Computation of Hele-Shaw free boundary problems near obstacles

The time-dependent evolution of source-driven Hele-Shaw free boundary flows in the presence of an obstacle is computed numerically. The Baiocchi transformation is used to convert the Hele-Shaw Laplacian growth problem into a free boundary problem for a streamfunction-like variable u(x, y, t) governed by Poisson’s equation (with constant right-hand side) with the source becoming a point vortex of strength linearly dependent on time. On the free boundary, both u and its normal derivative vanish, and on the obstacle, the normal derivative of u vanishes. Interpreting u as a streamfunction, at a given time, the problem becomes that of finding a steady patch of uniform vorticity enclosing a point vortex of given strength such that the velocity vanishes on the free boundary and the tangential velocity vanishes on the obstacle. A combination of contour dynamics and Newton’s method is used to compute such equilibria. By varying the strength of the point vortex, these equilibria represent a sequence of source-driven growing blobs of fluid in a Hele-Shaw cell. The practicality and accuracy of the method is demonstrated by computing the evolution of Hele-Shaw flow driven by a source near a plane wall; a case for which there is a known exact solution. Other obstacles for which there are no known exact solutions are also considered, including a source both inside and outside a circular boundary, a source near a finite-length plate and the interaction of an infinite free boundary impinging on a circular disc.