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Selection of a Hele-Shaw Bubble via Exponential Asymptotics
SIAM Journal on Applied Mathematics ( IF 1.9 ) Pub Date : 2020-01-30 , DOI: 10.1137/18m1220868
Christopher J. Lustri , Christopher C. Green , Scott W. McCue

SIAM Journal on Applied Mathematics, Volume 80, Issue 1, Page 289-311, January 2020.
The well-studied selection problems involving Saffman--Taylor fingers or Taylor--Saffman bubbles in a Hele-Shaw channel are prototypical examples of pattern selection. Exact solutions to the corresponding zero-surface-tension problems exist for an arbitrary finger or bubble speed, but the addition of surface tension leads to a discrete set of solution branches, all of which approach a single solution in the limit in which the surface tension vanishes. In this sense, the surface tension selects a single physically meaningful solution from the continuum of zero-surface-tension solutions. Recently, we provided numerical evidence to suggest that the selection problem for a bubble propagating in an unbounded Hele-Shaw cell behaves in an analogous way to other finger and bubble problems in a Hele-Shaw channel; however, the selection of the ratio of bubble speeds to background velocity appears to follow a very different surface tension scaling to the channel cases. Here we apply techniques in exponential asymptotics to solve the selection problem analytically, confirming the numerical results, including the predicted surface tension scaling laws. Further, our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. These results have ramifications for exotic-shaped Saffman--Taylor fingers as well as for the time-dependent evolution of bubbles propagating in Hele-Shaw cells.


中文翻译:

通过指数渐近法选择Hele-Shaw气泡

SIAM应用数学杂志,第80卷,第1期,第289-311页,2020年1月。
经过深入研究的选择问题涉及Hele-Shaw通道中的Saffman-Taylor手指或Taylor-Saffman气泡,是模式选择的典型示例。对于任意的手指或气泡速度,都存在对相应的零表面张力问题的精确解,但是表面张力的增加导致离散的解分支集,所有这些分支都在表面张力的极限内接近单个解消失。从这个意义上讲,表面张力从零表面张力解的连续体中选择一个物理上有意义的解。最近,我们提供了数值证据,表明在无界Hele-Shaw单元中传播的气泡的选择问题的行为与Hele-Shaw通道中的其他手指和气泡问题类似。然而,气泡速度与背景速度之比的选择似乎遵循与通道情况截然不同的表面张力比例。在这里,我们采用指数渐近技术来解析选择问题,从而确认了数值结果,包括预测的表面张力缩放定律。此外,我们的分析揭示了沿着溶液分支的气泡形状的多个尖端,这似乎是由于在共形映射平面中跨斯托克斯线的指数小波动贡献的打开和关闭而引起的。这些结果对异形Saffman-Taylor手指以及Hele-Shaw细胞中传播的气泡随时间的演变产生了影响。在这里,我们采用指数渐近技术来解析选择问题,从而确认了数值结果,包括预测的表面张力缩放定律。此外,我们的分析揭示了沿溶液分支的气泡形状的多个尖端,这似乎是由于在共形映射平面中跨斯托克斯线的指数小波状贡献的打开和关闭而引起的。这些结果对异形Saffman-Taylor手指以及Hele-Shaw细胞中传播的气泡随时间的演变产生了影响。在这里,我们采用指数渐近技术来解析选择问题,从而确认了数值结果,包括预测的表面张力缩放定律。此外,我们的分析揭示了沿溶液分支的气泡形状的多个尖端,这似乎是由于在共形映射平面中跨斯托克斯线的指数小波状贡献的打开和关闭而引起的。这些结果对异形Saffman-Taylor手指以及Hele-Shaw细胞中传播的气泡随时间的演变产生了影响。我们的分析揭示了沿着溶液分支的气泡形状的多个尖端,这似乎是由于在共形映射平面中跨斯托克斯线打开和关闭指数小波状贡献而引起的。这些结果对异形Saffman-Taylor手指以及Hele-Shaw细胞中传播的气泡随时间的演变产生了影响。我们的分析揭示了沿着溶液分支的气泡形状的多个尖端,这似乎是由于在共形映射平面中跨斯托克斯线的指数小波浪状贡献的打开和关闭而引起的。这些结果对异形Saffman-Taylor手指以及Hele-Shaw细胞中传播的气泡随时间的演变产生了影响。
更新日期:2020-01-30
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