Thermal convection during the directional solidification of a pure liquid with variable viscosity

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Abstract
The onset of a buoyancy-driven instability during the directional solidification of a pure liquid with a strongly temperature-dependent viscosity and an arbitrary Prandtl number is investigated using linear stability theory. The Rayleigh number for this system contains the lengthscale Ls defined as the ratio of the thermal diffusivity of the liquid and the solidification velocity times the density ratio of the two phases. It is independent of the actual depth of the liquid and it reflects the fact that increasing the solidification velocity stabilizes the system. The theory also shows that the difference in material properties between the two phases and the properties of the solidifying interface itself cause the interface to look like a boundary of finite conductivity measured by a wavenumber-dependent Biot number. For large viscosity variations, convection occurs below a stagnant layer which forms just beneath the interface where the liquid is immobilized by its very large viscosity. The thickness of this layer is measured by the natural logarithm of the viscosity contrast in the liquid times the lengthscale Ls. In this limit, the influence of the solidifying boundary is shielded from the bulk liquid by the stagnant layer and so the effect of the Biot number on the critical Rayleigh number is small. However, inertial effects, being associated with the bulk liquid, are very important for small Prandtl numbers of the fluid far from the interface. The model has applications to the solidification of magma chambers or lava lakes and to the material processing of polymeric liquids.