长气泡如何收缩：非强迫Hele-Shaw流的数值方法
How a Long Bubble Shrinks: a Numerical Method for an Unforced Hele-Shaw Flow
摘要：
我们开发了一种数值方法来解决一个自由边界问题，描述形状松弛，表面张力，一个长而薄的气泡的无粘流体被困在一个粘性流体中的Hele-Shaw细胞。外狄利克雷问题的解决方法采用了经典的边界积分公式。我们的版本的数值方法是特别有利的一个非常长的和薄的气泡，最近已经开发了一个渐近标度理论的动态。由于气泡的纵横比非常大，直接实施边界积分算法将是不切实际的。我们通过引入一个新的近似的积分出现在Fredholm积分方程和在气泡界面处的压力的正常导数的积分表达式的算法进行修改。新的近似允许一个大大减少节点的数量在几乎平坦的部分的气泡界面，同时保持良好的精度。新近似的另一个好处是它消除了调和共轭的切向导数的积分的数值发散。接口的位置是提前在时间上使用显式节点跟踪，而较大的节点间距使人们能够使用更大的时间步长。该算法进行了测试两个模型的问题，近似解析解。
英文摘要：
We develop a numerical method for solving a free boundary problem which describes shape relaxation, by surface tension, of a long and thin bubble of an inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. The method of solution of the exterior Dirichlet problem employs a classical boundary integral formulation. Our version of the numerical method is especially advantageous for following the dynamics of a very long and thin bubble, for which an asymptotic scaling theory has been recently developed. Because of the very large aspect ratio of the bubble, a direct implementation of the boundary integral algorithm would be impractical. We modify the algorithm by introducing a new approximation of the integrals which appear in the Fredholm integral equation and in the integral expression for the normal derivative of the pressure at the bubble interface. The new approximation allows one to considerably reduce the number of nodes at the almost flat part of the bubble interface, while keeping a good accuracy. An additional benefit from the new approximation is in that it eliminates numerical divergence of the integral for the tangential derivative of the harmonic conjugate. The interface's position is advanced in time by using explicit node tracking, whereas the larger node spacing enables one to use larger time steps. The algorithm is tested on two model problems, for which approximate analytical solutions are available.
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