This special issue of the European Journal of Applied Mathematics is devoted to papers on free boundary problems, largely for Hele-Shaw or Stokes flows. The majority of the papers were presented at a meetingSupport from the London Mathematical Society is gratefully acknowledged. on these topics held in Oxford in August 1998, 100 years after H. S. Hele-Shaw first described his cell [2], and 40 years on from the classic experiment of Saffman & Taylor [4]. The conference delegates sent a message of good wishes to P. Ya Polubarinova-Kochina (below), whose paper in 1945 [3], together with that of Galin in the same year [1], may be said to have initiated the modern study of the Hele-Shaw free boundary problem. We learned with great sadness of Polubarinova-Kochina's death this year, at the age of 100; her active scientific career spanned some 75 years.

As part of the preparation for the conference, with the assistance of K. A. Gillow we have assembled a 600-paper bibliography on Hele-Shaw and Stokes flow; it can be found at www.maths.ox.ac.uk/˜howison/Hele-Shaw/. We hope it will be useful to the many researchers in these areas, which even after a century of investigation will clearly retain their mathematical and practical interest for many years to come.

Small surface tension is known to select a discrete family from the continuum of Saffman–Taylor finger solutions with zero surface tension. Here a method developed recently in Chapman et al. [2] is employed in which a naive perturbation expansion in powers of the surface tension parameter is optimally truncated to show exponentially small terms being switched on across Stokes lines. These terms are responsible for finger selection.

We consider the two-dimensional problem of crystal growth in a forced flow. A dendrite is placed in a Hele-Shaw cell with insulating walls and grows due to undercooling. We neglect the surface energy in the Gibbs–Thomson relation. The problem is formulated in terms of analytic functions similarly to closely related work on the viscous fingering problem of Saffman and Taylor. We derive a solvability condition for the existence of a steady-state needle-like solidification front in the limit of small Peclet number, Pe = V∞l/a, where V∞ is the characteristic velocity of the melt, 2l is the channel width, and a is the thermal diffusivity of the liquid. The velocity of the crystallization front is directly proportional to the hydrodynamic velocity V∞ and undercooling, while the dendrite width ld does not depend upon the physical parameters, and indeed, ld = l.

In this note we modify and extend the work of Howison & King [12] to describe the situation of flow around a wedge of arbitrary angle in a Hele-Shaw cell. An ingenious complex-variable method due to Polubarinova-Kochina is used to construct an explicit solution to the zero-surface tension problem.

In this paper, a complex variable model for Hele-Shaw moving boundary problems with kinetic undercooling regularization is studied. This model consists of an abstract Cauchy–Kovalevsky problem and of a Riemann–Hilbert–Poincaré problem for holomorphic functions. The local existence of holomorphic solutions is shown. Some numerical results for this model complete the paper.

A moving boundary problem for one phase Hele-Shaw flow with surface tension is considered. The fluid domain is unbounded and its boundary has an infinite length, and a finite number of suction (or injection) points are given. This is a mathematical model of the ‘fingering phenomenon’. We prove the existence of a unique solution locally in time.

Short-time existence and uniqueness results in Sobolev spaces are proved for Hele-Shaw flow with kinetic undercooling and for Stokes flow without surface tension. In both cases, the flow is driven by arbitrarily distributed sources and sinks in the interior of the liquid domain. The proofs are based on a general approach consisting of the reformulation of the problem as a Cauchy problem for a nonlinear, nonlocal evolution equation on the unit sphere, quasilinearization by equivariance, investigation of the linearization, and Galerkin approximations. In the situation discussed here, the linearized evolution operator is a first-order differential operator, and thus the evolution equation is of hyperbolic type. Finally, a brief survey of the properties of the evolution equations that arise from Hele-Shaw flow and Stokes flow with and without regularization is given.

Following the historical development of the theory of exact solutions for singularity-driven Hele-Shaw flows, this note demonstrates that the problem of two-dimensional viscous sintering preserves quadrature identities. This provides a unified theoretical perspective in which to understand the two separate free boundary problems. The result is established directly from the equations of motion without appeal to conformal mapping theory, although the result underlies the existence of exact conformal mapping solutions. The formulation leads to a concise, closed-form representation of the evolution equations for the parameters in the conformal mapping function. Some examples are given.

We discuss the application of complex variable methods to Hele-Shaw flows and two-dimensional 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.

In this article the complex variable theory of two-dimensional Stokes flow as developed by Richardson [22], and modified by Howison & Richardson [16], is reviewed. The analysis of [16] is extended to a new solution driven by a point sink, which uses a cubic polynomial conformal mapping (with real coefficients) from the unit disk onto the fluid domain. This solution is analysed in the limit of small surface tension. An apparent ‘stability paradox’ (where two equivalent flow geometries are found, one of which is ‘stable’ and the other unstable) is resolved by allowing the coefficients to take complex values.