拉普拉斯增长II：过滤燃烧中无表面张力的Saffman-Taylor问题：通道宽度一半的一指形
Laplacian Growth II: Saffman - Taylor Problem Without Surface Tension in Filtration Combustion: Formation of One Finger with Half of the Channel Width
摘要：
过滤燃烧用无表面张力的拉普拉斯增长描述。这些方程有优雅的解析解，用复平面中极点运动的简单微分方程代替了复杂的积分微分运动方程。这样的解决方案的主要问题是有限时间奇点的存在。为了防止这种奇异性，通常采用非零表面张力，然而，非零表面张力在过滤燃烧中不存在，这破坏了解析解。然而，存在一种更优雅的方法来解决这个问题。首先，我们可以向系统引入少量极点噪声。其次，为了使问题正规化，我们抛弃了所有能产生有限时间奇点的新极点。可以严格证明这样的系统的渐近解是单指。此外，定性的考虑表明，具有1/2的沟道宽度的指是统计稳定的。因此，这样的解决方案的所有性质是完全相同的解决方案与非零表面张力下的数值噪声。无表面张力的ST问题的解类似于气体混合物燃烧情况下的蜂窝火焰方程的解。
英文摘要：
Filtration combustion is described by Laplacian growth without surface tension. These equations have elegant analytical solutions that replace the complex integro-differential motion equations by simple differential equations of pole motion in a complex plane. The main problem with such a solution is the existence of finite time singularities. To prevent such singularities, nonzero surface tension is usually used.However, nonzero surface tension does not exist in filtration combustion, and this destroys the analytical solutions. However, a more elegant approach exists for solving the problem. First, we can introduce a small amount of pole noise to the system. Second, for regularisation of the problem, we throw out all new poles that can produce a finite time singularity. It can be strictly proved that the asymptotic solution for such a system is a single finger. Moreover, the qualitative consideration demonstrates that a finger with 1/2 of the channel width is statistically stable. Therefore, all properties of such a solution are exactly the same as those of the solution with nonzero surface tension under numerical noise. The solution of the ST problem without surface tension is similar to the solution for the equation of cellular flames in the case of the combustion of gas mixtures.
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