THE ONSET OF CONVECTION IN A ROTATING MULTICOMPONENT FLUID LAYER. Jyoti Prakash, Virender Singh, Rajeev Kumar, Kultaran Kumari

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 2, pp , Warsaw 26 DOI:.5632/jtam-pl THE ONSET OF CONVECTION IN A ROTATING MULTICOMPONENT FLUID LAYER Jyoti akash, Virender Singh, Rajeev Kumar, Kultaran Kumari Department of Mathematics and Statistics, Himachal adesh University, Shimla, India jpsmaths67@gmail.com The onset of convective instability is analysed in a rotating multicomponent fluid layer in which density depends on n stratifying agentsone of them is heat) having different diffusivities. Two problems have been analysed mathematically. In the first problem, a sufficient condition is derived for the validity of the principle of the exchange of stabilities. Further, when the complement of this condition holds good, oscillatory motions of neutral or growingamplitudecanexist,andthusitisimportanttoderiveupperboundsforthecomplex growthrateofsuchmotionswhenatleastoneoftheboundingsurfacesisrigidsothatexact solutions of the problem in closed form are not obtainable. Thus, as the second problem, bounds for the complex growth rates are also obtained. Above results are uniformly valid for quite general nature of the bounding surfaces. Keywords: multicomponent convection, principle of exchange of stabilities, oscillatory motions, complex growth rate, concentration Rayleigh number, Lewis number. Introduction Whendensityofafluidisdeterminedbytwostratifyingagents,suchasheatandsaltdiffusing at different rates, the fluid at rest can be unstable even if its density increases downward. This convective phenomenon is known as thermosolutal convection or more generally as double diffusive convection. This phenomenon has, now, been extensively studied. For review on the subject of double diffusive convection one may be referred toturner, 973, 974, 985; Brandt and Fernando, 996; Radko, 23; Sekar et al., 23). Although the subject of double diffusive convection is still an important area of research Sekar et al., 23; Kellner and Tilgner, 24; Nield and Kuznetsov, 2; Schmitt, 2), there are many fluid systems where more than two components are presentturner, 985; Griffiths, 979b). Examples of such systems include the solidification of molten alloys, Earth core, geothermally heated lakes, sea water, magmas and their laboratory models. The presence of more than one salt in fluid mixtures is very often requested for describing natural phenomena such as contaminant transport, acid rain effects, underground water flow and warming of the stratosphere. The subject of more than two stratifying agents has attracted many researchersgriffiths, 979a,b; Pearlstein et al., 989; Rionero, 23a,b, 24; Lopez et al., 99; Terrones, 993; Poulikakos, 985; Shivakumara and Naveen Kumar, 24). In double diffusive convectionturner, 974) or, more generally, in multicomponent convectionturner, 985; Griffiths, 979a) instabilitymayoccurintwokinds:firstinformofsteadyorstationary)convectionwhichiscalled as salt finger modes and the second in form of oscillatory motions of growing amplitudeor overstability) which is called as diffusive convection. When a warm and saltier fluid lies above acoldandfreshfluidthenstationaryconvectionispreferred,andwhenacoldandfresherfluid lies above a warm and saltier fluid then oscillatory motions are preferred. The essence of these researchers is that small concentrations of the third diffusing component with a smaller mass

2 478 J. akash et al. diffusivity can have a significant effect upon the nature of diffusive instabilities and diffusive convection. The salt finger modes are simultaneously unstable under a wide range of conditions when the density gradients due to components with the greatest and smallest diffusivity are of the same signs even if the overall density stratification is hydrostatically stable. These researchers also notice some fundamental differences between doubly and triply diffusive convection. Oneisthatifthegradientsoftwoofthestratifyingagentsareheldfixed,thenthreecritical values of the Rayleigh number of the third agent are sometimes required to specify the linear stability criteriain double diffusive convection only one critical Rayleigh number is required). Theotherdifferenceisthattheonsetofconvectionforthecaseoffreeboundariesmayoccur via quassiperiodic bifurcation from the motionless basic state. Now the triply diffusive convection despite its complexities has also been well studied. But, to the author knowledge, not many investigations have been conducted on stability theory when more than three components are present, which may be, perhaps, due to the complexities involved in mathematical calculations and numerical computations. Some worth researches which may be referred here are due to Terrones and Pearlstein989) who derived analytical results for n components and numerical results for n = 5 using dynamically free boundary conditions. Later Lopez et al.99) predicted that the results of triply diffusive convection may be extended to multicomponent convection with n components for rigid surfaces also. Further significant contributions to multicomponent convection are due to Ryzhkov and Shevtsova27, 29) and Ryzhkov23). The establishment of nonoccurrence of any slow oscillatory motions which may be neutral or unstable implies the validity of the principle of the exchange of stabilitiespes). The validity of this principle in stability problems eliminates unsteady terms from linearized perturbation equations which results in notable mathematical simplicity since the transition from stability to instability occurs via a marginal state which is characterized by the vanishing of both real and imaginary parts of the complex time eigenvalue associated with the perturbation. Pellew and Southwell94) proved the validity of PESi.e. occurrence of stationary convection) for the classical Rayleigh-Benard instability problem. akash et al.24a) established such a criterion for the triply diffusive convection problem. To study the effect of rotation on a multicomponent fluid layer is an interesting topic. akash et al.24b) derived a sufficient condition for the occurrence of stationary convection and upper boundsakash et al., 25) for the complex growth rate of an arbitrary oscillatory motion of neutral or growing amplitude in rotatory hydrodynamic triply diffusive convection. The further extension of these results to the problem of the onset of convection in a multicomponent fluid layer in the domains of astrophysics and terrestrial physics, wherein the liquid concerned has the property of electrical conduction and the magnetic field and rotation are prevalent, is very much sought after in the present context. In the present work, we analyse the onset of buoyancy driven convection in a multicomponent fluid layer in the presence of uniform vertical rotation. We generalize the existing results of the rotatory hydrodynamic triply diffusive convection problem concerning the validity of the principle of the exchange of stabilitiesakash et al., 24b) and arresting the complex growth rate of oscillatory motionwhen it occurs)akash et al., 25) which are important especially whenatleastoneboundaryisrigidsothatexactsolutionsintheclosedformarenotobtainable. To the authors knowledge, no such results have been obtained so far for the hydrodynamical systems with more than three components. The results derived herein are uniformly valid for any combination of the rigid and free boundaries and the results of doubly diffusivebanerjee et al., 98; Gupta et al., 986) and triply diffusive convectionakash et al., 24a,b,c, 25) follow as a consequence. Further, the importance of the results obtained herein lies in that these results may be used for any rotatory hydrodynamic multicomponent system where no mathematical calculation or numerical computation is possible.

3 The onset of convection in a rotating multicomponent fluid layer Mathematical formulation and analysis A viscous finitely heat conducting Boussinesq fluid of infinite horizontal extension is statistically confined between two horizontal boundaries z = and z = d which are respectively maintained atuniformtemperaturest andt <T )anduniformconcentrationss,s 2,...,S n ) and S <S ),S 2 <S 2 ),...,S n ) <S n ) )intheforcefieldofgravityandinthepresence of uniform vertical rotationas shown in Fig. ). It is assumed that the cross-diffusion effects of the stratifying agents can be neglected. Fig.. Physical configuration The basic equations that govern the motion of the rotatory hydrodynamic multicomponent fluid layer are as followsakash et al., 24b; Terrones and Pearlstein, 989) u j = x j u i t +u j u i = P x j x i ρ 2 Ω r 2) + + δρ + δρ + δρ δρ ) n ρ ρ ρ ρ X i +2ε ijk u j Ω k +ν 2 u i T t +u T j =κ 2 T x j S t +u S j =κ 2 S x j S 2 t +u S 2 j =κ 2 2 S 2 x j 2.). S n S n +u j =κ n 2 S n t x j whereρisdensity;tistime;x j j=,2,3)arecartesiancoordinatesx,y,z;u j j=,2,3)are velocitycomponents;x i i=,2,3)arecomponentsoftheexternalforceinthex,y,zdirections, respectively;p ispressure;µisviscosity;ωisangularvelocity;t istemperature,κisthe coefficientofthermaldiffusivity;s,s 2,...,S n aren concentrationsandκ,κ 2,...,κ n arerespectivelythecoefficientsofmassdiffusivityofs,s 2,...,S n withκ >κ 2 >...>κ n. The above basic equations must be supplemented by the equation of state ρ=ρ [+αt T ) α S S ) α 2 S 2 S 2 )... α n S n ) S n ) )]2.2) whereα,α,α 2,...,α n arerespectivelythecoefficientsofvolumeexpansionduetotemperaturevariationandconcentrationvariationsforn concentrationcomponentss,s 2,...,S n.

4 48 J. akash et al. ρ isthevalueofρatz=.ν=µ/ρ iskinematicviscosityand P ρ 2 Ω r 2 ishydrostatic pressure. The basic state is assumed to be stationary, and the standard linear stability analysis procedureasoutlinedinthestudiesofakashetal.24b)isfollowedtoobtainthefollowing non-dimensional stability equations D 2 a 2 ) D 2 a 2 σ ) w=raa 2 θ R a 2 φ R 2 a 2 φ 2... R n a 2 φ n +TaDζ D 2 a 2 σ)θ= w )φ = w D 2 a 2 σ Le Le D 2 a 2 σ ) φ 2 = w Le 2 Le 2. D 2 a 2 σ Le n ) φ n = w Le n 2.3) and D 2 a 2 σ ) ζ= Dw 2.4) respectively. Equations2.3) and2.4) are to be solved using the following appropriate boundary conditions: w==θ=φ =φ 2 =...=φ n onboththehorizontalboundarieswhichareat z= and z= 2.5) andonrigidboundary Dw=ζ= 2.6) oronfreeboundary D 2 w=dζ= 2.7) themeaningofthesymbolsinvolvedineqs.2.3)and2.4)fromthephysicalpointofvieware asfollows:zistheverticalcoordinate,d=d/dzisdifferentiationw.r.t.z,a 2 >issquareofthe wavenumber,>isthethermalandtlnumberwhichisameasureofrelativeimportance ofheatconductionandviscosityofthefluidandvariesfromfluidtofluid.forair=.7 approximately), for water = 7approximately), for mercury =.44approximately) and for glycerine = 725. The andtl number of some fluidsparticularly water) depends considerablyontemperature.le >,Le 2 >,...,Le n >arethelewisnumbersfor n concentrationss,s 2,...,S n,respectively,ta>isthetaylornumber,ra>is thethermalrayleighnumber,r >,R 2 >,...,R n >aretheconcentrationrayleigh numbers for the n concentration components. A concentration Rayleigh number is the ratio of the buoyancy forceswhich drive free convective transport of solute) to dispersive/viscous forceswhich disperse solute and dissipate free convective transport). In the present problem, these have stabilizing effect on the onset of instability. w is vertical velocity, θ is temperature andφ,φ 2,...,φ n arerespectiveconcentrationsofthen components.σ=σ r +iσ i isthe complexgrowthratewhereσ r andσ i arerealconstants.forσ r <,thesystemisalwaysstable whileforσ r >,thesystembecomesunstable.whenσ=,thesystemismarginallystable

5 The onset of convection in a rotating multicomponent fluid layer 48 ensuringthevalidityoftheprincipleoftheexchangeofstabilities.whenσ r andσ i, the overstability of periodic motion is possible and oscillatory motions of growing or neutral amplitude occur. It may further be noted that equations2.3) and2.4) describe an eigenvalue problem for p and govern rotatory hydrodynamic multicomponent convection for quite general nature of the bounding surfaces. Theorem:Ifw,θ,φ,φ 2,...,φ n,ζ,σ),ra>,r >,R 2 >,...,R n >,Ta>, σ r isasolutiontoeqs.2.3)and2.4)togetherwithboundaryconditions2.5)-2.7) and R 2Le 2 π 4+ R 2 2Le 2 2π R n 2Le 2 n π 4+Ta π 4 thenσ i =.Inparticular, σ r = σ i = if R R 2 R n 2Le 2 π4+ 2Le 2 2 π Le 2 n π4+ta π 4 oof:multiplyingeq.2.3) byw thesuperscript henceforthdenotescomplexconjugation) throughout and integrating the resulting equation over the vertical range of z, we have w D 2 a 2 ) D 2 a 2 σ ) R 2 a 2 wdz=raa 2 w φ 2 dz... R n a 2 w θdz R a 2 w φ n dz+ta w Dζdz w φ dz 2.8) MakinguseofEqs.2.3) 2 6 and2.4),wecanwrite w D 2 a 2 ) D 2 a 2 σ ) wdz= Raa 2 +R a 2 Le +R n a 2 Le n φ D 2 a 2 σ Le ) φ dz+r 2a 2 Le 2 θd 2 a 2 σ )θ dz φ n D 2 a 2 σ Le n ) φ n dz+ta φ 2 D 2 a 2 σ Le 2 ) φ 2 dz+... ζ D 2 a 2 σ ) ζ dz IntegratingthevarioustermsofEq.2.9)bypartsforanappropriatenumberoftimesand utilizing boundary conditions2.5)-2.7), we obtain D 2 w 2 +2a 2 Dw 2 +a 4 w 2 )+ σ =Raa 2 Dw 2 +a 2 w 2 )dz Dθ 2 +a 2 θ 2 +σ θ 2 )dz R a 2 Le Dφ 2 +a 2 φ 2 + σ Le φ 2) dz 2.9) R 2 a 2 Le 2 Dφ 2 2 +a 2 φ σ φ 2 2 )dz... 2.) Le 2

6 482 J. akash et al. R n a 2 Le n Ta Dφ n 2 +a 2 φ n 2 + σ Le n φ n 2) dz Dζ 2 +a 2 ζ 2 + σ ζ 2) dz EquatingtheimaginarypartsofbothsidesofEq.2.)andcancellingσ i )throughoutfrom the resulting equation, we have Dw 2 +a 2 w 2 )dz= Raa 2 +R 2 a 2 φ 2 2 dz+...+r n a 2 θ 2 dz+r a 2 φ n 2 dz+ Ta φ 2 dz ζ 2 dz 2.) Now,fromequation2.3) 3 wederivethat D 2 a 2 σ ) φ D 2 a 2 σ ) φ Le Le dz= Le 2 w 2 dz 2.2) IntegratingthevarioustermsonthelefthandsideofEq.2.2)bypartsforanappropriate numberoftimesandmakinguseoftheboundaryconditionsonφ,itfollowsthat D 2 φ 2 +2a 2 Dφ 2 +a 4 φ 2 )dz+ 2σ r Dφ 2 +a 2 φ 2 )dz Le + σ 2 Le 2 φ 2 dz= Le 2 w 2 dz 2.3) Sinceσ r,itfollowsfromeq.2.3)that 2a 2 Dφ 2 dz Le 2 w 2 dz 2.4) Now, since φ,φ 2,...,φ n and w satisfy the boundary conditions φ ) = = φ ), φ 2 )==φ 2 ),...,φ n )==φ n ),w)==w),wehavefromtherayleigh-ritz inequalityschultz, 973) Dφ 2 dz π 2 Dφ 2 2 dz π 2 φ 2 dz φ 2 2 dz 2.5). Dφ n 2 dz π 2 φ n 2 dz

7 The onset of convection in a rotating multicomponent fluid layer 483 and Dw 2 dz π 2 w 2 dz 2.6) respectively. Utilizinginequalities2.5) and2.6)ininequality2.4),weget a 2 φ 2 dz 2Le 2 π 4 Dw 2 dz 2.7) Inthesamemanner,weobtainfromEqs.2.3) 4 6 theinequalities a 2. a 2 φ 2 2 dz 2Le 2 2π 4 φ n 2 dz 2Le 2 n π4 Dw 2 dz Dw 2 dz 2.8) respectively. Nowforthecaseofrigidboundaries,ζ)==ζ),againfromtheRayleigh-Ritzinequality Schultz, 973), we obtain Dζ 2 dz π 2 ζ 2 dz 2.9) MultiplyingEq.2.4)byζ andintegratingovertheverticalrangeofz,wegetfromthereal part of the final equation Dζ 2 +a 2 ζ 2 +σ r ζ 2 )=R ζ Dw dz ζ Dwdz ζ Dw dz using Schwartz inequality) which implies that Dζ 2 dz ζ 2 dz Dw 2 dz ζ 2 dz ζ Dwdz Dw 2 dz ζ Dw dz 2.2) andthususinginequality2.9)forthecaseofrigidboundariesandtheresult Dζ 2 dz= π 2 ζ 2 dzforthecaseoffreeboundariesbanerjeeetal.,995),weobtain π 2 ζ 2 dz ζ 2 dz Dw 2 dz

8 484 J. akash et al. which gives ζ 2 dz π 2 which implies that ζ 2 dz π 4 Dw 2 dz Dw 2 dz 2.2) Now using inequalities2.7),2.8) and2.2) in Eq.2.), we obtain [ R R 2 R )] n 2Le 2 π4+ 2Le 2 2 π Le 2 n π4+ta π 4 Dw 2 dz + a2 σ w 2 dz+raa 2 θ 2 dz< 2.22) whichclearlyimpliesforσ i )that R R 2 R n 2Le 2 π 4+ 2Le 2 2π Le 2 n π 4+Ta π4> 2.23) R Henceif + R R n + Ta,thenwemusthaveσ 2Le 2 π4 2Le 2 2 π4 2Le 2 n π4 π 4 i =. This proves the theorem. TheessentialcontentofTheoremfromthephysicalpointofviewisthatfortheproblem of rotatory hydrodynamic multicomponent convection, an arbitrary neutral or unstable mode of the system is definitely non-oscillatory in character and, in particular, the principle of the R exchangeofstabilities isvalidif + R R n + Ta 2Le 2 π4 2Le 2 2 π4 2Le 2 n π4 π.further,theabove 4 result is uniformly valid for quite general nature of the boundaries. Specialcases:ItfollowsfromTheoremthatanarbitraryneutralorunstablemodeisnon oscillatory in character, and in particular PES is valid for:.rayleigh-benardconvectionr =R 2 =...=R n =Ta=)PellewandSouthwell, 94) 2.RotatoryRayleigh-BenardconvectionR =R 2 =...=R n =)ifta/π 4 Gupta et al., 986) 3.Rotatory thermohaline convection R >, R 2 =... = R n =, T a > ) if R 2Le 2 π 4+Ta π 4 Guptaetal.,986) 4.ThermohalineconvectionR >,R 2 =...=R n =Ta=)if al.,986) R 2Le 2 π4 Guptaet 5.RotatoryhydrodynamictriplydiffusiveconvectionR >,R 2 >,R 3 =...=R n =, R Ta>)if R 2 2Le 2 π4+ 2Le 2 2 π4+ta π 4 akashetal.,24b) 6.Triply diffusive convection R >, R 2 >, R 3 =... = R n = Ta = ) if R R 2 2Le 2 π4+ 2Le 2 2 π4 akashetal.,24a)

9 The onset of convection in a rotating multicomponent fluid layer 485 oceeding in this manner, we can obtain conditions for stationary convection for all configurationswith3,4,...,n concentrationcomponents,respectively. Since the complement of the above result implies the occurrence of oscillatory motions, thus it is important to derive the bounds for the complex growth rate of oscillatory motions. We prove the following theorem in this direction. Theorem2:IfR a >,R >,R 2 >,...,R n >,Ta>,σ r andσ i,thena necessaryconditionfortheexistenceofanontrivialsolutionw,θ,φ,φ 2,...,φ n,ζ,σ) to Eqs.2.3) and2.) together with boundary conditions2.5)-2.7) is that { σ <max R +R R n ), } Ta oof: Rewriting equation2.) for ready reference, we have Dw 2 +a 2 w 2 )dz= Raa 2 +R 2 a 2 φ 2 2 dz+...+r n a 2 Nowsinceσ r,itfollowsfromeq.2.3)that θ 2 dz+r a 2 φ n 2 dz+ Ta φ 2 dz ζ 2 dz φ 2 dz σ 2 w 2 dz 2.24) Similarly,fromEqs.2.3) 4 and2.3) 6,byadoptingthesameprocedure,weget φ 2 2 dz σ 2 w 2 dz. 2.25) φ n 2 dz σ 2 w 2 dz respectively. Multiply Eq.2.4) by its complex conjugate, integrating the resulting equation by parts for an appropriate number of times and using boundary conditions2.5)-2.7), we have D 2 ζ 2 +2a 2 Dζ 2 +a 4 ζ 2 )dz+ 2σ r + σ 2 2 ζ 2 dz= Dw 2 dz Dζ 2 +a 2 ζ 2 )dz 2.26) Since,σ r,itfollowsfromeq.2.26)that ζ 2 dz 2 σ 2 Dw 2 dz 2.27)

10 486 J. akash et al. Now making use of inequalities2.24),2.25) and2.27), in Eq.2.), we have Ta2 ) σ 2 +Raa 2 Dw 2 dz+ a2 [ R +R R n )] σ 2 θ 2 dz< w 2 dz 2.28) which clearly implies that { σ <max R +R R n ), } Ta This establishes the desired result. Theabovetheoremmaybestatedinanequivalentformas:thecomplexgrowthrate of an arbitrary, neutral or unstable oscillatory perturbation of growing amplitude in a rotatory hydrodynamic multicomponent fluid layer heated from below must lie inside a semicircleintherighthalfofthep r,p i )-planewhosecentreisattheoriginandradiusequals R max{ +R R n ), } Ta.Further,itisprovedthatthisresultisuniformly valid for quite general nature of the bounding surfaces. Special cases: The following results may be obtained from Theorem 2 as special cases:.forrotatoryrayleigh-benardconvectionr ==R 2 =...=R n =,Ta>) σ <Ta Banerjee et al., 98) 2.ForthermohalineconvectionR >,R 2 =...=R n =Ta=) σ < R Banerjee et al., 98) 3.For rotatory Thermohalineconvection oftheveronistypeturner,985) R >, R 2 =...=R n =,Ta>) σ <max{ R, } Ta Gupta et al., 983) 4.FortriplydiffusiveconvectionR >,R 2 >,R 3 =...=R n =Ta=) σ < R +R 2 ) akash et al., 24c) oceeding in this manner, we can obtain bounds for the complex growth rate for all configurationswith3,4,...,n concentrationcomponents,respectively.

11 The onset of convection in a rotating multicomponent fluid layer Conclusions The present analysis generalizes the previous published results for rotatory hydrodynamic singly, doubly and triply diffusive convection. The mathematical analysis carried out here yields a sufficient condition for the validity of the principle of the exchange of stabilities in rotatory hydrodynamic multicomponent convection. Since the complement of this condition implies the occurrence of oscillatory motions, the bounds for the complex growth rate are also obtained as the second problem. It is further proved that the results obtained herein are uniformly applicable for quite general nature of bounding surfaces. Acknowledgment The authors would like to thank the learned reviewer for his valuable comments, which helped the authors to bring the manuscript in the present form. First authorj. akash) also acknowledges the recently awarded financial assistance by UGC, New Delhi in the form of major research projectgrant number, 43-42/24SR)). References. Banerjee M.B., Katoch D.C., Dube G.S., Banerjee K., 98, Bounds for growth rate of perturbation in thermohaline convection, oceedings of the Royal Society of London, A, 378, Banerjee M.B., Shandil R.G., Lal P., Kanwar V., 995, A mathematical theorem in rotatory thermohaline convection, Journal of Mathematics Analysis and Applied, 89, Brandt A., Fernando H.J.S., 996, Double diffusive convection, American Geophysical Union, Washington, DC 4.GriffithsR.W.,979a,Anoteontheformationofsaltfingeranddiffusiveinterfacesinthree component systems, International Journal of Heat and Mass Transfer, 22, Griffiths R.W., 979b, The influence of a third diffusing component upon the onset of convection, Journal of Fluid Mechanics, 92, Gupta J.R., Sood S.K., Bhardwaj U.D., 986, On the characterization of non-oscillatory motions in a rotatory hydromagnetic thermohaline convection, Indian Journal of Pure and Applied Mathematics, 7,, Gupta J.R., Sood S.K., Shandil R.G., Banerjee M.B., Banerjee K., 983, Bounds for the growth of a perturbation in some double-diffusive convection problems, Journal of Australian Mathematics Society, Ser. B, 25, Kellner M., Tilgner A., 24, Transition to finger convection in double diffusive convection, Physics of Fluids, 26, Lopez A.R., Romero L.A., Pearlstein A.J., 99, Effect of rigid boundaries on the onset of convective instability in a triply diffusive fluid layer, Physics of Fluids A, 2, 6, Nield D.A., Kuznetsov A.V., 2, The onset of double-diffusive convection in a nano fluid layer, International Journal Heat Fluid Flow, 32, 4, Pearlstein A.J., Harris R.M., Terrones G., 989, The onset of convective instability in a triply diffusive fluid layer, Journal of Fluid Mechanics, 22, Pellew A., Southwell R.V., 94, On the maintained convective motion in a fluid heated from below, oceedings of the Royal Society of London, A, 76, PoulikakosD.,985,Theeffectofathirddiffusingcomponentontheonsetofconvectionina horizontal porous layer, Physics of Fluids, 28,,

12 488 J. akash et al. 4. akash J., Bala R., Vaid K., 24a, On the characterization of nonoscillatory motions in triply diffusive convection, International Journal of Fluid Mechanics Research, 4, 5, akashJ.,BalaR.,VaidK.,24b,Ontheprincipleoftheexchangeofstabilitiesinrotatory triply diffusive convection, oceeding National Academy Science, Physical Sciences, India, 84, 3, akashJ.,BalaR.,VaidK.,24c,Upperlimitstothecomplexgrowthratesintriply diffusive Convection, oceeding Indian National Science Academy, 8,, akashJ.,BalaR.,VaidK.,25,OnarrestingthecomplexgrowthratesinRotatoryTriply Diffusive convection, communicated for publication 8. Radko T., 23, Double-Diffusive Convection, Cambridge university ess 9. Rionero S., 23a, Multicomponent diffusive-convective fluid motions in porous layers ultimately boundedness, absence of subcritical instabilities, and global nonlinear stability for any number of salts, Physics of Fluids, 25, 544, Rionero S., 23b, Triple diffusive convection in porous media, Acta Mechanics, 224, Rionero S., 24, Onset of convection in rotating porous layers via a new approach, Discrete and Continuous Dynamical Systems, Ser. B, 9, 7, Ryzhkov I.I., 23, Long-wave instability of a plane multicomponent mixture layer with the soret effect, Fluid Dynamics, 4, 48, Ryzhkov I.I., Shevtsova V.M., 27, On thermal diffusion and convection in multicomponent mixtures with application to the thermogravitational column, Physics of Fluids, 9, 27, Ryzhkov I.I., Shevtsova V.M., 29, Long wave instability of a multicomponent fluid layer with the soret effect, Physics of Fluids, 2, 42, Schmitt R.W., 2, Thermohaline convection at density ratios below one: A new regime for salt fingers, Journal of Marine Research, 69, Schultz M.H., 973, Spline Analysis, entice-hall Inc. Englewood Cliffs NJ 27. Sekar R., Raju K., Vasanthakumari R., 23, A linear analytical study of Soret-driven Ferro thermohaline convection in an anisotropic porous medium, Journal of Magnetism and Magnetic Materials, 33, Shivakumara I.S., Naveen Kumar S.B., 24, Linear and weakly nonlinear triple diffusive convection in a couple stress fluid layer, International Journal of Heat and Mass Transfer, 68, Terrones G., 993, Cross-diffusion effects on the stability criteria in a triply diffusive system, PhysicsofFluidsA,5,9, Terrones G., Pearlstein A.J., 989, The onset of convection in a multicomponent fluid layer, PhysicsofFluidsA,,5, Turner J.S., 973, Buoyancy Effects in Fluids, Cambridge University ess 32. Turner J.S., 974, Double diffusive phenomenon, Annual Review of Fluid Mechanics, 6, Turner J.S., 985, Multicomponent convection, Annual Review of Fluid Mechanics, 7, -44 Manuscript received January 2, 25; accepted for print September 5, 25