A characterization theorem in hydromagnetic triply- diffusive convection

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1 Available online at Advances in Applied Science Research, 4, 5(6):45-5 ISSN: CODEN (USA): AASRFC A characteriation theem in hydromagnetic triply- diffusive convection Hari Mohan and Sada Ram Department of Mathematics, International Centre f Distance Education and Open Learning (ICDEOL), Himachal Pradesh University, Summer Hill Shimla (HP), India ABSTRACT The problem of triply diffusive magnetoconvection is considered in the present paper. An attempt is made to establish the relationship between various energies in Veronis type configurations. The analysis made brings out that f Veronis type configuration, the total kinetic energy associated with a disturbance exceeds the sum of its total magnetic and concentration energies in some particular parameter regime. Further, this result is valid f any combination of dynamically free rigid boundaries that are either perfectly conducting insulating. Key wds: Triply Diffusive Convection; Rayliegh Numbers; Chandrasekhar Number; Prandtl Number; Magnetic Prandtl Number; Lewis Numbers. MSC No: 76E99, INTRODUCTION Thermohaline convection me generally double diffusive convection has matured into a subject possessing fundamental departure from its counterpart, namely single diffusive convection, and is of direct relevance in the fields of oceanography, astrophysics, liminology and chemical engineering etc. F a broad and a recent view of the subject one may be referred to []. Two fundamental configurations have been studied in the context of thermohaline instability problem, the first one by [] wherein the temperature gradient is stabiliing and the concentration gradient is destabiliing and the second one by [3] wherein the gradient is destabiliing and the concentration gradient is stabiliing. The main results derived by Stern and Veronis f their respective configurations are that both allow the occurrence of a stationary pattern of motions oscillaty motions of growing amplitude provided the destabiliing concentration gradient the temperature gradient is sufficiently large. However, stationary pattern of motion is the preferred mode of setting in of instability in case of Stern s configuration whereas oscillaty motions of growing amplitude are preferred in Veronis configuration. Me complicated double-diffusive phenomenon appears if the destabiliing thermalconcentration gradient is opposed by the effect of magnetic field rotation. [4]investigated the problem of thermohaline convection coupled with crossdiffusions f the Veronis type configuration and derived a semi-circle theem that prescribed upper limits f the complex growth rate of oscillaty motions of neutral growing amplitude in such a manner that it naturally culminates in sufficient conditions precluding the non- existence of such motions. [5] investigated the Rayleigh- Tayl instability of a Newtonian viscous fluid overlying Walters B viscoelastic fluid through pous medium. [6] have considered the hydromagnetic instability of the plane interface between two unifm, superposed and streaming Rivlin-Ericksen viscoelastic fluids through pous medium. 45

2 Hari Mohan and Sada Ram Adv. Appl. Sci. Res., 4, 5(6): 45-5 All the above researchers have considered the case of two component systems. However, it has been recognied later on by [7], [8] that there are many situations wherein me than two components are present. Examples of such multiple diffusive convection fluid systems include the solidification of molten alloys, geothermally heated lakes, magmas and their labaty models and sea water.[9], Pearlstein et al. []and []have theetically studied the onset of convection in a hiontal layer, of infinite extension of a triply diffusive fluid (where the density depends on three independently diffusing agencies with different diffusivities).these researchers found that small concentrations of a third component with a smaller diffusivity can have a significant effect upon the nature of diffusive instabilities and oscillaty and direct 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 same signs.some fundamental differences between the double and triply convection are noticed by these researchers diffusive. Among these differences, one is that if the gradients of two of the stratifying agencies are held fixed,then three critical values of the Rayleigh number of the third agency are sometimes required to specify the linear stability criteria(only one critical number is required in double diffusive convection ). Another difference is that the onset of convection may occur via a quassiperiodic bifurcation from the motionless basic state. [] studied the effect of cross-diffusion on the stability criteria in a triply diffusive system. [3] studied the case of multicomonent mixture with application to thermogravitational column. [4] also studied the longwave instability of a multicomponent fluid with Set effect. [5] studied a triply convective diffusive fluid mixture saturating a pous hiontal layer, heated from below and salted from above and obtained sufficient conditions f inhibiting the onset of convection and guaranteeing the global nonlinear stability of the thermal conduction solution. [6] also investigated the multicomponent diffusive convection in pous layer f the me general case when heated from below and salted by m salts partly from above and partly from below.[7] investigated the problem of triply diffusive convection in Maxwell fluid saturated pous layer and obtained the criterion f the onset of stationary and oscillaty convection. [8] investigated the bifurcation analysis of a triply diffusive coupled stress fluid in terms of a simplified model consisting of seven nonlinear dinary differential equations. [9] have studied the linear and weakly nonlinear triple diffusive convection in a couple stress fluid layer. Chandrasekhar [] in his investigation of magneto hydrodynamic simple Be nard convection problem sought unsuccessfully the regime in terms of the parameters of the system alone, in which the total kinetic energy associated with a disturbance exceeds the total magnetic energy associated with it, since these considerations are of decisive significance in deciding the validity of the principle of exchange of stabilities. However, the solution f w ( = cons tan t(sin )) is not crect mathematically (and Chandrasekhar was aware of it).banerjee et. al. until 985 did not pursue their investigation in this direction and consequently did not see this connection. This gap in the literature on magnetoconvection has been completed by []who presented a simple mathematical proof to establish that Chandrasekhar s conjecture is valid in the regime Q and further this result is unifmly applicable f any combination of a dynamically free rigid boundary when the region outside the liquid are perfectly conducting insulating. [] showed that in the parameter regime the total kinetic energy associated with a disturbance is greater than the total magnetic energy associated with it. [] further extended these energy considerations to a me general problem, namely, magnetohydrodynamic thermohaline convection problem, of Veronis type and established that in the parameter regime RS +, the total kinetic energy associated with a disturbance exceeds the sum of its total magnetic 4 and thermal energies. A similar characteriation theem in magnetothermohaline convection of the Veronis type was also established by Banerjee et. al in the subsequent year.[3] derived a characteriation theem in hydromagnetic double diffusive convection and established that the total kinetic energy associated with a disturbance is greater than the sum of its total magnetic and concentration energies in the parameter regime, RS

3 Hari Mohan and Sada Ram Adv. Appl. Sci. Res., 4, 5(6): 45-5 The present analysis extends these energies considerations to another complex problem, namely, triply diffusive magnetoconvection problem (analogous to magnetothermohaline convection of the Veronis type) wherein one destabiliing heat component and two stabiliing concentration components have been considered. We establish R here that in the parameter regime + S RS +, the total kinetic energy associated with a disturbance exceeds the sum of its total magnetic and concentration energies. Further, this result is valid f any combination of dynamically free rigid boundaries that are either perfectly conducting insulating. MATHEMATICAL FORMULATION AND ANALYSIS A viscous and finitely heat conducting Boussinesq fluid is statically confined between two hiontal boundaries = and =d of infinite hiontal extension and finite vertical depth which are respectively maintained at unifm temperatures T and T ( T > T ) and unifm concentrations S, S and S( < S ), S(, < S ) in the presence of unifm vertical magnetic field H r. Following [7]and[]), the relevant governing equations and boundary conditions f the triply diffusive magnetoconvection in their non-dimensional fm are given by: p ( D a ) D a w = RT a θ RS a φ RS a φ QD( D a ) h ( D a p) θ = w D D p w p w φ =, (), () a φ =, (3) a, (4) and p D a h = Dw. (5) with w = = θ = φ = φ on both the boundaries, D w = on a tangent stress free boundary everywhere, Dw = on a rigid boundary, h = on both the boundaries if the regions outside the fluid are perfectly conducting, Dh = ah at = if the regions outside the fluid are insulating. Dh = ah at = (6) d In the above equations () (6), is real independent variable such that, D = is differentiation w.r.t, d w is the vertical velocity, θ is the temperature, φand φ are two concentrations, h is the vertical magnetic field, a is the square of the wave number, is the Prandtl number, is the magnetic Prandtl number, and are the Lewis numbers f two concentration components respectively, R > is the thermal Rayliegh number, R S > and T 47

4 Hari Mohan and Sada Ram Adv. Appl. Sci. Res., 4, 5(6): 45-5 R S > are the concentration Rayliegh numbers f the two concentration components respectively, p = p r + ip i is complex constant in general. We now prove the following theem: Theem : If (p, w, θ, φ, h ), p = p r + ip i, p r is a solution of () (5) together with boundary conditions (6) R with RT > R S > R S > and + S RS +, then ( Dw + a w ) d > Q ( Dh + a h ) d + RS a φ + RS a φ d. Proof: Multiplying (5) by h (the complex conjugate of h ), integrating the resulting equation over the range of by parts a suitable number of times, and making use of the boundary conditions (6) we get p ( Dh + a h ) d + h d = am + w Dh, (7) where M {( h ) + ( h ) } =. Equating the real part of (7), we get p r am + Dh + a h d + ( ) h d = Real part of w Dh d w Dh d w Dh d w d Dh d. (8) (using Schwart inequality) Since p, therefe from (8), we get Dh r d < w d d < Dh d Dh w d. (9) 48

5 Hari Mohan and Sada Ram Adv. Appl. Sci. Res., 4, 5(6): 45-5 Using (9), it follows from (8) that ( Dh a h ) d w d. < + () Since w () = = w (), therefe using[5], we get w d < Dw d. () It follows from () and () that ( Dh + a h ) d < Dw d ( Dw a w ) < + d ( Dh + a h ) d + RS a φ d + RS a φ d ( Dw + a w ) d + RS a φ d + RS a < φ d () Multiplying () by the complex conjugate of () and integrating by parts over the vertical range of f an appropriate number of times and making use of the boundary conditions (6) fφ we get ( Dφ a ) 4 D φ + a Dφ + a φ d + pr + φ d + p Since, φ d = w d p r, therefe, from (3), we get. (3) 4 D φ + a Dφ + a φ d < w d. (4) Since φ () = = φ (), therefe using Rayleigh-Rit inequality [973], we get φ d < Dφ d and also 4 φ d D φ d. (using Schwart inequality) (5) It follows from (4) and (5) that 49

6 Hari Mohan and Sada Ram Adv. Appl. Sci. Res., 4, 5(6): 45-5 ( + a ) φ d w d ( + a ) a d < φ d < a φ < 4 4 w d Dw + d, a ( a w ) ( + a ) since the minimum value of f a > is 4. a R S RS a φ d < ( Dw + a w )d. (6) 4 4 Following the same procedure, we have from equation (4), that RS RS a φ d < ( Dw + a w )d (7) 4 4 Now from (), (6) and (7), we get ( Dh + a h ) d + RS a φ d + RS a φ d RS < Therefe, if R S ( Dw a w ) + S R S R + + then from (8), we get d. (8) ( Dw + a w ) d > Q ( Dh + a h ) d + RS a φ + RS a φ d and this completes the proof of the theem. (9) We note that the left hand side of (9) represents the total kinetic energy associated with a disturbance while the right hand side represents the sum of its total magnetic and concentration energies, and Theem may be stated in the following equivalent fm: At the neutral unstable state in the triply diffusive magnetoconvection problem of the Veronis' type configuration, the total kinetic energy associated with a disturbance is greater than the sum of its total magnetic and concentration R energies in the parameter regime S RS + + and this result is unifmly valid f any combination of dynamically free rigid boundaries that are either perfectly conducting insulating. 5

7 Hari Mohan and Sada Ram Adv. Appl. Sci. Res., 4, 5(6): 45-5 CONCLUSION In the present paper, the hydromagnetic triply diffusive convection problem of Veronis type configuration is considered. The analysis made brings out the following main conclusion: At the neutral unstable state in the hydromagnetic triply diffusive convection problem of the Veronis type configuration, the total kinetic energy associated with a disturbance is greater than the sum of its total magnetic and R concentration energies in the parameter regime S RS + +, and this result is unifmly valid f any combination of dynamically free rigid boundaries that are either perfectly conducting insulating. REFERENCES [].Brandt, A., Fernando H.J.S. Double Diffusive Convection.American Geophysical Union Washington, DC [] Stern, M.E. The Salt Fountain and Thermohaline Convection. Tellus. 96., 7 [3] Veronis, G. J.Mar. Res ,. [4] Mohan, H. Application and Applied Mathematics-An International Journal (AAM).. 5(), 48. [5] Kumar, P., Singh.J. G. Application and Applied Mathematics-An International Journal (AAM)., 5(),. [6] Kumar, P., Mohan, H. Application and Applied Mathematics-An International Journal (AAM).. 7(), 4. [7] Griffiths, R.W.. J.Fluid Mech. 979, 9,659. [8] Turner,J.S., Ann. Rev. Fluid Mech.985,7,. [9] Pearlstein, A.J., Harris, R.M., J. Fluid Mech. 989,., 443. [] Lope, A.R., Romero, L.A., Pearlstein, A.J. Physics of Fluids. 99. (6), 897. [] Terrones, G. Phys. Fluids A5. 7. [3] Ryhkov, I., I., Shevtsova, V.M. Phys. Fluids. 7. 9,. [4] Ryhkov, I., I., Shevtsova, V.M. Phys. Fluids [5] Rionero, S. Triple Diffusive Convection in Pous Media.Acta Mech.3a...4, 447. [6] Rionero, S. Phys Fluids. 3b. 5,. [7] Zhao, M., Wang, S., Zhang, Q. Applied Mathematical Modelling , 35. [8] Shivkumara, I.S., Kumar, S.B.N. International Journal of Engineering Research and Applications. 3.3 (6), 37. [9] Shivkumara, I.S., Kumar, S.B.N. International Journal of Heat and Mass Transfer , 54. [] Chandrasekhar, S. Philos. Mag.vol. 95, 43, 5. [] Banerjee, M.B., Katyal S.P. J. Math. Anal. Appl , 383. [] Banerjee, M.B., Gupta, J.R., Katyal, S.P. Math. Anal. Appl ,4. [3] Mohan, H., Kumar, P., Devi, Pushpa.. Ganita. 6.57(), 49. [4] Schult, M.H. Spline Analysis, Prentice-Hall, Englewood Cliffs, N.J