Size-Dependent Rayleigh--B\'{e}nard Problem

Problems of thermoviscous flows are of prime importance for many physical processes. Here the classical Boussinesq equations are modified by including couple stresses, which account for size-dependency. This size-dependency is specified by a material length scale $l$, which becomes increasingly important as the characteristic geometric dimension of the problem decreases. The modified two-dimensional linear momentum equations become \[ \rho \left( {\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}} \right)=-\frac{\partial p}{\partial x}+\mu \nabla^{2}u-\mu l^{2}\nabla^{2}\nabla^{2}u \] \[ \rho \left( {\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}} \right)=-\frac{\partial p}{\partial y}+\mu \nabla^{2}v-\mu l^{2}\nabla^{2}\nabla^{2}v-\rho \alpha \left( {T-T_{0} } \right) \] The stability of natural convection for the Rayleigh--B\'{e}nard problem is studied numerically and we consider the onset of convective instability and multiple stable steady states arising within specific ranges of Rayleigh and Prandtl numbers and $l$.