Ajaev VS, Davis SH
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Feb;61(2):1275-84. doi: 10.1103/physreve.61.1275.
Directional solidification of a dilute binary alloy in a Hele-Shaw cell is modeled by a long-wave nonlinear evolution equation with zero flux and contact-angle conditions at the walls. The basic steady-state solution and its linear stability criteria are found analytically, and the nonlinear system is solved numerically. Concave-down (toward the solid) interfaces under physically realistic conditions are found to be more unstable than the planar front. Weakly nonlinear analysis indicates that subcritical bifurcation is promoted, the domain of modulational instability is expanded and transition to three-dimensional patterns is delayed due to the contact-angle condition. In the strongly nonlinear regime fully three-dimensional steady-state solutions are found whose characteristic amplitude is larger than that for the two-dimensional problem. In the subcritical regime secondary bifurcation to stable solutions is promoted.
通过一个长波非线性演化方程对Hele-Shaw单元中稀二元合金的定向凝固进行建模，该方程在壁面处具有零通量和接触角条件。通过解析方法得到了基本稳态解及其线性稳定性判据，并对非线性系统进行了数值求解。发现在物理现实条件下向下凹（朝向固体）的界面比平面前沿更不稳定。弱非线性分析表明，由于接触角条件，亚临界分岔得到促进，调制不稳定性区域扩大，向三维模式的转变延迟。在强非线性区域，找到了完全三维的稳态解，其特征振幅大于二维问题的特征振幅。在亚临界区域，促进了向稳定解的二次分岔。
