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.
Highlights
In the standard Hele-Shaw free boundary problem for fluid with zero surface tension, the pressure, or velocity potential, satisfies Laplace’s equation in the fluid region, is constant on its boundary and the normal velocity of the interface is given by the gradient of the velocity potential in the normal direction
This formulated, but nonlinear, two-dimensional free boundary problem has a remarkable variety of applications occurring over a wide range of lengthscales including oil recovery, flow in porous media and injection moulding
It seems likely that the general valley geometry problem is more complicated with the shape of the valley being determined by both a Poisson growth problem for the groundwater depth outside the valley itself, and a Laplacian growth problem inside the valley governing erosion of the valley walls [19]
Summary
In the standard Hele-Shaw free boundary problem for fluid with zero surface tension, the pressure, or velocity potential, satisfies Laplace’s equation in the fluid region, is constant on its boundary and the normal velocity of the interface is given by the gradient of the velocity potential in the normal direction. Lacey [11] discusses the behaviour of Schwarz function singularities in the case of Hele-Shaw flow with a time-dependent gap for the case when there is no hydrodynamic forcing In particular he derives an explicit solution for the behaviour of an elliptical blob being squeezed between the plates of a Hele-Shaw cell. As noted in [7] there is a connection between the Poisson growth problem and the behaviour of a blob of fluid in a rotating Hele-Shaw cell of constant width This is immediately evident by comparing the Schwarz function equations in each case.
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