Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Poisson growth

Poisson growth The two-dimensional free boundary problem in which the field is governed by Poisson’s equation and for which the velocity of the free boundary is given by the gradient of the field—Poisson growth—is considered. The problem is a generalisation of classic Hele-Shaw free boundary flow or Laplacian growth problem and has many applications. In the case when the right hand side of Poisson’s equation is constant, a formulation is obtained in terms of the Schwarz function of the free boundary. From this it is deduced that solutions of the Laplacian growth problem also satisfy the Poisson growth problem, the only difference being in their time evolution. The corresponding moment evolution equations, a Polubarinova–Galin type equation and a Baiocchi-type transformation for Poisson growth are also presented. Some explicit examples are given, one in which cusp formation is inhibited by the addition of the Poisson term, and another for a growing finger in which the Poisson term selects the width of the finger to be half that of the channel. For the more complicated case when the right hand side is linear in one space direction, the Schwarz function method is used to derive an exact solution describing a translating circular blob with changing radius. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Loading next page...
 
/lp/springer-journals/poisson-growth-8RFkOzszRj

References (16)

Publisher
Springer Journals
Copyright
Copyright © 2014 by The Author(s)
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-014-0094-9
Publisher site
See Article on Publisher Site

Abstract

The two-dimensional free boundary problem in which the field is governed by Poisson’s equation and for which the velocity of the free boundary is given by the gradient of the field—Poisson growth—is considered. The problem is a generalisation of classic Hele-Shaw free boundary flow or Laplacian growth problem and has many applications. In the case when the right hand side of Poisson’s equation is constant, a formulation is obtained in terms of the Schwarz function of the free boundary. From this it is deduced that solutions of the Laplacian growth problem also satisfy the Poisson growth problem, the only difference being in their time evolution. The corresponding moment evolution equations, a Polubarinova–Galin type equation and a Baiocchi-type transformation for Poisson growth are also presented. Some explicit examples are given, one in which cusp formation is inhibited by the addition of the Poisson term, and another for a growing finger in which the Poisson term selects the width of the finger to be half that of the channel. For the more complicated case when the right hand side is linear in one space direction, the Schwarz function method is used to derive an exact solution describing a translating circular blob with changing radius.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Nov 15, 2014

There are no references for this article.