Abstract The two-phase Hele-Shaw problem when two immiscible fluids have different viscosities is considered. A method of solution using the Schwarz function approach is discussed. Examples of different scenarios of the evolution of an elliptical interface in Hele-Shaw cells with both constant and a uniformly changing distance between the plates are shown. In these examples, the sinks and sources are not only points, but also line distributions located in the exterior and the interior domains. The latter drastically enriches the family of exact solutions for zero surface tension compared to the literature. Asymptotic solutions for nonzero surface tension when the eccentricity of an ellipse is small are presented.

中文翻译：

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具有椭圆界面的两流体 Hele-Shaw 问题

摘要 考虑两种不混溶流体具有不同粘度时的两相Hele-Shaw问题。讨论了使用 Schwarz 函数方法的求解方法。显示了 Hele-Shaw 单元中椭圆界面演化的不同场景的示例，其中板之间的距离恒定且均匀变化。在这些示例中，汇点和源点不仅是点，而且是位于外部和内部域中的线分布。与文献相比，后者极大地丰富了零表面张力的精确解系列。给出了椭圆偏心率较小时非零表面张力的渐近解。