# Poisson growth

## 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.

### Keywords

Laplacian growth Hele-Shaw flows## 1 Introduction

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 simply 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. Because of this, and with its rich mathematical structure becomingly increasingly evident, Laplacian growth, as it is also known, has attracted much attention in the literature e.g. [9]. As a consequence, powerful mathematical techniques, naturally formulated in terms of complex variables, have been developed leading to the derivation of a variety of exact solutions for Laplacian growth. These formulations include the Polubarinova–Galin equation [8, 20], Richardson’s demonstration of an infinity of conservation laws for the moments of the fluid [21], and a formulation in terms of the Schwarz function for the free boundary e.g. [4].

The mathematical problem in which Laplace’s equation is replaced by Poisson’s equation has received less attention, despite having important applications. When the right hand side of Poisson’s equation is constant the model is equivalent to the evaporation of thin liquid films [2]. Moment-based methods were used by [2] to derive exact solutions, some of which displayed typical Laplacian growth type cusp formation on the boundary.

When the upper plate of a Hele-Shaw cell is raised or lowered, the thickness of the thin gap between the plates is time-dependent and this leads to the pressure inside the blob satisfying Poisson’s equation with a time-dependent right-hand side. This time-dependence can be scaled out, so that the right-hand side becomes constant. Shelley et al. [22] studied this variable-gap Hele-Shaw problem and showed when the upper plate is raised instability arises, this being analogous to the evaporation case of [2]. Shelley et al. [22] examined the existence and regularity of solutions and found exact and numerical solutions with and without surface tension, including examples illustrating bubble fission and cusp formation. The multiply connected problem has been tackled by Crowdy and Kang [3]. They used associated quadrature domain identities to find explicit solutions which demonstrate that the usual well-posed ‘squeeze’ case for singly connected domains can break down in finite time in the multiply connected case. Another physical manifestation of Poisson growth is the debonding of adhesively joined surfaces e.g. [12]. The thin gap between the surfaces acts like a Hele-Shaw cell and air fingers penetrate the adhesive as the surfaces are separated.

In addition to Laplacian growth, several authors, e.g. [10, 13, 14], have considered other elliptic PDEs governing the interior of the fluid, as occurs, for example, in flow of fluid in inhomogeneous porous media. Few explicit solutions are known for these more general elliptic operators, one difficulty being their lack of conformal invariance.

The Poisson growth problem is formulated in Sect. 2 and its associated Schwarz function equation derived. This equation is then used to derive the moment evolution equations, a Polubarinova–Galin type equation and an associated Baiocchi-type transformation. Examples of explicit solutions are given in Sect. 3. The case when the right hand side of Poisson’s equation is linear in one of the space variables is considered in Sect. 4. The Schwarz function equation is derived and used to find an exact solution consisting of a propagating circular blob with time-dependent radius.

## 2 Schwarz function formulation of the free boundary problem

### 2.1 Derivation of Schwarz function equation

Equation (1) implies that \(\nabla .\mathbf{u}=-\beta \); that is, depending on the sign of \(\beta \) there is a source (\(\beta <0\)) or sink (\(\beta >0\)) distributed uniformly over the extent of the fluid blob. The free boundary problem (1) is a generalisation of the standard Laplacian growth or Hele-Shaw problem (\(\beta =0\)) and is referred to here as the Poisson growth problem.

Strictly, the derivation of (3) applies on the free boundary only, but by analytic continuation it applies everywhere in the fluid blob \(\Omega (t)\) where the terms of (3) are defined. If \(\beta =0\) then (3) reduces to the well-known Schwarz function relation governing Laplacian growth e.g. [4]. Since the nature of the singularities of \(g\) and \(w\) in (3) are independent of \(\beta \), an immediate implication is that solutions of the Poisson growth free boundary problem share the same geometries as the standard Hele-Shaw problem although they will in general have different evolution.

### 2.2 Moments

### 2.3 Polubarinova-Galian type equation

### 2.4 A related Baiocchi-type transform

## 3 Examples

### 3.1 Circular blob

### 3.2 Stability of a circular blob and bubble

A similar analysis for a perturbed circular bubble in an infinite fluid (as opposed to a finite blob of fluid) using the map \(z=a(\zeta ^{-1}+\epsilon \zeta ^n)\) from the interior of the unit \(\zeta \)-disk to the exterior of the perturbed bubble and considering singular behaviour as \(z\rightarrow \infty \) shows that a bubble is stable if \(Q<\beta \pi a^2\).

### 3.3 Limaçon

### 3.4 A finger solution

If \(\beta <0\) the finger solution breaks down in finite time as it either becomes an infinitely thin finger when \(\lambda (0)<1/2\), or occupies the full width of the channel if \(\lambda (0)>1/2\).

### 3.5 Application to the geometry of valleys

## 4 Linear variation in the mass forcing

## 5 Conclusions

The Poisson growth problem has been formulated in terms of the Schwarz function of the free boundary for the cases when the right hand side is either constant or varies linearly in one space direction. Simple examples of exact solutions have been derived. Although not presented here, it is straightforward to find the Schwarz function equation when the right hand side of Poisson’s equation depends only on the radial coordinate \(r^2=z{\bar{z}}=zg\) on \(\partial \Omega \), in which case a trivial solution is a non-translating circular blob with time-dependent area. It is of interest to explore the possibility of formulating Schwarz function equations and finding explicit solutions for other forms of functional dependence of the right hand side of Poisson’s equation.

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