Viscoplastic Saffman-Taylor fingers with and without wall slip

The Saffman-Taylor instability for the flow of a Herschel-Bulkley fluid through a Hele-Shaw cell is explored theoretically and experimentally. The theoretical analysis adopts conventional Hele-Shaw approximations, but generalized to account for a Herschel-Bulkley rheology and to include a model for the effective slip of the fluid over smooth walls. A linear stability analysis is presented for fingering instabilities on both planar and axisymmetrical interfaces. The linear instability of a planar interface is continued numerically into the nonlinear regime. It is found that certain finger widths are selected and controlled by the yield stress. Stresses also fall sufficiently behind the fingertips to allow the yield stress to block the cell to either side. Experiments are conducted using aqueous suspensions of Carbopol pumped into a Hele-Shaw cell through a circular vent. Instabilities are created by first pumping a disk of Carbopol into the cell, then either pumping air into the fluid-filled cell or withdrawing the Carbopol through the vent. In both cases, the fingers forming on the retreating air-Carbopol interface are interrogated as a function of flux, gap size and the type of cell walls. The instability is very different for cells with either rough or smooth walls, an effect that we attribute to effective slip. The trends observed in the experiments are in broad agreement with theoretical predictions