Analytical solutions for Hele-Shaw moving boundary flows in the presence of a circular cylinder

Analytical solutions for the one-phase, zero surface tension Hele-Shaw problem describing the source-driven growth of a blob of viscous fluid in both the exterior and the interior of a circular cylinder are constructed explicitly. These are stated in the form of conformal maps which parameterise the region occupied by the blob. The results are derived by first applying the Baiocchi transform to the governing equations, and then seeking solutions to the transformed problem in terms of the Schwarz function of the free section of the blob's boundary. A pivotal role is played by an auxiliary function which gives the modulus of the parameterising map on the pre-image of this free boundary section. On determining this function, the map is found via the solution of a related Schwarz problem, expressed in an integral form. All stages of the growth of the blob are considered. For the exterior flow, this includes after the blob has fully engulfed the cylinder. During this stage, the region occupied by the blob is doubly connected and the parameterising map is constructed in terms of automorphic functions. Also considered, in the exterior case, is when the source is at infinity. To illustrate the results, several examples are computed. Comparisons are also made with existing numerical results.