Selection of convective planform orientation by boundary anisotropy

Abstract

Thermal instability of a fluid layer confined between isotropic horizontal solid walls leads to convection cells having no preferred horizontal direction. For thermally anisotropic walls, we find that certain planform orientations are preferred, in that convection sets in at a smaller Rayleigh number (Ra) for some orientations than for others, thus providing a means by which a regular planform may be established in a large-aspect-ratio layer. We consider horizontal layers of two Boussinesq, Newtonian fluids separated by a rigid, thermally anisotropic plate of constant thickness. The upper and lower fluid layers are bounded above and below, respectively, by rigid, thermally anisotropic plates of arbitrary thickness. When the bounding surfaces are thermally anisotropic, the horizontal wavevector (a) of the resulting convective flow has two distinct components. Thus, instead of a neutral curve in the (Ra, a)-plane, there is a neutral surface, and Ra depends on both components of a, or alternatively, on |a| and the planform orientation angle Φrε[0, 2π]. In the isotropic case, the neutral surface is axisymmetric (i.e. invariant with respect to Φr consistent with the known dependence on |a| only.For anisotropic walls, axisymmetry is replaced by π- periodicity in the Φr direction, corresponding to invariance with respect to a 180° rotation, and the neutral surface has an even number of local minima. We study the dependence of Φr on the middle plate orientation (Φp) Several different Φr−Φp topologies are found. When the number of local minima exceeds two, discontinuous Φr−Φp plots may occur. The dependence of Φr on the thicknesses and conductivities of the plates and fluids and on the orientation of the plates is discussed, with special reference to the transitions between different Φr−Φp topologies.