Effect of Stefan number on thermoelastic instabilities in unidirectional solidification

Abstract

During the casting process, thermoelastic distortion of the partially solidified material affects the contact pressure at the solid/mold interface, which in turn can affect the thermal contact resistance, thus coupling the heat transfer and thermomechanical problems. This coupled system has the potential for instability. In this paper, the effect of Stefan number on the stability of unidirectional solidification is investigated, under the simplifying assumption that the solidified material is linear elastic. The Stefan number is a measure of the influence of thermal capacity on the solution and previous analyses have generally been restricted to the case of zero Stefan number, corresponding to a solidifying material of zero thermal capacity.This generalization necessitates a numerical solution, which is here implemented using the finite difference method. However, since the growth of the perturbation is linear, the two-dimensional stability problem is reduced to two one-dimensional numerical problems which can be solved sequentially.The results show that, in all cases, an initial sinusoidal perturbation grows to a maximum amplitude in the solidification front and then decays, the maximum being reached when the mean solidified layer thickness is about half the wavelength of the perturbation.In general, increasing Stefan number has a stabilizing effect on the process. This effect is most noticeable in cases where the zero Stefan number approximation predicts substantial growth of an initial perturbation, e.g. where the thermal contact resistance is very sensitive to pressure.