具有动态过冷度的Saffman-Taylor指
Saffman-Taylor fingers with kinetic undercooling
摘要：
Hele-Shaw通道中稳定传播的Saffman-Taylor指的数学模型可应用于二维相互作用的周期性排列的流注放电。在拖缆的情况下，界面上的相关正则化不是由表面张力提供的，而是被假定为涉及相当于动力学过冷的机制，其作用是惩罚高速度并防止未正则化溶液的爆破。以前的渐近结果的Hele-Shaw手指问题与动力过冷度表明，对于一个给定的值的动力过冷度参数，有一个离散的一组可能的手指形状，每个分析在鼻子和占用不同的分数的通道宽度。在动力学过冷参数消失的极限中，每个家庭的分数接近1/2，这表明动力学过冷对1/2的这种“选择”在定性上类似于众所周知的具有表面张力的类似物。我们对待计算这些萨夫曼-泰勒手指动力学过冷，这原来是更微妙的比类似物与表面张力，因为动力学过冷允许手指的形状是角落自由，但不是分析的数值问题。我们通过建立一个动力学过冷度和表面张力的问题，并以数值计算的极限，表面张力消失的选择机制提供了数值证据。
英文摘要：
The mathematical model of a steadily propagating Saffman-Taylor finger in a Hele-Shaw channel has applications to two-dimensional interacting streamer discharges which are aligned in a periodic array. In the streamer context, the relevant regularisation on the interface is not provided by surface tension, but instead has been postulated to involve a mechanism equivalent to kinetic undercooling, which acts to penalise high velocities and prevent blow-up of the unregularised solution. Previous asymptotic results for the Hele-Shaw finger problem with kinetic undercooling suggest that for a given value of the kinetic undercooling parameter, there is a discrete set of possible finger shapes, each analytic at the nose and occupying a different fraction of the channel width. In the limit in which the kinetic undercooling parameter vanishes, the fraction for each family approaches 1/2, suggesting that this 'selection' of 1/2 by kinetic undercooling is qualitatively similar to the well-known analogue with surface tension. We treat the numerical problem of computing these Saffman-Taylor fingers with kinetic undercooling, which turns out to be more subtle than the analogue with surface tension, since kinetic undercooling permits finger shapes which are corner-free but not analytic. We provide numerical evidence for the selection mechanism by setting up a problem with both kinetic undercooling and surface tension, and numerically taking the limit that the surface tension vanishes.
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