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 thi s phenomenon is important in a wide variety of applications involving stron gly non-Newtonian fluids - in other words, fluids that exhibit yield stress - we undertake a full theoretical examination of Saffmann-Taylor instabili ty in this type of fluid, in both longitudinal and radial flows in Hele-Sha w cells. In particular, we establish the detailed form of Darcy's law for y ield-stress fluids. Basically the dispersion equation for both flows is sim ilar to equations obtained for ordinary viscous fluids but the viscous term s in the dimensionless numbers conditioning the instability contain the yie ld stress. As a consequence the wavelength of maximum growth can be extreme ly small even at vanishing velocities. Additionally an approximate analysis shows that the fingers which are left behind at the beginning of destabili zation should tend to stop completely. Fingering of yield-stress fluids the refore has some peculiar characteristics which nevertheless are not suffici ent to explain the fractal pattern observed with colloidal systems.