On nonlinear convection in mushy layers. Part 2: Mixed oscillatory and stationary modes of convection

Abstract

This paper presents part 2 of a study of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. Part 1 dealt with only the oscillatory modes of convection (Riahi, J. Fluid Mech. vol. 467, 2002, pp. 331–359). In the present paper we consider the particular range of parameters where the critical values of the scaled Rayleigh number R for the onset of oscillatory and stationary convection are close to each other, and we develop and analyse a nonlinear theory in such a parameter regime which takes into account those mixed stationary and oscillatory modes of convection with common wavenumber vectors. Under a near-eutectic approximation and in the limit of large far-field temperature, we first determine a number of weakly nonlinear solutions, and then the stability of these solutions is investigated. The most interesting result is the preference for a mixed solution composed of standing and stationary hexagonal modes over a relatively wide range of the parameter values and for R just above its lowest subcritical value where convection is possible. Such a preferred solution has properties mostly in agreement with the experimental results due to Tait et al. (Nature, vol. 359, 1992, pp. 406–408) in the sense that the flow is downward at the cell centres, upward at the cell boundaries and there is some tendency for channel formation at the cell nodes.