Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Abstract

We discuss the application of complex variable methods to Hele-Shaw flows and twodimensional Stokes flows, both with free boundaries. We outline the theory for the former, in the case where surface tension effects at the moving boundary are ignored. We review the application of complex variable methods to Stokes flows both with and without surface tension, and we explore the parallels between the two problems. We give a detailed discussion of conserved quantities for Stokes flows, and relate them to the Schwarz function of the moving boundary and to the Baiocchi transform of the Airy stress function. We compare the results with the corresponding results for Hele-Shaw flows, the principal consequence being that for Hele-Shaw flows the singularities of the Schwarz function are controlled in the physical plane, while for Stokes flow they are controlled in an auxiliary mapping plane. We illustrate the results with the explicit solutions to specific initial value problems. The results shed light on the construction of solutions to Stokes flows with more than one driving singularity, and on the closely related issue of momentum conservation, which is important in Stokes flows, although it does not arise in Hele-Shaw flows. We also discuss blow-up of zero-surface-tension Stokes flows, and consider a class of weak solutions, valid beyond blow-up, which are obtained as the zero-surface-tension limit of flows with positive surface tension.

Cell motility: A viscous fingering analysis of active gels

“…For a simple cell geometry as the circular one, the shape equation can be derived by assuming the radial geometry for Ψ as done in [5]. Nevertheless, sometimes this geometry, which is assumed a priori, cannot allow to satisfy all the equations because we face a free-boundary problem, which is especially difficult to analyse for Darcy and Stokes flows [6]. Free boundary problems can be solved either by complex analysis [6] or with the help of the Green function techniques.…”

“…The mathematical solutions of these equations with these boundary conditions lead to a free-boundary problem where the cell shape cannot be fixed a priori. Treating the disorientated cortex at the cell border as a boundary layer in both cases (static and motile) with an isotropic core (static case) or a fully oriented core (motile case), we succeed to formulate mathematically the shape equation with the Schwarz function [6], a technique used for shape drops in hydrodynamics and wetting [7]. The actin cortex, once treated as a boundary layer, induces a modification of the boundary conditions for the hydrodynamic flow.…”

“…First we show that the process of polymerization-depolymerisation induces a flow which can be approximated by some Darcy flow when the friction on the substrate is important and we establish the boundary conditions of Neumann and Dirichlet type at the origin of the free-boundary problem. In section 3 we present the complex analysis formulation [6] and prove mathematically that the Callan et al [5] approach cannot explain the cell motility. Section 4 is a reminder of the theory of active gels [8,9,10] with application to the polar geometry (2D cylindrical geometry).…”

Mathematical analysis of some multi-dimensional tissue-growth models

“…The analysis above glosses over the fact that in this ill-posed case the solution will typically cease to exist in finite time; there is substantial discussion of such matters in the literature on the Hele-Shaw problem (see, for example, Cummings et al [7] and references therein) and we shall not comment further here. The tissue interface can thus be expected to develop a complicated fingering morphology (familiar in the Hele-Shaw context), which would correlate with rapid penetration into the surrounding tissue; tumour cells around the tip of a finger might then be prone to break off from the main tumour mass, possibly resulting in metastatic spread, attaching to (and growing at) other parts of the body.…”

“…The system (4.2), (4.5) can be regarded as a Stokes-flow analogue of the Hele-Shaw squeeze film problem and it warrants further investigation by complex variable methods and so forth (cf. Franks [7], for example), as does the converse one-phase problem which is described below; in two dimensions, writing v = (u, v) and…”