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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
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
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
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 =
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
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
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
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
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
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
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 =
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
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
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
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
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
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
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 =
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
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
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
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
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 =
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
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
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
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
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
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
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
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
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
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
[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