TY - JOUR

T1 - Laplacian growth without surface tension in filtration combustion

T2 - Analytical pole solution

AU - Kupervasser, Oleg

N1 - Publisher Copyright:
© 2016 Wiley Periodicals, Inc.

PY - 2016/5/1

Y1 - 2016/5/1

N2 - Filtration combustion (FC) 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 FC, 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 regularization 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 12 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 Saffman-Taylor problem without surface tension is similar to the solution for the equation of cellular flames in the case of the combustion of gas mixtures.

AB - Filtration combustion (FC) 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 FC, 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 regularization 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 12 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 Saffman-Taylor problem without surface tension is similar to the solution for the equation of cellular flames in the case of the combustion of gas mixtures.

KW - Filtration combustion

KW - Final time singularity

KW - Hele-Shaw cell

KW - Laplacian growth

KW - Pole solution

KW - Saffman-Taylor problem

KW - Zero surface tension

UR - http://www.scopus.com/inward/record.url?scp=84968862812&partnerID=8YFLogxK

U2 - 10.1002/cplx.21627

DO - 10.1002/cplx.21627

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AN - SCOPUS:84968862812

SN - 1076-2787

VL - 21

SP - 31

EP - 42

JO - Complexity

JF - Complexity

IS - 5

ER -