Dufour-Driven Thermosolutal Convection of the Veronis Type

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1 JOURNAL OF MATEMATICAL ANALYI AND APPLICATION 8, ARTICLE AY Dufour-Driven Thermosolutal Convection of the Veronis Tye ari Mohan Deartment of Mathematics, imachal Pradesh Uniersity, ummer ill, himla, 75, India ubmitted by Maria Clara Nucci Received Aril 7, 997 The resent aer mollifies the nastily behaving governing equations of Dufourdriven thermosolutal convection of the Veronis tye by the construction of an aroriate linear transformation extends the results of M B Banerjee et al Ž Proc Roy oc London er A 378, 98, 3; J Math Anal Al 79, 993, 37 concerning the linear growth rate the behaviour of oscillatory motion 998 Academic Press INTRODUCTION The stability roerties of binary fluids are quite different from ure fluids because of oret Dufour 5 effects An externally imosed temerature gradient roduces a chemical otential gradient the henomenon, known as the oret effect, arises when the mass flux contains a term that deends uon the temerature gradient The analogous effect that arises from a concentration gradient deendent term in the heat flux is called the Dufour effect Although it is clear that the thermosolutal oretdufour roblems are quite closely related, their relationshi has never been carefully elucidated They are in fact, formally identical this is done by means of a linear transformation that takes the equations boundary conditions for the latter roblem into those for the former Banerjee et al 3 formulated a novel way of combining the governing equations boundary conditions for Veronis thermosolutal configuration derived a semi-circle theorem rescribing uer limits for the comlex growth rate of an arbitrary oscillatory erturbation neutral or X98 $5 Coyright 998 by Academic Press All rights of reroduction in any form reserved

2 57 ARI MOAN unstable Guta et al 6 derived sufficient conditions for the validity of the rincile of exchange of stabilities Ž PE in the Veronis thermosolutal configuration A close critical examination of the roofs of the results of Banerjee et al Guta et al yields the following drawbacks in their analysis Ž i Many ositive definite integrals have been deleted in deriving the final results Ž ii Banerjee et al have combined the governing equations of the roblem in such a manner that the final equation from where the conclusion is drawn involves the growth rate in an aroriate combination with the thermosolutal concentration Rayleigh number the Prtl number owever, this does not lead to a sufficient condition for the validity of PE Ž iii Guta et al have deleted the ositive definite integrals have obtained an uer estimate for the term involving the concentration in such a way that the final inequality involves only the arameters of roblem but not the comlex growth rate the wave number thereby recluding the ossibility of obtaining an uer estimate for the growth rate ince the Dufour-driven thermosolutal convection roblem thermosolutal roblem are formally identical therefore the above inherent drawbacks in the analysis of Banerjee et al Guta et al can be taken into account while deriving the uer limit for the comlex growth rate of an arbitrary oscillatory erturbation, neutral or unstable for the Dufourdriven thermosolutal roblem of the Veronis tye Further, a recently established characterisation theorem of Banerjee et al 4, disroving the existence of neutral or unstable oscillatory motions in an initially bottom heavy thermosolutal convection configuration of the Veronis tye with a quite general nature of the bounding surfaces whenever the thermosolutal Rayleigh number is less than a critical value, has brought a fresh outlook to the subject matter of thermosolutal convection aved the way for further theoretical exerimental investigation in this field of enquiry The aim of the resent aer is to extend the results of Banerjee et al 3, 4 to the Dufour-driven thermosolutal convection roblem of the Veronis tye owever, the governing equations in the resent case are not amenable to the analysis followed by Banerjee et al on account of couling between,, which was not the case for the roblem considered by Banerjee et al Therefore, there arises the non-trivial need to mollify the nastily behaving governing equations by the construction of an aroriate linear transformation derive the desired extensions

3 DUFOUR-DRIVEN TERMOOLUTAL CONVECTION 57 MATEMATICAL FORMULATION AND ANALYI The relevant governing equations boundary conditions of Dufourdriven thermosolutal convection of the Veronis tye with a slight change in notations are easily seen to be given by 7, 8 ž / Ž D a D a RTaRa, Ž Ž D a R3Ž D a, Ž D a, Ž 3 where RT gd 4, Rs gd 4,,, R3, D C, Ž D C is called the Dufour coefficient, with or or D at z z Ž both boundaries dynamically free Ž 4 D at z z Ž both boundaries rigid Ž 5 D at z D at z or Ž lower boundary dynamically free uer boundary rigid Ž 6 D at z D at z Ž lower boundary rigid uer boundary dynamically free, Ž 7 where z is the real indeendent variable such that z, D ddz is the differentiation with resect to z, a is a constant, is a constant, is a constant, is a constant, RT R are ositive constants,

4 57 ARI MOAN r ii is a comlex constant in general such that r i are real constants as a consequence the deendent variables Ž z rž z iiž z, Ž z rž z i Ž z, Ž z rž z iiž z are comlex valued functions of the real variable z The meanings of the symbols from a hysical oint of view are as follows: z is the vertical coordinate, ddz the differentiation along the vertical direction, a the square of the wave number, the Prtl number, the Lewis number, is here referred to as the Dufour number, RT the Rayleigh number, R the concentration Rayleigh number, the comlex growth rate, the vertical velocity, the temerature, the concentration It may further be noted that Eqs ŽŽ 7 describe an eigenvalue roblem for govern Dufour-driven thermosolutal instability for any combination of dynamically free rigid boundaries We rove the following theorems TEOREM If Ž,,,, r i i, r, i,, R, R, is a solution of Eqs ŽŽ T 3 with either of the boundary conditions ŽŽ 4 7 then where R 4 4, Ž R 6R Ž s Ž RTR3 Rs R Proof Using the transformations R 3 Ž 9 Eqs ŽŽ 7 assume the forms ž / T Ž D a D a R ar a, Ž Ž D a B, Ž Ž D a, Ž

5 DUFOUR-DRIVEN TERMOOLUTAL CONVECTION 573 with or or or D at z z Ž 3 D at z z, Ž 4 D at z D at z 5 Ž 5 D at z 5, Ž 6 D at z where R R R Ž, R R R, BŽ R T T 3 T 3, the sign has been omitted for simlicity Multilying Eq Ž by * Ž * indicates comlex conjugation throughout integrating the resulting equation over the vertical range of z, we get * Ž D a D a dz ž / T R a *dz R a *dz Ž 7 Ž Ž Making use of Eqs the fact that,we can write R T T B R a *dz a D a * *dz, Ž 8 / * Ra *dz Ra ž D a *dz Ž 9 Combining Eqs 7 9, we have * Ž D a D a dz ž / R T B a D a * *dz / * Ra ž D a *dz Ž

6 574 ARI MOAN Integrating the various terms of Eq Ž by arts for an aroriate number of times making use of either of the boundary conditions Ž 3 Ž 6, it follows that 4 D a D a dz D a dz R T Ž B a D a * dz / * R a D a ž dz Ž Equating the real imaginary arts of both sides of Eq Ž cancelling i throughout from the imaginary art, we get 4 D a D a dz D a dz R T Ž r B a D a dz r R a D a dz, ž / R T Ž D a dz a dz R a dz Ž 3 B Multilying each of Eqs Ž Ž by their comlex conjugate integrating by arts over the vertical range of Z for an aroriate number of times making use of either of the boundary conditions Ž 3 Ž 6 we get I dz B dz, Ž 4 I dz dz, Ž 5

7 DUFOUR-DRIVEN TERMOOLUTAL CONVECTION 575 where 4 I D a a D dz r D a dz Ž 6 r 4 I D a a D dz D a dz Ž 7 We note that since, I I are ositive definite r Combining Eqs 3, 4, 5, we have a R a B T Ž D dz dz dz R a R a I R T sa I dz Ž 8 Now it follows from Eqs Ž 3, Ž 4, Ž 5, Ž 8 that a dz D a dz, Ž 9 R I B dz, Ž 3 4 dz a D dz a dz Ž 3 dz dz, Ž 3 4 I a D dz a dz Ž 33 We first note that since,, satisfy Ž Ž, Ž Ž, Ž Ž, we have by the RayleighRitz inequality 9 D dz dz, Ž 34 D dz dz, Ž 35 D dz dz, Ž 36

8 576 ARI MOAN 4 D dz dz Ž 37 Using inequalities 34, 37, 9 33 aroriately, we get 4 4 R I dz, Ž 38 9a a D dz dz dz Ž 39 3 R R s a 8 a 3 4 a dz dz dz Ž 4 3 R R It now follows from Eq 8 inequalities 3, 38, 39, 4 that Rs a R4 6R dz Ž 4 which clearly imlies that R 4 4 Ž R 6R Ž Ž This comletes the roof of the theorem Theorem from the hysical oint of view imlies that the comlex growth rate of an arbitrary oscillatory motion of growing amlitude in the Dufour-driven thermosolutal convection of the Veronis tye lies inside a semicircle in the uer half of the r i lane whose centre is at origin radius is ½ 5 R R R COROLLARY If Ž,,,, r i i, r, i,,, R, R is a solution of Eqs ŽŽ T 3 with either of the boundary ŽŽ 4 conditions 4 7 if R 4, then 43 i

9 DUFOUR-DRIVEN TERMOOLUTAL CONVECTION 577 Proof This follows from Theorem The corollary imlies that for the roblem of Dufour-driven thermosolutal convection of the Veronis tye if R s4 4 then an arbitrary neutral or unstable mode is definitely non-oscillatory in character in articular PE is valid TEOREM If RT, R,,, r, i, R Ž 74 4 Ž Ž R B 3 then a necessary condition for the existence of a non-triial solution Ž,,, of Eqs Ž Ž 3 with either of the boundary conditions ŽŽ 4 7 is that R R Ž 44 T Proof We write Eq in the alternative form 4 r D a D a dz D a R a D a dz Ra D a ½ 5 T s B r T s a R a dz R dz, Ž 45 derive the validity of the theorem from the resulting inequality obtained by relacing each one of the terms of this equation by its aroriate estimate Utilizing inequalities Ž 34 Ž 37, we obtain 4 D a D a dz Ž a dz Ž 46 ince, we have r r D a dz Ž 47 Next, multilying Eq by * throughout integrating the various terms on the left h side of the resulting equation by arts for an aroriate number of times by making use of the boundary conditions on

10 578 ARI MOAN Ž Ž, namely, we have from the real art of the final equation Ž r ž / D a dz dz Real art of B * dz B * dz B * dz B * wdzb dz ½ 5 ½ 5 B dz dz Ž utilizing the chwartz inequality Combining this inequality with inequality Ž 35 the fact that r, we get which imlies that thus ½ 5 ½ 5 a dz B dz dz, B ½ 5 a ½ 5 dz dz, B Ž a D a dz dz Ž 48 Further utilizing inequality 36, we have R a D a dz R a a dz, therefore a D a dz Ž utilizing Eq Ž 3 a dz since ž / R a D a dz a dz Ž 49

11 DUFOUR-DRIVEN TERMOOLUTAL CONVECTION 579 Also from Eq 3 the fact that, we have r ½ 5 r T a R dz R dz Ž 5 Now, if ermissible, let R R T Then in that case, we derive from Eq Ž 45 inequalities Ž 46 Ž 5 that ½ ž / 5 R R3Ba Ž a dz, Ž 5 Ž Ž a which imlies that thus we necessarily have 3 Ž a ž / R, abr 3 7 Ž Ž 4 R, 4 BR 3 Ž 3 4 since the minimum value of a a for a is 74 ence if 7 Ž Ž 4 R, 4 R B 3 then we must have R R Ž 5 T This comletes the roof of the theorem Theorem imlies from the hysical oint of view that Dufour-driven thermosolutal convection of the Veronis tye cannot manifest oscillatory motion of growing amlitude in an initially bottom heavy configuration if the thermosolutal Rayleigh number R, the Lewis number, the Prtl number, the Dufour number, R, a ositive number, satisfy the 3 inequality 7 Ž Ž 4 R 4 R B Further this result is uniformly valid for the quite general nature of the bounding surface 3

12 58 ARI MOAN ACKNOWLEDGMENT The author exresses his sincere thanks to Professor J R Guta, Deartment of Mathematics, imachal Pradesh University, himla for his helful discussions guidance during the rearation of the aer REFERENCE R de Groot P Mazur, Nonequilibrium Thermodynamics, 73, North-oll, Amsterdam, 96 G Veronis, On finite amlitude instability in theomohaline convection, J Marine Res 3 Ž 965, 3 M B Banerjee, D C Katoch, G Dube, K Banerjee, Bounds for growth rate of a erturbation in thermohaline convection, Proc Roy oc London er A 378 Ž 98, 3 4 M B Banerjee, J R Guta, Jyoti Parkash, On theormohaline convection of the Veronis, tye, J Math Anal Al 79 Ž 993, 37 5 D D Fitts, Non-equilibrium Thermodynamics, McGrawill, New York, 96 6 J R Guta, K ood, U D Bhardwaj, On characterization of nonoscillatory motions in rotatory hydromagnetic thermohaline convection, Indian J Pure Al Math 7 Ž 986, 7 D Gutkowicz-Krusin, M A Collins, J Ross, RayleighBernard instability in nonreactive binary fluids, Phys Fluids Ž 979, D A Neild, The thermohaline RayleighJaffreys Problems, J Fluid Mech 9 Ž 967, M chultz, aline Analysis, Prentice all, Englewood Cliffs, New Jersey, 973