Monday 19 Nov 2012: Poisson growth and the geometry of valleys
Robb Mcdonald - University College London
Harrison 103 15:00-16:00
The two-dimensional free boundary problem in which the field variable is governed by Poisson's
equation and for which the velocity of the free boundary is given by its
gradient--Poisson growth--is considered. The problem can be considered a
generalisation of classic Hele-Shaw free boundary flow or Laplacian growth
problem. 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. Some simple examples are given.
The formulation is used to find an explicit solution for the shape of the
valley or ravine formed by groundwater seepage. This is a flow in
porous media problem which is modelled as a Poisson growth free boundary
problem. The resulting geometry
compares favourably to alternative theories of valley shape.