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 Saffman–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.