On weakly nonlinear evolution of convective flow in a passive mushy layer

The problem of weakly nonlinear convective flow in a mushy layer, with a permeable mush–liquid interface and constant permeability, is studied under operating conditions for an experiment. A Landau type nonlinear evolution equation for the amplitude of the secondary solutions, which is based on the Landau theory and formulation for the Rayleigh, $R$, number close to its critical value, $R_c$, is developed. Using numerical and analytical methods, the solutions to the evolution equation are calculated for both supercritical and subcritical conditions. We found, in particular, that for $R<R_c$, the amplitude of the secondary solutions decays with time. For $R>R_c$, the tendency for chimney formation in the mushy layer increases with time. In addition, in such a supercritical regime, the basic flow is linearly unstable and we see the presence of steady flow for large values of time. These results suggest a possible slow transition to turbulence in such a flow system.