Saffman–Taylor instability in yield stress fluids

Abstract

Pushing a fluid with a less viscous one gives rise to the well known Saffman–Taylorinstability. This instability is important in a wide variety of applications involving stronglynon-Newtonian fluids that often exhibit a yield stress. Here we investigate theSaffmann–Taylor instability in this type of fluid, in longitudinal flows in Hele–Shaw cells. Inparticular, we study Darcy’s law for yield stress fluids. The dispersion equation for the flowis similar to the equations obtained for ordinary viscous fluids but the viscous terms inthe dimensionless numbers conditioning the instability now contain the yieldstress. This also has repercussions on the wavelength of the instability as it followsfrom a linear stability analysis. As a consequence of the presence of yield stress,the wavelength of maximum growth is finite even at vanishing velocities. Westudy Darcy’s law and the fingering patterns experimentally for a yield stress fluidin a linear Hele–Shaw cell. The results are in rather good agreement with thetheoretical predictions. In addition we observe different regimes that lead to differentmorphologies of the fingering patterns, in both rectangular and circular Hele–Shaw cells.