Self-similar relaxation dynamics of a fluid wedge in a Hele-Shaw cell

Let the interface between two immiscible fluids in a Hele-Shaw cell have, at t=0, a wedge shape. As a wedge is scale-free, the fluid relaxation dynamics are self-similar. We find the dynamic exponent of this self-similar flow and show that the interface shape is given by the solution of an unusual inverse problem of potential theory. We solve this problem analytically for an almost flat wedge, and numerically otherwise. The wedge solution is useful for analysis of pinch-off singularities.