Symbolic Computational Approach to the Marangoni Convection Problem With Soret Diffusion

A recently reported solution for stationary stability of a thermosolutal system with Soret diffusion is re-derived and examined using a symbolic computational package. Symbolic computational languages are well suited for such an analysis and facilitate a pragmatic approach that is adaptable to similar problems. Linearization of the equations, normal mode analysis, and extraction of the final solution are performed in a Mathematica ® notebook format. An exact solution is obtained for stationary stability in the limit of zero gravity. A closed form expression is also obtained for the location of asymptotes in relevant parameter, (Smc,Mac), space. The stationary stability behavior is conveniently examined within the symbolic language environment. An abbreviated version of the Matheraatica ® notebook is given in the Appendix.t NOMENCLATURE C concentration c disturbance concentration Din, n diffusivity elements surface tension DI2 Y1 Dm Dufour coefficient, ---DI] Y2 d depth of fluid layer h surface heat transfer coefficient h s surface mass transfer coefficient J mass flux from surface Ma Marangoni number, 71dAT Dllkt hd Nu surface Nusselt number, -pcpDII Pr Prandtl number, v/Dll AP pressure difference Ap disturbance pressure difference Q heat flux from surface Sh surface Sherwood number, hs--_-dsurface tension D21 T2 D22 Sm Sorer coefficient, __m 022 "/1 T temperature t time velocity vector U x-component velocity disturbance velocity vector V y-component velocity W z-component velocity w z-component disturbance velocity x,y,z Cartesian coordinates (see Fig. 1) Greek Symbols _t wavenumber T surface tension variation 0 disturbance temperature normal mode temperature _. eigenvalue kt dynamic viscosity v kinematic viscosity p density a surface tension "_ diffusivity ratio, D22]DI 1 X normal mode concentration Subscripts 0 value at z = 0 1 value at z = 1 b basic state c critical value oo asymptotic value "tThe unabridged Mathematica ®notebook is available from the author or on the internet at http://zeta.lerc.nasa.gov/6712/people/skarda.htm NASA/TM--1998-206634