Saffman-Taylor fingers and directional solidification at low velocity

Abstract

We examine the McLean-Saffman equations for viscous fingering in the limit where the finger fills almost completely the Hele-Shaw channel (λ≃1). We find an infinite countable set of solutions. For each branch of solutions, λ increases toward 1 as (U-$U_n^{\mathrm{*}}$ ${)}^{3/2}$, when the velocity U of the finger approaches a lower value $U_n^{\mathrm{*}}$ that we calculate. We then discuss the connections of these results with directional solidification at small Péclet numbers. Our analysis does not reveal any sign of wavelength selection for steady-state cells by a solvability condition, contrary to recent numerical findings.