Effect of anisotropy and boundary conditions on Darcy and Brinkman porous penetrative convection

Abstract

We investigate the effects of anisotropic permeability and changing boundary conditions upon the onset of penetrative convection in a porous medium of Darcy type and of Brinkman type. Attention is focussed on the critical eigenfunctions which show how many convection cells will be found in the porous layer. The number of cells is shown to depend critically upon the ratio of vertical to horizontal permeability, upon the Brinkman coefficient, and upon the upper boundary condition for the velocity which may be of Dirichlet type or constant pressure. The critical Rayleigh numbers and wave numbers are determined, and it is shown how an unconditional threshold for nonlinear stability may be derived. Highlights Shows how number of convection cells depends upon the temperature of the upper layer and the anisotropy of the permeability Shows how number of convection ceels depends upon the temperature of the upper layer and the Brinkman coefficient Shows how number of convection cells patters depends upon the upper boundary condition on the velocity or the ambient pressure

Thermal convection in a higher-gradient Navier–Stokes fluid

“…Great care has to be taken to ensure the correct form of boundary conditions is chosen to properly describe the physical problem in hand, cf. [27][28][29][30][31][32][33][34][35][36][37][38][39].…”

“…Let • and (•, •) be the norm and inner product on L 2 (V ). System (39) together with any of the boundary conditions I, II or III is now written in the form…”

“…80-87]. Here A diag{1, 1, 1, Pr}, N(U) consists of u j u i, j and u i θ , i , while L is represented by the right hand side of (39).…”

“…The right hand side of ( 41) allows us to introduce a bilinear form on H. Let U (u 1 , v 1 , w 1 , θ, π 1 ) T and V (u 2 , v 2 , w 2 , φ, π 2 ) T be solutions to (39) subject to boundary conditions of case I, II or III. Let d i j , ω i j be the symmetric and skew-symmetric parts of u 1 i, j , and let e i j , ζ i j be the symmetric and skew-symmetric parts of u 2 i, j .…”

“…To find the critical Rayleigh numbers for instability the usual procedure is to linearize (39) and look for solutions like…”