Hele-Shaw flows with free boundaries in a corner or around a wedge Part I: Liquid at the vertex

Consider a Hele-Shaw cell with the fluid (liquid) confined to an angular region by a solid boundary in the from of two half-lines meeting at an angle alphapi; if 0 < alpha less than or equal to 1 we have flow in a corner, while if I < alpha less than or equal to 2 we have now around a wedge. We suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment emanating from the vertex, so we have liquid at the vertex, and contemplate such a situation that has been produced by injection at a number of points into an initially empty cell. We show that, if we assume the pressure to be constant along the free boundary, the region occupied by the fluid is the image of a semidisc (a domain bounded by a semicircle and its diameter) in the zeta-plane under a conformal map given by a function of the form zeta(alpha) times a rational function of zeta. The form of this rational function can be written down, and the parameters appearing in it then determined as the solution to a set of algebraic equations. Examples of such flows are given (including one which shows that, in a certain sense, injection can produce a cusp), and the limiting situation in the wedge configuration as one injection point is moved to infinity is also considered.