Investigated in this paper is the stability of the gravity-driven flow of a liquid layer laden with soluble surfactant down a heated incline. A mathematical model incorporating variations in surface tension with surfactant concentration and temperature has been formulated. A linear stability analysis is carried out both asymptotically for small wavenumbers and numerically for arbitrary wavenumbers. An expression for the critical Reynolds number has been derived which accounts for thermocapillary and solutocapillary effects, and reduces to known documented results for special cases. Also, a nonlinear reduced model has been derived using weighted residuals, and solved numerically to simulate the instability of the equilibrium flow and the development of permanent surface waves that arise. The nonlinear simulations were found to be in good agreement with the linear stability analysis.

中文翻译：

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与加热的载有表面活性剂的降膜相关的 Marangoni 不稳定性

本文研究了载有可溶性表面活性剂的液体层在加热斜坡上的重力驱动流动的稳定性。已经建立了包含表面张力随表面活性剂浓度和温度变化的数学模型。对小波数渐近地进行线性稳定性分析，对任意波数进行数值分析。已推导出临界雷诺数的表达式，该表达式解释了热毛细管和溶质毛细管效应，并简化为特殊情况下已知的记录结果。此外，还使用加权残差导出了非线性简化模型，并通过数值求解来模拟平衡流的不稳定性和出现的永久表面波的发展。