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In this article, an investigation is conducted to analyze the marginal stability with and without magnetic field in a mushy layer. During alloy solidification, such mushy layer, which is adjacent to the solidification front and composed of solid dendrites and liquid, is known to produce vertical chimneys. Here, we carry out numerical investigation for particular range of parameter values, which cover those of available experimental studies, to determine the convective flow at the onset of motion. The governing coupled non-linear partial differential equations are non-dimensionalised and solved to get the steady basic-state solution. The thickness of the mushy layer is determined as a part of the solution. Using multiple shooting technique, we determine the steady-state solutions in a range of critical Rayleigh number. We analyse the effect of several parameters, Chandrasekhar number Q, and Robert's number τ on the problem. It was found that an increase in Q has a stabilizing effect on solidification and the critical Rayleigh number increases on increasing Q. It was also found that for moderate or small values of Robert's number τ the critical Rayleigh number is mostly insensitive.

This present study considers the problem of steady magneto-convection in a horizontal mushy layer with variable permeability and an impermeable mush-liquid interface during directional solidification of binary alloys. We model the flow by introducing a uniform magnetic field in the mushy layer which is considered as a porous medium where Darcy's law holds and the permeability is a function of the local solid volume fraction. Basic-state solutions are obtained analytically using the no-flow condition. With the help of multiple shooting techniques, we obtain numerical solutions to the linear perturbation system for non-magnetic and magnetic cases. Numerical results are presented showing the effects of the magnetic field and the permeability of the layer. These results demonstrate that the application of an external magnetic field has stabilizing effects on the convection and can reduce the tendency for chimney formation in the mushy layer. In addition, variable permeability, which corresponds to an active mushy layer, indicates more stable and realizable flow system as compared to the case of constant permeability.

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We consider the nonlinear effect of convective flow in a horizontal mushy layer during solidification. The mushy layer that we consider has a permeable mush-liquid interface and is treated as an active porous medium with variable permeability. The nonlinear partial differential equations involved in this system are conservation equations for flow momentum, mass, temperature, and concentration. Numerical solutions to the resulting weakly nonlinear equations are obtained using a fourth-order Runge-Kutta integration scheme via a shooting technique. An evolution equation of Landau type is derived in terms of linear and first-order solutions by introducing an adjoint operator. The Landau constant is calculated for both cases: constant permeability and variable permeability. The analysis reveals that there is a slow transition of the flow to a steady state with smaller amplitude for an active mushy layer.