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Dissertation Defense

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Laplacian Growth: Interface Evolution in a Hele-Shaw Cell, by Khalid Malaikah

  • Seminar
When Apr 23, 2013
from 01:00 PM to 03:00 PM
Where 322 Morton Hall
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Abstract: Laplacian growth is an interface dynamics where the normal component of velocity of a free boundary is proportional to the normal derivative of a harmonic function defined in a moving domain. The interface evolution in a Hele-Shaw cell is governed by the Laplacian growth model. Laplacian growth is a non-linear complex dynamics which has been attracting an attention of mathematicians and physicists for centuries. In this study we are interested in the derivation of the governing equations for the free boundary in terms of the Schwarz function for some specific Hele-Shaw flows.

The Structure of the thesis is as follows.

In Chapter 1, we give an introduction to the history of the problem; discuss the methods and the state of the art.

Chapter 2 is devoted to the Schwarz function equation for the two-phase Hele-Shaw flows with a fixed gap. Here we re-derive the equations earlier obtained by D. Crowdy using a slightly different method. Our derivation is based on an introduction of a single-valued complex velocity potential.

In Chapter 3, we derive the Schwarz function equation for a class of generalized Hele-Shaw flows and apply it to the case of an interior problem in a cell with the time-dependent gap.

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