The effect of surface tension on the shape of fingers in a Hele Shaw cell
The effect of surface tension on the shape of fingers in a Hele Shaw cell
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DOI：
10.1017/s0022112081002735
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发表时间：
1981-01
影响因子：
3.7
通讯作者：
J. McLean;P. Saffman
中科院分区：
文献类型：
--
作者：
J. McLean;P. Saffman
Saffman & Taylor（1958）和Pitts（1980）在Hele Shaw单元中指法的实验结果是通过在界面处包含表面张力效应的二维势流来模拟的。利用自由流线技术，将自由曲面的形状表示为非线性积分微分方程的解。对方程进行了数值求解，并与实验结果进行了比较。轮廓的形状可以很好地预测，但手指宽度对表面张力的依赖在定量上不准确，尽管定性行为是正确的。注意到数值与形式奇异摄动分析之间的冲突，但没有解决。本文还研究了稳定手指对小扰动的稳定性。线性稳定性分析表明，二维手指不受表面张力效应的稳定，这与实验结果不一致。提出了理论与实验不一致的可能原因。
The experimental results of Saffman & Taylor (1958) and Pitts (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the interface. Using free streamline techniques, the shape of the free surface is expressed as the solution of a nonlinear integro-differential equation. The equation is solved numerically and the solutions are compared with experimental results. The shapes of the profiles are very well predicted, but the dependence of finger width on surface tension is not quantitatively accurate, although the qualitative behaviour is correct. A conflict between the numerics and a formal singular perturbation analysis is noted but not resolved. The stability of the steady finger to small disturbances is also examined. Linearized stability analysis indicates that the two-dimensional fingers are not stabilized by the surface-tension effect, which disagrees with the experimental observations. A possible reason for the discrepancy between theory and experiment is suggested.