Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 11401021, 51971031)
Received Date:21 October 2020
Accepted Date:16 November 2020
Available Online:12 March 2021
Published Online:20 March 2021
Abstract:During directional solidification of binary alloy mixtures, instability in the solid/liquid interface appears due to constitutional undercooling. As a result of this instability, a reactive porous medium, namely mushy layer, is formed, and it separates the liquid phase from the solid phase completely. The intrinsic structure of the mushy layer is of fine-scale dendritic crystal that shelters solute in the interstitial fluid. In a gravitational field, the rejection of lighter solute components from an advancing solidification front brings about unstable density gradient. Ensuing convective motions in the mush are driven by a density difference. The convection can change the solid matrix of the mushy layer. Hence, the dynamic response of the mushy layer is driven by interaction among heat transfer, solute transport and convection. As a contactless control tool, external magnetic field can change the heat and solute transport, which has a significant effect on the phase change process. Therefore, when magnetic field, thermal diffusion, solute transport and buoyancy convection are considered simultaneously in the phase transformation process, the mechanism of mushy region will become more complex and interesting. In this paper, the effect of external magnetic field on the stability of mushy layer during binary alloy solidification is studied. The coupling effects of magnetic field, temperature field, concentration field and convection are considered in the model. Including the direct mode and the oscillation mode, the resulting dispersion relation reveals the influence of magnetic field on the stability of mushy layer through linear stability analysis. It is found that the Lorentz force can reduce the instability effect which is caused by buoyancy convection. In the oscillation mode, an external magnetic field brings about a stabilizing effect on the mushy layer, but in the direct mode, the effect of external magnetic field on stability of the mushy layer is uncertain. In conclusion, the finding in this paper provides an important theoretical reference for improving products quality by applying an external magnetic field in the metals processing industry. Keywords:mushy layer/ magnetohydrodynamics/ stability
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2.1.控制方程
考虑二元合金在磁场中的凝固问题, 认定糊状层的上下边界是等温的、不变形的和不渗透的. 这使得糊状层与系统的其他部分动态隔离, 具体描述见图1. 假定系统以恒定速率${V^*}$向上凝固, 在$0 < z^* < {H^*}$的区域内存在一个厚度为${H^*}$的糊状层. 糊状层顶部和底部的温度及浓度分别为$T_{{\rm{top}}}^{{*}}$, $T_{{\rm{bot}}}^{{*}}$, $C_{{\rm{top}}}^{{*}}$, $C_{{\rm{bot}}}^{{*}}$, 同时假设它们为常数. 为了便于分析, 假设固相和液相的比热和热导率相同. 图 1 以恒定速度${V^*}$凝固的糊状层示意图 Figure1. Schematic representation of mushy layer system, which advancing upwards with a constant solidification speed ${V^*}$.