A study of the Saffman-Taylor instability in viscoplastic fluids

This thesis aims to explore and understand the Saffman-Taylor instability at the interface of a Hershel-Bulkley fluid and air. We describe the flow in a Hele-Shaw cell theoretically and experimentally. For our theoretical analysis, we follow the usual Hele-Shaw approximations for incompressible flows with a Herschel-Bulkley constitutive law. We extend this model by allowing the fluid to slip on the smooth walls of the cell. We conduct a linear stability analysis for both planar and circular geometries and predict that effective slip stabilizes the interface. We also use numerical computations to examine the non-linear regime of a single finger in the planar geometry. We find that slip generates wider fingers and makes the plugged-up area at the roots of the fingers smaller and appear at later times. We conduct experiments using corn syrup as a control case and aqueous suspensions of Carbopol as a model Herschel-Bulkley fluid. The fluids are pumped into a Hele-Shaw cell through a circular vent, forming an initial disk in the cell. Then, we either pump air into the center of the disk, creating an expanding annulus, or we withdraw the disk through the vent. In both cases, the interface is in an unstable Saffman-Taylor configuration causing fingers to appear. We look at the effect of the type of cell wall, the gap size, the flux and the size of the initial disk. The instability and the observed patterns are very different between a cell with smooth or rough walls, confirming our theoretical prediction on the influence of effective slip. The trends with the other parameter variations are in reasonable agreement with the theoretical predictions.Science, Faculty ofMathematics, Department ofGraduat