Saffman-Taylor instability in yield-stress fluids

When a fluid is pushed by a less viscous one the well-known Saffman-Taylor instability phenomenon arises, which takes the form of fingering. Since this phenomenon is important in a wide variety of applications involving strongly non-Newtonian fluids - in other words, fluids that exhibit yield stress - we undertake a full theoretical examination of Saffmann-Taylor instability in this type of fluid, in both longitudinal and radial flows in Hele-Shaw cells. In particular, we establish the detailed form of Darcy's law for yield-stress fluids. Basically the dispersion equation for both flows is similar to equations obtained for ordinary viscous fluids but the viscous terms in the dimensionless numbers conditioning the instability contain the yield stress. As a consequence the wavelength of maximum growth can be extremely small even at vanishing velocities. Additionally an approximate analysis shows that the fingers which are left behind at the beginning of destabilization should tend to stop completely. Fingering of yield-stress fluids therefore has some peculiar characteristics which nevertheless are not sufficient to explain the fractal pattern observed with colloidal systems.