A theoretical investigation of the convective instability problem in the thin horizontal layer of a magnetic fluid heated from below is carried out. The effects of the relaxation time tau and the vortex (rotational) viscosity xi are considered and discussed. The Chebyshev pseudospectral method is employed to solve the eigenvalue problems and numerical calculations are carried out for a number of magnetic fluids and in full range of the magnetic field. A variety of results under gravity-free conditions are also presented and the critical temperature gradient are determined for a variety of situations. It is shown that the consideration of (xi,tau), in the stability analysis, is most effective in the thin layer of the fluid and at low values of Langevin parameter alpha(L).
The stability of a viscoelastic fluid in a densely packed horizontal porous layer heated from below is considered using an Oldroyd model. Critical Rayleigh number, wave number, and frequency for overstability are determined by applying the linear stability theory. It is shown that the critical Rayleigh number is invariant under all relevant boundary combinations. Also, it is found that the effect of elasticity of the fluid is to destabilize the system and that of porosity is to stabilize the same. The limiting case of very high Prandtl number and the degenerate case corresponding to the Maxwell model are analyzed in some detail.
A theoretical study of the problem of steady nonlinear double-diffusive convection through a porous medium is presented. The Brinkman᎐Forchheimer model is used to represent the porous medium. A variational formulation is given to deal with the weak solution and the existence, regularity, and uniqueness results are discussed. ᮊ
Introduction.In comparison to the large amount of work that has been carried out on the analysis of Saint-Venant's principle for solutions of various elliptic boundary problems in elasticity (see [6,8], and the references therein), the related studies to investigate the spatial decay of solutions for flow problems in fluid dynamics are still considerably fewer in the literature.Horgan and Wheeler [11] presented Saint-Venant type results within the general framework of the Navier-Stokes equations governing the steady laminar flow of an incompressible viscous fluid in a cylindrical pipe of arbitrary cross section. These authors studied the classical entry flow problem of viscous laminar flow theory and showed that the decay rate, which depended upon the Reynolds number, the prescribed entry profile of the base flow, and the cross section of the pipe, was exponential.A two-dimensional version of this problem with more explicit results was also investigated by Horgan [9]. Ames and Payne [2] readdressed the problem studied in [11] with a somewhat different approach in formulation of the problem and methodology. They relaxed the assumption that the flow is fully developed at the exit section and obtained more explicit estimates for a weighted energy integral associated with the flow.It is well known that the stream function in two-dimensional Stokes flow satisfies a biharmonic equation; hence the study of the spatial evolution of stationary Stokes flows in a semi-infinite parallel plate channel has direct relevance to the results of Saint-Venant's principle in plane elastostatics, where the latter problems have been well studied by many authors [7,13,15,17,19]. On the other hand, it was shown by a number of investigators [3,4,10,14] that for some initial boundary value problems, the solutions of parabolic equations enjoy the spatial decay behavior of their counterparts for related elliptic equations. A natural question then arises as to whether or not the same exponential spatial decay behavior is present for the solution of transient Stokes flow. The investigations of this question have only been carried out recently. Lin [16] presented a study dealing with
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