Stick-slip instability for viscous fingering in a gel

– The growth dynamics of an air finger injected in a visco-elastic gel (a PVA/borax aqueous solution) is studied in a linear Hele-Shaw cell. Besides the standard Saffmann-Taylor instability, we observe with increasing finger velocities the existence of two new regimes: (a) a stick-slip regime for which the finger tip velocity oscillates between 2 different values, producing local pinching of the finger at regular intervals, (b) a “tadpole” regime where a fracture-type propagation is observed. A scaling argument is proposed to interpret the dependence of the stick-slip frequency with the measured rheological properties of the gel. Introduction. – When a fluid is injected into a more viscous liquid, the invading front is unstable at early stages and splits into multiple fingers. This process, known as the SaffmannTaylor (ST) problem [1], has been extensively studied as a simple example of non-linear dynamical instability. The width of the fingers arises from a competition between the surface tension, which tends to widen them, and the viscous stress, which tends to make them thinner. In a standard ST experiment, the fluids are confined between two parallel plates separated by a gap of thickness b much smaller than any other length scale. For air driven into a Newtonian liquid, the stationary width w of the fingers relative to the gap thickness b only depends on the capillary number Ca = ηU/γ where η is the liquid viscosity, U the bubble tip velocity and γ the liquid surface tension. When the bubble is forced to grow in a linear channel of width W << b, the ratio w/W decreases as Ca is increased until it reaches a limiting value equal to one-half. In most practical situations however, the liquid phase (typically polymer solutions or complex multi-phase systems) does not show a purely Newtonian rheology. Many experimental and theoretical studies have thus attempted to understand the modifications of the ST results associated with more complicated rheological characteristics of the invaded phase [2–4]. Two situations may be distinguished. For liquids with shear rate dependent viscosity, the shape of the finger is slightly modified. This can be understood by considering the inhomogeneous (∗) Université Louis Pasteur and Institut Universitaire de France