Stability and perturbation analyses for nonlinear convection in a mushy layer and in the presence of viscous heating

This paper studies nonlinear steady convection in a horizontal mushy layer and in the presence of viscous dissipation in the temperature equation, which often is referred to as viscous heating. We consider the appropriate system of equations and the associated boundary conditions for the flow in the mushy layer. Under certain assumptions and conditions, we determine the weakly nonlinear solutions of the resulting system and their stability by using perturbation and stability analyses. We find, in particular, that for a wide range of values of the viscous heating parameter and sufficiently small amplitude of the flow, the only stable convective flow is in the form of subcritical down-hexagons with down-flow at the cells’ centers and up-flow at the cells’ boundaries. This result appears to agree with the available experimental results for the flow pattern near the onset of motion in a mushy layer.