Saffman-Taylor fingers and directional solidification at low velocity

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-Uₙ*{)}3/2$, when the velocity U of the finger approaches a lower value ${U}ₙ*$ that we calculate. We then discuss the connections of these results with directional solidification at small P\'eclet numbers. Our analysis does not reveal any sign of wavelength selection for steady-state cells by a solvability condition, contrary to recent numerical findings.