Dissertation Defense
Laplacian Growth: Interface Evolution in a HeleShaw Cell, by Khalid Malaikah
What 


When 
Apr 23, 2013 from 01:00 PM to 03:00 PM 
Where  322 Morton Hall 
Contact Name  Tanya Savin 
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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 HeleShaw cell is governed by the Laplacian growth model. Laplacian growth is a nonlinear 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 HeleShaw 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 twophase HeleShaw flows with a fixed gap. Here we rederive the equations earlier obtained by D. Crowdy using a slightly different method. Our derivation is based on an introduction of a singlevalued complex velocity potential.
In Chapter 3, we derive the Schwarz function equation for a class of generalized HeleShaw flows and apply it to the case of an interior problem in a cell with the timedependent gap.