Conditions of boundary layer separation for Boussinesq equations

In this paper, we analyze structural bifurcation of solutions to 2-D incompressible Boussinesq equations, where no-slip boundary condition for velocity and nonhomogenous Dirichlet boundary condition for temperature are considered. We get two conditions for boundary layer separation by Taylor expansion of the functions in Boussinesq equations and structural bifurcation theory for flows with Dirichlet boundary conditions. Furthermore, the conditions, determined by initial values, the external force and the temperature on the boundary, can predict when and where boundary layer separation of the Boussinesq equations will occur. The basic theory on boundary layer separation in this manuscript comes from the book Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics written by Ma and Wang.