[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY THERMOSOLUTAL CONVECTION IN A HETEROGENEOUS VISCO-ELASTIC (OLDROYDIAN) FLUID LAYER IN A POROUS MEDIUM Suhir Kumar Associate Professor, S.D. (PG) College, Muzaffarnagar (UP). ABSTRACT The paper critically examines, within the framework of linear stability analysis, the thermosolutal convection in a heterogeneous visco-elastic (Olroyian) flui layer in a porous meium. In the present paper, stationery, oscillatory an non-oscillatory convection have been iscusse in etails. A variational principle is establishe for the present problem. Some Results are iscusse numerically also. Also, principle of exchange of stabilities is not vali in the present problems. KEYWORDS: Thermosolutal instability, Heterogeneous flui, Olroyian flui, Porous Meium. INTRODUCTION The problem of onset of convection in a horizontal layer of flui heate from below, when buoyancy forces arise from ensity ifference ue to variation in temperature, was stuie by Benar [] an Rayleigh [] an the problem, uner varying assumptions of hyromagnetics, has been treate in etail by Chanrashekhar [3], Investigations of thermosolutal convection, when buoyancy forces arise also from variations in solute concentration apart from those ue to variation in temperature, are motivate by its irect relevance to the hyroynamics of oceans, as well as its interesting complexities, as a ouble-iffusion phenomenon. Stomell et al. [4] i the pioneering work in this irection. Since then the problem of thermohaline or thermosolutal convection has been stuie in three basic configurations by Stern [5], Veronis [6] an Niel [7] respectively, when the flui layer heate an solute from above, heate an solute from below an heate from below an solute form above. An experimental emonstration by Toms an Strabrige [8] has reveale that a ilute solution of methyl methacrylate in n-butyl acetate agrees well with the theoretical moel of Olroyian visco-elastic flui propose by Olroy [9]. Sharma [] has stuie the instability of the plane interface between two Olroyian visco-elastic superpose conucting fluis in the presence of a uniform magnetic fiel. In view of the fact that the stuy of visco-elastic flui in a porous meium fins, applications in geophysics an chemical technology, a number of researchers have contribute in this irection. However, the thermosolutal convection in a heterogeneous visco-elastic (Olroyian) flui layer in a porous meium seems, to the best, of our knowlege, uninvestigate so far. In this paper, therefore, we have examine the stability of a visco-elastic [Olroyian] flui layer heate an solute from below in a porous meium first stuie by Khare an Sahai [] leaing to an averse temperature graient an a solute concentration graient with free bounaries when the initial non-homogeneity is present in the flui. Hence it can be looke upon as an extension of thermosolutal convection in a homogenous flui layer in porous meium iscusse by Khare an Sahai. CONSTITUTIVE EQUATIONS AND THE EQUATIONS OF MOTION Let T ij ij, e ij ij, p, q i, T an C enote respectively the total stress tensor, shear stress tensor, rate of strain tensor, Kronecker elta, scalar pressure, velocity vector of the flui, viscosity, stress relaxation time, strain retaration time, temperature fiel an concentration fiel. Then the Olroyian visco-elastic flui is escribe by the constitutive equations [578]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 Tij pij ij ij eij t t () q q i j an eij x j xi where q. is the total erivative being the sum of local an convective erivatives. t t Let us consier a horizontal layer of saturate porous meium of thickness between two free-bounaries z = an z =, z-axis being vertically upwar. Let the interstitial flui (flui in the porous meium) be visco-elastic, f(z), where f(z) is a monotonic function of z with f() = an is such that f z is constant. The layer is infinite in horizontal irections an is heate an solute from below leaing to an averse temperature graient uniform solutal graient (T T ) an a (S S ), where T an T are the constant temperatures of the lower an upper bounaries with T > T an also S an S are the constant solute concentrations of the lower an upper surfaces with S > S. The effective ensity is the superposition of inhomogeneity escribe by (a) f(z) an (b) T) S)] which is cause by temperature an solute graients. This leas to the effective ensity [f(z) (T T) (S S)]. () When the flui flows through a porous meium, the gross effect is represente by Darcy's law. As a result, the usual viscous term is replace by the resistance term permeability an the filter velocity. Hence, the basic equations are Dq V, where k an V enote respectively the meium k t t [ t p X i] k, t q (3). q (4) an ( q. ), t (5) T (. )T k q T T, t (6) C (. )C k q S C, t (7) [579]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 where q an are respectively the velocity an kinematic viscosity of the flui respectively, k is the intrinsic permeability of the meium an k correspons to non-porous meium. BASIC STATE AND THE PERTURBATION EQUATIONS The initial stationary state, whose stability we wish to examine is that of an incompressible, viscous, visco-elastic (Olroyian) flui arrange in a horizontal strata in a non-homogeneous an isotropic porous meium. The system is acte upon by a temperature T, concentration C an the gravity fiel g (,, g). The initial state whose stability we wish to examine is thus characterize by z q (,, ), T T z, C C z, [f(z) z z]an p p gz. (8) where p. Following the usual proceure an normal moe technique given by F(z) exp.[i(k x k y) nt], (9) x y where k kx ky is the real wave number of propagation an n is the frequency of arbitrary isturbance. We get Ra [ FP ] [ (D a )(D a P )][ (D a ) P ]w [D a P ] P [ (D a ) P ]w R a [ (D a ) P ]w R a (D a P ) w P [ F P ][D a P ][ (D a ) P ]w. () where 4 g R kt is the thermal Rayleigh number, P is the thermal Prantl number, kt 4 g k R kt is the concentration Rayleigh number, S is the Lewis number, kt 4 f g k F T z is the elastic parameter, R,P an. kt k Equation () can also be written as [ FP ][ P (D a )(D a P )][ (D a ) P ]w Ra [D a P ] [ (D a ) P ]w P Ra [ (D a ) P ]w P R a (D a P ) w P P [ F P ][D a P ][ (D a ) P ]w. () The solution of the equation () is to be obtaine uner the following bounary conitions : w = D an z =. () [58]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 RESULTS AND DISCUSSION (A) Stationary Convection Hence for stationary convection, equation () becomes R a a ) w =. (3) Multiplying equation (3) by w* an integrating over the range of z, we have Ra (D a ) ww *z 4 or Ra ( D w a a Dw )z (4) In view of the bounary conitions () an following Chanrashekhar [3] only possible solutions. Thus, we observe that the hypothesis that initial state solutions are perturbe is contraicte. Therefore, the instability can not set in as stationary convection, or in other wors the Principle of Exchange of Stabilities (PES) is not vali for the problem uner investigation. (B) Oscillatory Convection Now for the proper solution of equation () for w belonging to the lowest moe, we follow Chanrashekhar [3] an assume that solution w satisfying the bounary conitions is given by w = w Equation () yiels [ FP ] [ P (D a )(D a P )][ (D a ) P ]w Ra [D a P ] [ (D a ) P ]w P Ra [ (D a ) P ]w P R a (D a P ) w P P [ F P ](D a )[D a P ][ (D a ) P ]w. Let I [D a ][D a P ][ (D a ) P ]w, I 3 [ (D a ) P ]w an I 4 [D a P ]w. Substituting w = w, I, I 3 an I 4, we get I [D a ][D a P ][ (D a ) P ]w sin z [ ( a ) P ][ a P ][ a ]wsin z I [D a P ][ (D a ) P ]wsin z, [ ( a ) P ][ a P ]wsin z, I 3 [ (D a ) P ]wsin z [ ( a ) P ] an I 4 [D wsin z a wsin z P wsin z], [ a P ]wsin z Substituting for I, I, I 3 an I 4 from (6) to (9) into equation (5), we have (5) I [D a P ][ (D a ) P ]w, (6) (7) (8) (9) P [ FP ][ a ][ a P ][ ( a ) P ] R a [ FP ][ a P ] [58]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 [ ( a ) P ] P Ra [ ( a ) P ][ FP ] P R a [ a P ] [ FP ] P P [ FP ][ a ][ a P ][ ( a ) P ]. () Moreover, equation () can be rewritten as Ra Ra [ a P ] [ a ][ a P ] [ a P ] Ra P [ ( a ) P ] P ( a )[ a P ][ FP ] [ FP ] RX 3 R4X[ X P ] [ X][ X P ] [ X P ] RX P [ ( X) P ] P ( X)[ X P ][ FP ] () [ FP ] where R R R a F R 3,R 4,R 4 4,X, an F 4. As iscusse earlier, the Principle of Exchange of Stabilities being not vali for the present problem, the marginal i i i in equation (), the equation at the marginal state is obtaine as RX [ X][( X)i 3 i i P ] [i[ X] ip ] RX ip R4X[ ( X) i P ] iip [ X][ ] PX[ X] [ [ X] i P ] [ F P i ] 3 3 [[ X] [ X]F P i F P i ( ) i( i P [ X]F P i ( ) F P i ] () The real part of equation () is given by R4X[ ( X) i P ] R [ X] i P R3X X [ [ X] ip ] P [ X] [[ X] [ X]F P i F P i ( )] (3) [ F P i ] Also, the imaginary part of equation () is given by R3X[ X] R4Xi P [ X][ ] [ X] i ip i [ [ X] ip ] 3 3 P [ X][ i P i [ X]F P i( ) F P i ] [ F P i ] This leas to the following sixth egree equation in i : 6 4 i i i A B C D, (4) [58]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 3 where, A ( X) P F [( X) P P F ] 3 4 3 4 B ( X) P ( X) F P R3X( X)F P 4 3 4 R4 X[( X)( )F P ( ) ( X) PF P ( ), 4 3 C ( X) P R3X( X) [P F P ] R4P X( X)( ) PP ( X)[ F ( X)( )]. An D = R 3 X[ + X] 3. i is given by equation (4). We now iscuss the existence of overstable marginal state uner various situations. Case : The case when f (i) R 3 > z An (iii) Observe that (i) an (ii) R 3 > i for which (ii) k k i S T is positive (see equation (4)). It follows that the overstability cannot occur at the marginal state. However, the situation contrary to our assume conitions in this case, in general, oes not automatically guarantee the occurrence of overstability. In fact, for R 3 (i) may exist. (ii) If R 3 -elastic parameter F is so large that it makes either of B or C negative an satisfying the inequality 4[BD C ][AC B ] > [AD BC]. Thus, we see that the visco-elasticity has an effective role in instability criteria as there is a sufficient room for the existence of marginal state even if R 3 Case : When R 3 <, one of the roots of equation (4) is always positive irrespective of the other parameters. Therefore the marginal state an overstability essentially occur. (C) Nature of Non-Oscillating Moes : For R 3 >, k S > k T non- i r r r an w = w 5 4 3 D r D r D r D3 r D4 r D5, where D P ( a )F, 3 D P ( a )[ P F P F( a )( )], 3 3 D3 FP a ( a )[R( ) R R ] P P ( a )[F ( a ) ( )] (5) D FP a (R R R) ( a ) P [( a ) P F ( )] P P ( a )], [583]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 3 P ( a ) P a (R R R ), D4 P a ( a )[R R R( )] P ( a )[a RF P ( a )] an D5 Ra ( a ). r with real coefficients an thus has five roots, which may be real. Since R 3 5 in the characteristic equation being negative, hence has at least one positive real root an thus making the system unstable. Thus, we conclue that non-oscillatory moes are unstable in nature. VARIATIONAL PRINCIPLE A variational principle can be establishe for the present problem following Chanrashekhar [3]. Let one of the characteristic values be n i an let the corresponing solutions be enote by a subscript i, then ( n) k L n Dw k ( n) g f [ an n] DL nw w g g w, n z k [ n] (6) (7) From equation (6), we have, g f [ n ] DL n w w g g w, i i i i i i i ni z k [ n i] i Also from equation (7) ( n) k L i n Dwi (9) k ( n) Let n j be a characteristic value ifferent from n i, an let subscript j istinguishes the corresponing solutions. We multiply equations (8) an (9) respectively by w j an DW j an integrate them with respect to z from z = an z = using the bounary conitions w D w X Dz at z = an z =. (3) We have g f ( n i) (DL i)w jz ni wiw jz ni z k ( n i) g w z g w z (3) i j i j ( n i) an k L i(dw j)z ni DwiDw jz. (3) k i ( n i) Substituting the characteristic values n j, w j j i i respectively, integrating the same from z = an z = uner the bounary conitions (3) an substituting the results in equation (3), we get g f ( n i) L i(dw j)z ni wiw jz ni z k ( n i) (8) [584]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 g [k T D i D j k T k ij n ji j ]z g [ksdid j ksk i j n ji j]z (33) Also integrating the L.H.S. of equation (3) by parts an using the bounary conitions (3), we get ( n i) k L i(dw j)z ni DwiDw jz. (34) k ( n i) Putting i = j an suppressing the subscript, equations (33) an (34) yiel ( n) nw (Dw) z (w (Dw) z k ( n) k k g f g gkt w z n z [(D ) k ( ) ]z n z g gks z [(D ) k ( ) ]z (35) Equation (35) provies a basis for the variational formulation of the problem as iscusse below: Let J w (Dw) z, J w (Dw) z, k k k f g J3 w z, J4 z, z (36) g gk T J5 z,j 6 [(D ) k ( ) ]z, gk S an J 7 [(D ) k ( ) ]z, With the help of equation (36), equation (35) can be written as ( n) g nj J J 3 (J4 J 5) n J6 J7 ( n) n g n n[j J4 J 5] J3 J J6 J7 (37) n n n in respectively. ' the corresponing changes in Js, i enote by gives ' Ji s. We can analyse these changes with the help of equation which [585]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 g ( ) g n n J J4 J5 J 3 J J J4 J5 J3 J n ( n) n n J6 J7 (38) ' ' We, now use the expressions for Ji sgiven by (36) to evaluate Ji s. Integrating by parts a suitable number of times an using (3), we fin J (D w )wz, J (D k ) w z, k kk f g J 3 (w).wz, J 4 ( ) z, z (39) g gkt J 5 ( ) z, J 6 (D k ) z, gk S an J 7 (D k ) z. Combining equations (38) an (39) an using equations (3) to (37) in it an rearranging the terms, we get n g J J4 J6 J 3 J g (w w) z n ( n) g (w w) z (4) Multiplying equation of magnetic fiel by an integrating w.r.t. z from z = to z =, we get nz [ktk k T(D ) ]z wz. (4) an z =, we get n z nz [ktk ktd ]z wz. (4) Subtracting equation (4) from equation (4) an integrating by parts, using bounary conitions (3), we get after some rearrangement of the terms g [w w]z nj4 (43) Proceeing similarly, we get g [w w]z nj5 (44) Using equations (43) an (44) equation (4) reuces to n g ( ) J J4 J6 J 3 J. (45) n ( n) [586]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 Now, it is evient from equation (37) that the quantity within [ ] on the L.H.S. of equation (45) can not vanish. Therefore, equation (35) provies a basis for the variational formulation of the problem uner investigation. NUMERICAL COMPUTATIONS The effect of various parameters on the instability criteria is stuie with the help of numerical computations using variational principle. sin lz an sin lz, where w The substitution of the trial solution in equation (35) an its further simplification ultimately gives, in imensionless 5 4 3 3 4 5 where 3 A PF( y ), A P ( y )[ P P F( y )( )] P ( y )P P Fy (R R R5 p b )], 4 3 A 3 ( y ) z[ P PF P Fy ( )[R R Rb p ( )] P P ( y ) ( ) P y (R R R5b p ) 4 3 3 A 4 ( y ) P Rb y P ( y )( F( y ) ( )] y ( )(R R ) 5 an A5 Rb y ( y ). where the quantities have been non-imensionalize as k n k y, b,, P S,, l l k kt 4 4 3 g g g P, R, R an R 5. k kt kt The roots of equation (46) have been locate for ifferent values of b, y, R, R an P by making use of numerical A A A A A A, (46) stabilizing effect on the system. [587]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785.365 P =.5, F =, P = 3, t=.5 R =, R = 5, R 5=, b =.5, p = 3.459.36.355 Unstable a c.35.345 Stable.34.335..4.6.8. e Critical Wave Number a cvs. e Fig..4 P =.5, e=.5, P = 3, t=.5 R =, R = 5, R 5 =, b =.5, p = 3.459.35.3 Unstable.5 ac..5 Stable..5 4 6 8 Critical Wave Number a c Vs. F Fig. F CONCLUSION An analysis of the problem an the iscussions of the results lea to the conclusion that the Principle of Exchange of Stability is not vali for this problem an the frequency of oscillations an the Rayleigh number in the marginal state are given by equation (3) an equation (4). [588]
[Kumar*, 5(): February, 6] ISSN: 77-9655 (IOR), Publication Impact Factor: 3.785 Further, we fin for ensity istribution with positive graient an for k s >k T, the overstable marginal state oes not exist an we have only non-oscillatory moes which make the system unstable. For ensity istribution with negative graient, the marginal state an overstabse solution exist, irrespective of the values of other parameters. REFERENCES [] Benar, H. : Revue generale es sciences pures et apliques, : 6-7 an 39-8 (9). [] Rayleigh, L. : Phil. Mag. 3 : 59 (96). [3] Chanrashekhar, S. : Hyroynamic an Hyromagnetic Stability, Oxfor University Press, Lonon (96). [4] Stomell, H., Arones, A. B. an Bhanchor, D. : Deep Sea Res. 3 : 5 (956). [5] Stern, M.E. : Tellus : 7 (96). [6] Veronis, G. : J. Flui Mech. 3 : (965). [7] Niel, D.A. : J. Flui Mech. 9 : 545 (967). [8] Toms, B.A. an Strabrige, D. J. : Trans. Faraay Soc., 49 : 5 (953). [9] Olroy, J. G. : Proc. Roy. Soc. (Lonon), A45, 78-97 (958). [] Sharma, R. C. : J. Maths. Phys. Sci. : 63 (978). [] Khare, H. C. an Sahai, A. K. : Proc. Nat. Aca. Sci. Inia 6(A), IV, 673-689(99). [589]