Viscous fingering instabilities in carbon black gels*

Pattern formation in fluids occurs in numerous physical processes in which mechanical mixing, chemical reactions, evaporation and/or surface effects play a key role. When the pattern develops in a non-Newtonian fluid, the non-linear rheology interferes with the patterning process, which often generates a richer dynamics than that commonly observed for a Newtonian fluid. Here we focus on the viscous fingering instability in a time-dependent yield stress fluid. We study experimentally the flow of a carbon black gel sandwiched in a parallel plate geometry, for which the upper plate is being lifted up at constant velocity. We show the existence of a critical initial gap spacing and a critical lift velocity, above which the flow becomes unstable, leading to the growth of viscous finger originating from the Saffman-Taylor instability at the fluid-air interface. The resulting pattern in the gel consists in a tree-like branched structure, whose wavelength surprisingly follows the scaling established for Newtonian fluids. The signature of the fluid non-linear rheology lays in the spatial extent of the pattern, which is governed by the yield strain of the gel. Finally, we show that varying the shear history of the gel, one can produce patterns with a wealth of new morphologies.