# Laplacian growth without surface tension in filtration combustion: Analytical pole solution

*Complexity*21 (5):31-42 (2016)

**Abstract**

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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Archival date: 2017-07-21

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References found in this work BETA

Which Forces Reduce Entropy Production?Hubler, Alfred

Modeling Complex Systems Macroscopically: Case/Agent‐Based Modeling, Synergetics, and the Continuity Equation.Rajaram, Rajeev & Castellani, Brian

Classification of Emergence and its Relation to Self‐Organization.Halley, Julianne D. & Winkler, David A.

Nonextensive Model of Self‐Organizing Systems.Grabowski, Franciszek

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2016-06-30

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