Laplacian Growth II: Pole solutions in formation of the Saffman-Taylor Finger with one half of the channel width without surface tension

Details Laplacian Growth II: Pole solutions in formation of the Saffman-Taylor Finger with one half of the channel width without surface tension Laplacian Growth II: Pole solutions in formation of the Saffman-Taylor Finger with one half of the channel width without surface tension

2011-06-04

arxiv.org/abs/1106.0832v2

Nonlinear Sciences

Chaotic Dynamics

7 pages, 3 figures

Scientific paper

Laplacian growth without surface tension has nice analytical solutions which replace its complex integro-differential motion equations by simple differential equations of poles motion in a complex plane. The main problem of such solution is existing of finite time singularities. To prevent such singularities nonzero surface tension usually is used. But such nonzero surface tension destroys analytical solutions. However more elegant way exists to solve the problem. First of all, we can introduce some small poles noise to system. Secondary, for regularization of problem we throw out all new poles that can give finite time singularity. It can be strictly proved that asymptotic solution for such system is a single finger. Moreover the qualitative consideration demonstrate that finger with 1/2 of the channel width is statistically stable. So all properties of such solution are completely the same as for the solution with a nonzero surface tension under a numerical noise. Surprisedly, flame front propagation problem has the same pole solutions and qualitative behavior.

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