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LINEAR STABILITY ANALYSIS ON THE ONSET OF DDC IN A DPM SATURATED WITH CSF WITH INTERNAL HEATING

Kamble Shravan S.

Assistant Professor, Dept. of Mathematics, Sri Shivalingeshwara Govt. First Grade College, Aland, Karnataka-India ***

Abstract - The effect of rotation on the onset of double diffusive convection (DDC) in a horizontal couple stress fluid saturated porous layer with an internal heat source is investigated using linear stability analysis. Thelinearstability analysis is based on the classical normal mode technique. The extended Darcy model which includes thetimederivativeterm and Coriolis term has been employed in the momentum equation. The expressions for stationary and oscillatory Rayleigh number are obtained as a function of governing parameters such as internal Rayleigh number, couple stress parameter, Taylor number, normalized porosity and Lewis number and their effects on the stability of the system are shown graphically

Key Words: Rotation, couple stress fluid (CSF), Darcy Porous medium (DPM), Double diffusive convection (DDC), Internal Heat Source.

Nomenclature

a Wavenumber, 22lm 

c Specificheatofsolid

p c Specificheatoffluidatconstantpressure

C Couplestressparameter, 2 c d

d Heightoftheporouslayer

Da Darcynumber,   2 / Kd

g Gravitationalacceleration,   0,0, g

i Unitnormalvectorinthe x-direction

j Unitnormalvectorinthe y-direction

K Permeability

k Unitnormalvectorinthe z-direction

, lm Horizontalwavenumbers

Le Lewisnumber, TS

p Pressure

Pr Prandtlnumber, T

DPr Darcy-Prandtlnumber, PrDa

q Velocityvector,   ,, uvw

Q Internalheatsource

SRa SoluteRayleighnumber, S T gSKd  

TRa ThermalRayleighnumber, T T gTKd  

iR InternalRayleighnumber, 2 T Qd 

SRa SoluteRayleighNumber,   S T gSKd  

TRa ThermalRayleighNumber,   T T gTKd 

S Soluteconcentration

S Salinitydifferencebetweenthewalls

t Time

T Temperature Ta Taylornumber,

T Temperaturedifferencebetweenthewalls ,, xyz Spacecoordinates

Greek symbols

a Wavenumber, 22lm 

S Solutecoefficientofexpansion

T Thermalcoefficientofexpansion  Ratio of specific heats,

 Thermalanisotropyparameter, TxTz

 Dimensionless amplitude of temperature perturbation



 2 2 z K  Ω

       1 pp s ff ccc    

 Porosity  Normalizedporosity, 

Solutediffusivity T

Thermaldiffusivity  Dynamicviscosity

Couplestressviscosity

 Kinematicviscosity, 0

 Fluiddensity

0 Referencedensity

 Growthrate

 Dimensionless amplitude of concentration perturbation

contaminant transport in fluid saturated soils, liquid gas storage, and food processing [see [11] & [13]]. Double diffusive convection (DDC) in porous media has attracted manyauthorslike[2]&[14]duringthelastseveraldecades.

The effect of internal heat source is important in several applicationsthatincludereactorsafetyanalysis,metalwaste form development for nuclear fuel, fire and combustion studies,andstorageof radioactivematerials.The onset of convection due to internal heat source has become an interesting problem in various areas of geophysics and engineeringunderthesituationsofradioactivedecayora weekexothermicreactionwithintheporousmaterial.

0,0, Ω Other symbols

Ω Angularvelocity,

Subscripts

b Basicstate

c Critical

f Fluid

h Horizontal

m Porousmedium 0 Reference

s Solid

Superscripts

* Dimensionlessquantity

′ Perturbedquantity

Osc Oscillatory

St Stationary

1. INTRODUCTION

Doublediffusiveconvection(DDC)inporousmediahasbeen intensively studied because of its applications in different branchesofscienceandengineering,suchasunderground disposal of nuclear wastes, groundwater pollution,

Earlierstudiesofconvectiveflowsinporousmediawithin rectangularenclosures,withoutthelocalheatsourceeffects. Averylittleattentionhasbeendevotedtothisproblemwith non-Newtonian fluids. The corresponding problem in the caseofporousmediumhasalsonotreceivedmuchattention untilrecently.Withgrowingimportanceofnon-Newtonian fluidswithsuspendedparticlesinmoderntechnologyand industries,theinvestigationsofsuchfluidsaredesirable.The studies of such fluids have applications in number of processes that occur in industry, such as the extrusion of polymer fluids, solidifications of liquid crystals, cooling of metallicplateinabath,exoticlubricationandcolloidaland suspension solutions. In the category of non-Newtonian fluids couple stress fluids have distinct features, such as polareffects.

Themainaimfeatureofcouplestresseswillbetointroduce a size dependent effect that is not present in the classical viscousfluids.Thetheoryofpolarfluidsandrelatedtheories aremodelsforfluidswhosemicrostructureismechanically significant. The constitutive equations for couple stress fluidsweregivenby[1].

Anisotropy is generally a consequence of preferential orientation of symmetric geometry of porous matrix or fibers and is in fact encountered in numerous systems in industryandnature.Alsoartificialporousmatrixanisotropy can be made deliberately according to applications. [3] studied the combine defect of horizontal and vertical heterogeneityandanisotropyontheonsetofconvectionina porousmedium. [4]performedlinearandnonlinearstability analysisofdoublediffusiveconvectioninanisotropicporous mediaincludingSoreteffectandreportedthattheeffectof mechanicalanisotropicparametersistodestabilizeandof thermalanisotropicparametersistostabilizethesystem.

There are large number of practical situations in which convectionisdrivenbyinternalheatsourceintheporous media.Thewideapplicationsofsuchconvectionsoccurin nuclear reactions, nuclear heat cores, nuclear energy, nuclearwastedisposals,oilextractions,andcrystalgrowth. [5] investigated linear stability analysis for the onset of naturalconvectioninafluidsaturatedporousmediumwith uniform internal heat source and density maximum in an







D d dz 2 1 22 22 xy    2 2 2 1 2 z   



local thermal nonequilibrium model and predicted that internal heat source parameter advances the onset convection,[6]studiedtheonsetofstationaryconvectionin alowPrandtlnumberwithinternalheatsourceandfound that effect of internal heat source parameter is destabilization.Recently,[7-10]havestudiedtheproblemof thermal instability in porous media with internal heat source.Few authors have studied on rotation withcouple stressfluidinporousmedia[see[19],[20],[21],[22]&[23]].

Although few literatures on DDC in a porous medium saturatedbyordinaryfluidwithorwithoutaninternalheat source is available, no attention has been devoted to the studyofDDCinaporouslayersaturatedbyacouplestress fluids in the presence of an internal heat and rotation. Thereforeinthepresentworkweintendtoinvestigatethe onsetdoublediffusiveconvectioninarotatingcouplestress fluid saturated porous layer with an internal heat source employing a modified Darcy model using linear stability analyses.Ourobjectiveistostudyhowtheonsetcriterion forstationaryandoscillatoryconvectionareaffectedbythe Lewis number, solute Rayleigh number, Taylor number, Couple stress parameter, internal heat source and normalizedporosity.

2. GOVERNING EQUATIONS

We consider an infinite horizontal couple stress fluid saturatedporouslayerconfinedbetweentheplanes 0 z  and zd  with the vertically downward gravity force g actingonit.Aconstanttemperature 0TT and 0T with stabilizingconcentrations 0SS and 0S respectivelyare maintained between the lower and upper surfaces. A Cartesianframeofreferenceischosenwiththeorigininthe lowerboundaryand z -axisverticallyupwards.Theporous layer rotates uniformly about the z -axis with a constant angular velocity (0,0,) . The modified Darcy model, whichincludesthetimederivativeandtheCoriolistermis employed as a momentum equation [see [17]]. The basic governingequationsare

where, the variables and constants have their usual meaning, as given in the Nomenclature. Further ()/(),()(1)()(), mfmsf ccccc pp   c isthespecificheatofthesolidand p c isthespecificheatof thefluidatconstantpressurerespectively.

1.1 BASIC STATE

Thebasicstateofthefluidisassumedtobequiescentandis given

Thentheconductionstatetemperatureandconcentration aregivenby

Onthebasicstatewesuperposeinfinitesimalperturbations intheform

indicate

equationsaregiven

.0 q  (1) 00 0 2 2 () 1 () TS c pTSgq q K       , (2) 2 0 (.)() T T qTTQTT t     , (3) 2 (.) S T qSS t     , (4)

000

(z)      , (5)

temperature b T(z) , solute concentration b S(z)

b P(z) and density b(z)  ,

following equations 22 0 22 000 00 1 bb bTb bTbSb dP dTdS b g,Q(TT),, dzdzdz [(TT)(SS)]     , (6)

bbb bbb q(,,),PP(z),TT(z), SS(z),(z),

The

, pressure

satisfy the

0 1 b TSin(R(z/d)) i T(z)T SinRi   , 0 1 b S(z)S(z/d)S  (7) 1.2 PERTURBED

STATE

bbb bb ''' qqq,TTT,SSS, '' PPP,   , (8)

by 0 ' .q  , (9) 0 0 2 2 1 0 TS c ' q ''' p(TS)g t '' q()q K           , (10)

where,primes

perturbations.SubstitutingEq.(5) into Eqs. (1)- (4) and using Eqs. (5)- (7), the perturbed

Byoperatingcurltwice-onequation(10),weeliminate ' p fromitandthenrendertheresultingequationandtheEqs. (11) and (12) dimensionless using the following transformations.

Where l,m are horizontal wavenumbers and  is the growthrate.Infinitesimalperturbationsofthereststatemay either damped or grow depending on the value of the parameter . Substituting Eq. (18) into the linearized versionofEqs.(14)-(16),weobtain

Toobtain non-dimensional equationsas(on dropping the asterisksforsimplicity),

is the Taylor number and

istheinternalRayleghnumber,andallthe other non-dimensional parameters are as defined in the Nomenclature.

The boundary conditions are assumed to be stress free, isothermalandisohaline,theEqs.(14)-(16)aretobesolved fortheboundaryconditions

2. LINEAR STABILITY ANALYSIS

Wepredictthethresholdsofbothmarginalandoscillatory convections using linear theory. The Eigenvalue problem definedbyEqs.(14)-(16)subjecttotheboundaryconditions (17) is solved using the time-dependent periodic disturbances in a horizontal plane. Assuming that the amplitudesoftheperturbationsareverysmall,wewrite

 and

alm

. The boundary conditions(17)nowbecomes

We assume the solutions of Eqs. (19)-(21) satisfying the boundaryconditions(22)intheform

Themostunstablemodecorrespondsto 1 n  (fundamental mode).Therefore,substitutingEq.(23)with 1 n  intoEqs. (19)-(21),weobtainamatrixequation

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 02 | Feb 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page617 2 T '' TT''''' (q.)TwTQT td     , (11) 2 S '' SS'''' (q.)SwS td     , (12)

2 '''***'''*** T *'*'* T (x,y,z)(x,y,z)d,(u,v,w)(/d)(u,v,w), tt(d/),T(T)T,S(S)S     (13)

2 222 2 22 1 1 1 1 1 D TS D [(C)Ta]w Prtz (C)(RaRa) Prt       , (14) 2 0 i [R(q.)]Tw t    , (15) 2 1 0 (q.)Sw tLe     , (16) where, 2 2 K Ta()   

2 iTRQd/  

2 2 0 w wTS z    at 01z,  . (17)

(w,T,S)(W(z),(z),(z))exp[i(lxmy)t]   (18)

222222 222 222 1 1 10 D T D S D [(C(Da))(Da)TaD]W Pr aRa(C(Da)) Pr aRa(C(Da)) Pr         , (19) 22 0 i [(Da)R]W  , (20) 22 1 0 [(Da)] Le   , (21) Whrere,

2 2 0 W W z     at 01z,  . (22)

Dd/dz

222



000 (W(z),(z),(z))(W,,)Sin(nz),   123 (n,,,......)  (23)

22222 0 2 0 2 0 0 100 0 1 10 TS DDD i ()TaaRa()aRa() PrPrPr W (R) Le                           (24) where, 222 a 

2.1 STATE

Forthevalidityoftheprincipalofexchangeofstabilities(i.e., steadyoscillatory),wehave 0   (i.e. 0 ri  )atthe marginalofstability.ThentheRayleighnumberatwhichthe marginallystablesteadymodeexistsbecomes

This coincides with the results of [24]. Further

, Eq.(30)gives

which has the critical value

obtainedby[15]and[16].

2.2

We now set i i   in Eq. (25) and clear the complex quantitiesfromthedenominator,toobtain

Intheabsenceofheatsource 0 i (i.e.,R)  ,Eq.(26)reduces to

1 1 St TS (a)(C(a))Ta(a) RaLeRa aa(C(a))

Thisresultexactlycoincideswiththeonegivenby[17].

It is important to note that the critical wavenumber St c a dependsonthecouplestressparameterandTaylornumber. In the absence of Taylor number 0 (i.e.,Ta)  , Eq. (27) gives

2 1 1 St TS Ra(a)[C(a)]LeRa

Whichistheresultgivenby[12]. Forsinglecomponentfluid 0 S Ra,  inEq.(27)gives

Which is the one obtained by [18]. When 0 C  (i.e., Newtonianfluidcase),Eq.(29)reducesto

D w

Pr((R)(aLeRa)

Pr(aLeRaTa)(Le(R))))

2 2 2

DSi RaPr(R)(TaaLeRa)

Eq.

Rayleighnumberintheform 2 2 2222 22 Di S DD T D LePr(R)[() Le (Ta())a()Ra] PrPr Ra a(Le)(Pr)         , (25)

Theconditionofanon-trivialsolutionoftheabovesystemof homogeneous linear

(24) yields the expression

thermal

STATIONAR

222422 22 St iS T (R)(TaaLeRa) Ra a     ,

(26)

22222 222

     , (27)

 

22222

, (28)

22222222 2222

     , (29)

1 1 St T (a)(C(a))Ta(a)

aa(C(a))

222222 22 St T (a)Ta(a) Ra aa   (30)

Ta 

222 2 St T (a) Ra a    , (31)

Ra  

St c a  

2 4

for 2

OSCILLATORY STATE

12TiRai  , (32)

(

242

Since TRa isaphysicalquantity,itmustbereal.Hence,from Eq. (32) it follows that either 0 i  or 2 Le(PrR)

0  Le(PrR)           (33)

i ,   oscillatory onset). For oscillatory onset, setting 2 0  0 i ()   givesanexpressionforfrequencyofoscillationsin theform(ondroppingthesubscripti)

2 42 222222 42

iS Di DSi Di

222422 2 4222264 22222 2

Di D

Le w[(Le(Ta)PrR) Pr Pr(aLeRa(Le(R)))]w        (34)

NowEq.(32)with 2 0  ,gives

Osc TDiS

The analytical expression for the oscillatory Rayleigh numbergivenbyEq.(34)isminimizedwithrespecttothe wavenumber numerically, after substituting for 2 0 w()  fromEq.(33),forvariousvaluesofphysicalparametersin order to know their effects on the onset of stationary and oscillatoryconvection.

3. RESULT AND DISCUSSION

Theneutralstabilitycurvesinthe T Raa planeforvarious parameter values are as shown in Figs.1-6. We fixed the values for the parameters except the varying parameter. From these figures it is clear that the neutral curves are connectedinatopologicalsense.Thisconnectednessallows thelinearstabilitycriteriatobeexpressedintermsofthe criticalRayleighnumber TcRa ,belowwhichthesystemis stableandunstableabove.

InFig.1themarginalstabilitycurvesfordifferentvaluesof couplestressparameter C aredrawn.Itisobservedthat withtheincreaseof C thevaluesofRayleighnumberand the corresponding wavenumber for oscillatory mode decreases while those for stationary mode increases. Therefore, the effect of C is to advance the onset of oscillatory convection while its effect is to inhibit the stationaryconvection.

Fig.2depictstheeffectofTaylornumber Ta ontheneutral stabilitycurves.Wefindthattheeffectofincreasing Ta is toincreasethevalueoftheRayleighnumberforstationary andoscillatorymodesandthecorrespondingwavenumber. ThustheTaylornumber Ta hasastabilizingeffectonthe double diffusive convection in a horizontal couple stress fluidsaturatedporouslayerwithaninternalheatsource.

Fig.3indicatestheeffectofinternalRayleighnumber iR on the neutral stability curves for the fixed values of other parameters. It is observed that the value of the Rayleigh numberforstationaryandoscillatorymodeincreaseswith increasing iR ,indicatingthattheeffectof iR istoinhibitthe onsetofstationaryandoscillatoryconvection.

InFig.4themarginalstabilitycurvesfordifferentvaluesof Lewisnumber Le aredrawn.Itisobservedthatwiththe increase of Le the values of Rayleigh number and the correspondingwavenumberforoscillatorymodedecreases while those for stationary mode increases. Therefore, the effect of Le is to advance the onset of oscillatory convection while its effect is to inhibit the stationary convection.

Fig.5depictstheeffectofsoluteRayleighnumber SRa on the neutral stability curves for stationary and oscillatory modes. We find that the effect of increasing SRa is to

increase the critical value of the Rayleigh number for stationary and oscillatory modes and the corresponding wavenumber.ThusthesoluteRayleighnumber SRa hasa stabilizing effect on the double diffusive convection in a horizontalcouplestressfluidsaturatedporouslayerwithan internalheatsource.

Theeffectofnormalizedporosityparameter  isdepicted in the Fig. 6. We find that an increase in  decreases the minimum of the Rayleigh number for oscillatory mode, indicatingthattheeffectofincreasing  istoadvancethe onsetofoscillatoryconvection.

4 6 8 10 12 14 300 400 500 600 700 800 Stationary Oscillatory

(C=0.2,0.4,0.7,1,2) Ra S=100,=0.7,Ta=100,R=3,Le=20 Ra T a 2 4 6 8 10 12 150 300 450 600 750 Ra S=100,=0.7, Ta=100,Ri=3,Le=20

Ta Ra T a Stationary Oscillatory 200 100 50 Ta=10

Fig.1.NeutralstabilitycurvesfordifferentsvaluesofC.

Fig.2.Neutralstabilitycurvesfordifferentsvaluesof

5. CONCLUSION

The effect of rotation on the onset of double diffusive convection (DDC) in a horizontal couple stress fluid saturated porous layer with an internal heat source is investigated using linear stability analysis. The linear stability analysis is based on the classical normal mode technique.Thefollowingconclusionsaredrawn: TheTaylor number Ta hasastabilizingeffectonthedoublediffusive convection in a horizontal couple stress fluid saturated porous layerwithan internal heatsource. The effect of solute Rayleigh number SRa is todelayboth stationary

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 02 | Feb 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page620 1 2 3 4 5 6 200 400 600 800 1000 1200 1400 Ra S=100,=0.7,Ta=100,C=3,Le=20, Fig.3.Neutralstabilitycurvesfordifferentvaluesof internalRayleighnumberRi a Ra T Ri=(6,4,3,1) 2 4 6 8 10 500 1000 1500 2000 2500 3000 3500 Ra S=100,=0.7, Ta=100,C=3,Le=20, Stationary Oscillatory 10 15 20 Le=7 Fig.4.Neutralstabilitycurvesfordifferentvaluesof LewisnumberLe a Ra T 2 4 6 8 10 500 1000 1500 2000 2500 3000 Stationary Oscillatory R i=3,=0.7, Ta=100,C=3,Le=20, Fig.5.Neutralstabilitycurvesfordifferentvaluesof soluteRayleighnumberRa S a Ra T 300 200 100 Ra S =50

2 3 4 5 6 7 200 400 600 800 1000 1200 1400 Stationary Oscillatory R i=3,Ri=3, Ta=100,C=3,Le=20, Fig.6.Neutralstabilitycurvesfordifferentvaluesof normalizedporosity. a Ra T 0.9 0.7 0.2 0.4 =1

and oscillatory convection. And the effect of Lewis number Le is to delay the onset of stationaryconvection while it advances the oscillatory convection. The internal Rayleighnumber iR hasadestabilizingeffectonthedouble diffusive convection in a porous medium. The effect of couple stress parameter C is to advance the onset of oscillatory convection whereas its effect is to inhibit the stationaryonset.Thenormalizedporosityparameter  has adestabilizingeffectinthecaseofoscillatorymode.

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