On nonlinear transient state of modal package convection in a mushy layer

Abstract

Finite-amplitude transient convective flow with a continuous finite bandwidth of modes in a horizontal mushy layer during the solidification of binary alloys is investigated. Under a near-eutectic approximation and in the limit of large far-field temperature, we analyse the nonlinear transient convection for values of the Rayleigh number close to its critical value by using multiple scales and perturbation techniques. Applying the combined temporal and spatial evolution approach, we determine a set of coupled time-dependent partial differential equations for the amplitude functions of the convective modes. In the transient regime, we find a wide class of solutions to these equations in the form of transient rolls and transient hexagons with down-flow or up-flow at the cells’ centres that eventually either reach their steady-state form or decay to zero. We also apply a criterion of stability to examine the stability of these time-dependent solutions. Among the detected solutions, there are inclined large-scale transient rolls and large-scale transient hexagons that are realizable at higher values of their amplitude. Our main results agree with the available experimental observations.