Three-dimensional effects in directional solidification in Hele-Shaw cells

We consider the one-sided model of directional solidification of a binary alloy in a Hele-Shaw cell and study the linear stability and nonlinear evolution of an initially curved solid-liquid interface whose shape is determined by the contact-angle condition at the wall. For the values of the angle close to 1/2 pi we use a perturbation expansion to obtain the corrections to the known stability criterion of the planar front under various assumptions about the mon-dimensional thickness of the Hele-Shaw cell and both static and dynamic contact-line conditions. A boundary-integral method is used to find stability criteria for arbitrary contact angle. Weakly nonlinear analysis is also carried out; it is shown that subcritical bifurcation can be promoted and the appearance of three-dimensional structures can be delayed due to the contact-angle conditions.