 119892120572120573119889

4120592120581 thermal Rayleigh number

119876 = 120583119890119867211988924120587120588119898]120578 Chandrasekhar number

119872 = (1198881198674120587119873119890120578)2 Nondimensional number

according to Hall currents119891 The mass fraction120577 119885 Component of vorticity120585 119885 Component of current density119873119862

119877and 119873119862

119877 Critical Rayleigh numbers in the

absence and presence ofcompressibility

Greek Letters

120598 Medium porosity (m0 s0 k0)120583 Dynamic viscosity (kmminus1 sminus1)1205831015840 Fluid viscoelasticity (kmminus1 sminus1)

] Kinematic viscosity (m2 sminus1)]1015840 Kinematic viscoelasticity (m2 sminus1)120588 Density (kgmminus3)

Acknowledgment

The authors are grateful to the referees for their technicalcomments and valuable suggestions resulting in a significantimprovement of the paper

8 Journal of Fluids

References

[1] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publications New York NY USA 1981

[2] H Sato ldquoThe Hall effect in the viscous flow of ionized gasbetween parallel plates under transversemagnetic fieldrdquo Journalof the Physical Society of Japan vol 16 no 7 pp 1427ndash1433 1961

[3] I Tani ldquoSteady flow of conducting fluid in channels undertransverse magnetic field with consideration of Hall EffectrdquoJournal of Aerospace Science vol 29 pp 297ndash305 1962

[4] A Sherman and G W Sutton Magnetohydrodynamics North-western University Press Evanston Ill USA 1962

[5] A S Gupta ldquoHall effects on thermal instabilityrdquo Revue Rou-maine de Mathematique Pures et Appliquees pp 665ndash677 1967

[6] E A Spiegel andGVeronisrsquo ldquoOn the Boussinesq approximationfor a compressible fluidrdquoThe Astrophysical Journal vol 131 pp442ndash447 1960

[7] K Chandra ldquoInstability of fluids heated from belowrdquo Proceed-ings of the Royal Society A vol 164 pp 231ndash242 1938

[8] J W Scanlon and L A Segel ldquoSome effects of suspended par-ticles on the onset of Benard convectionrdquo Physics of Fluids vol16 no 10 pp 1573ndash1578 1973

[9] O M Phillips Flow and Reaction in Permeable Rocks Cam-bridge University Press Cambridge UK 1991

[10] D B Ingham and I Pop Transport Phenomena in PorousMedium Pergamon Press Oxford UK 1998

[11] D A Nield and A Bejan Convection in Porous MediumSpringer New York NY USA 2nd edition 1999

[12] C R B Lister ldquoOn the thermal balance of a mid-ocean ridgerdquoGeophysics Journal of the Royal Astronomical Society Continuesvol 26 pp 515ndash535 1972

[13] J A M McDonnell Cosmic Dust John Wiley amp Sons TorontoCanada 1978

[14] R C Sharma and P Kumar ldquoRayleigh-Taylor instability oftwo superposed conducting Walterrsquos B1015840 elastico-viscous fluidsin hydromagneticsrdquo Proceedings of the National Academy ofSciences A vol 68 no 2 pp 151ndash161 1998

[15] R C Sharma ldquoMHD instability of rotating superposed fluidsthrough porous mediumrdquo Acta Physica Academiae ScientiarumHungaricae vol 42 no 1 pp 21ndash28 1977

[16] S Sunil and T Chand ldquoRayleigh-Taylor instability of plasma inpresence of a variable magnetic field and suspended particlesin porous mediumrdquo Indian Journal of Physics vol 71 no 1 pp95ndash105 1997

[17] S Sunil R C Sharma and V Sharma ldquoStability of stratifiedWalterrsquos B1015840 visco-elastic fluid in stratified porous mediumrdquoStudia Geotechnica etMechenica vol 261 no 2 pp 35ndash52 2004

[18] S Sunil R C Sharma and S Chand ldquoHall effect on thermalinstability of Rivlin-Ericksen fluidrdquo Indian Journal of Pure andApplied Mathematics vol 31 no 1 pp 49ndash59 2000

[19] M Singh ldquoHall Current effect on thermosolutal instability ina visco-elastic fluid flowing in a porous mediumrdquo InternationalJournal of Applied Mechanics and Engineering vol 16 no 1 pp69ndash82 2011

[20] M Singh and P Kumar ldquoHydrodynamic and hydromagneticstability of two stratifiedWalterrsquosB1015840 elastico-viscous superposedfluidsrdquo International Journal of AppliedMechanics and Engineer-ing vol 16 no 1 p 233 2011

[21] U Gupta P Aggarwal and R K Wanchoo ldquoThermal con-vection of dusty compressible Rivlin-Ericksen viscoelastic fluid

with Hall currentsrdquo Thermal Science vol 16 no 1 pp 177ndash1912012

[22] D D Joseph Stability of Fluid Motion II Springer New YorkNY USA 1976

[23] K Walterrsquos ldquoThe solution of flow problems in case of materialswith memoryrdquo Journal of Mecanique vol 1 pp 469ndash479 1962

[24] E A Spiegel ldquoConveive instability in a compressible atmo-sphererdquo Journal of Astrophysics vol 141 pp 1068ndash1090 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Journal of Fluids

References

[1] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publications New York NY USA 1981

[2] H Sato ldquoThe Hall effect in the viscous flow of ionized gasbetween parallel plates under transversemagnetic fieldrdquo Journalof the Physical Society of Japan vol 16 no 7 pp 1427ndash1433 1961

[3] I Tani ldquoSteady flow of conducting fluid in channels undertransverse magnetic field with consideration of Hall EffectrdquoJournal of Aerospace Science vol 29 pp 297ndash305 1962

[4] A Sherman and G W Sutton Magnetohydrodynamics North-western University Press Evanston Ill USA 1962

[5] A S Gupta ldquoHall effects on thermal instabilityrdquo Revue Rou-maine de Mathematique Pures et Appliquees pp 665ndash677 1967

[6] E A Spiegel andGVeronisrsquo ldquoOn the Boussinesq approximationfor a compressible fluidrdquoThe Astrophysical Journal vol 131 pp442ndash447 1960

[7] K Chandra ldquoInstability of fluids heated from belowrdquo Proceed-ings of the Royal Society A vol 164 pp 231ndash242 1938

[8] J W Scanlon and L A Segel ldquoSome effects of suspended par-ticles on the onset of Benard convectionrdquo Physics of Fluids vol16 no 10 pp 1573ndash1578 1973

[9] O M Phillips Flow and Reaction in Permeable Rocks Cam-bridge University Press Cambridge UK 1991

[10] D B Ingham and I Pop Transport Phenomena in PorousMedium Pergamon Press Oxford UK 1998

[11] D A Nield and A Bejan Convection in Porous MediumSpringer New York NY USA 2nd edition 1999

[12] C R B Lister ldquoOn the thermal balance of a mid-ocean ridgerdquoGeophysics Journal of the Royal Astronomical Society Continuesvol 26 pp 515ndash535 1972

[13] J A M McDonnell Cosmic Dust John Wiley amp Sons TorontoCanada 1978

[14] R C Sharma and P Kumar ldquoRayleigh-Taylor instability oftwo superposed conducting Walterrsquos B1015840 elastico-viscous fluidsin hydromagneticsrdquo Proceedings of the National Academy ofSciences A vol 68 no 2 pp 151ndash161 1998

[15] R C Sharma ldquoMHD instability of rotating superposed fluidsthrough porous mediumrdquo Acta Physica Academiae ScientiarumHungaricae vol 42 no 1 pp 21ndash28 1977

[16] S Sunil and T Chand ldquoRayleigh-Taylor instability of plasma inpresence of a variable magnetic field and suspended particlesin porous mediumrdquo Indian Journal of Physics vol 71 no 1 pp95ndash105 1997

[17] S Sunil R C Sharma and V Sharma ldquoStability of stratifiedWalterrsquos B1015840 visco-elastic fluid in stratified porous mediumrdquoStudia Geotechnica etMechenica vol 261 no 2 pp 35ndash52 2004

[18] S Sunil R C Sharma and S Chand ldquoHall effect on thermalinstability of Rivlin-Ericksen fluidrdquo Indian Journal of Pure andApplied Mathematics vol 31 no 1 pp 49ndash59 2000

[19] M Singh ldquoHall Current effect on thermosolutal instability ina visco-elastic fluid flowing in a porous mediumrdquo InternationalJournal of Applied Mechanics and Engineering vol 16 no 1 pp69ndash82 2011

[20] M Singh and P Kumar ldquoHydrodynamic and hydromagneticstability of two stratifiedWalterrsquosB1015840 elastico-viscous superposedfluidsrdquo International Journal of AppliedMechanics and Engineer-ing vol 16 no 1 p 233 2011

[21] U Gupta P Aggarwal and R K Wanchoo ldquoThermal con-vection of dusty compressible Rivlin-Ericksen viscoelastic fluid

with Hall currentsrdquo Thermal Science vol 16 no 1 pp 177ndash1912012

[22] D D Joseph Stability of Fluid Motion II Springer New YorkNY USA 1976

[23] K Walterrsquos ldquoThe solution of flow problems in case of materialswith memoryrdquo Journal of Mecanique vol 1 pp 469ndash479 1962

[24] E A Spiegel ldquoConveive instability in a compressible atmo-sphererdquo Journal of Astrophysics vol 141 pp 1068ndash1090 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of