A Hele-Shaw problem and the second Painlevé transcendent

Abstract

We consider the time evolution of an unbounded interface between a viscous fluid and a displacing fluid of negligible viscosity in a Hele-Shaw geometry. Using a conformal mapping transformation z=F(ζ, t) from the lower half ζ-plane to the flow domain Ω(t) in the z-plane, the dynamics is formulated in terms of an integro-differential equation for F on the real ζ axis. This formulation is similar to earlier results for radial and channel geometries. Using a certain explicit solution it is shown that the zero surface tension problem is ill-posed in the sense of Hadamard. Furthermore, certain earlier formal asymptotic results for small but non-zero surface tension [47], are rigorously established. This is achieved by relating the similarity reduction of the Harry–Dym equation to the Painlevé II equation, and by studying this equation in the complex plane. In particular, it is shown that there exists a unique solution of Painlevé II equation which satisfies the appropriate far-field algebraic decay condition in a certain sector Ŝ of the complex plane and which in Ŝ is free of singularities. This solution asymptotes to certain elliptic functions in other sectors of the complex plane and contains a set of denumerably infinite number of singularities that approach the boundary of Ŝ from the outside at large distances from the origin (in the similarity variable). This is consistent with earlier numerical results.