Evidence for a universal saturation profile for radial viscous fingers

Complex fingering patterns develop when a low viscosity fluid is injected from a point source into the narrow space between two parallel plates initially saturated with a more viscous, immiscible fluid. We combine historical and new experiments with (a) a constant injection rate; (b) a constant source pressure; and (c) a linearly increasing injection rate, together with numerical simulations based on a model of diffusion limited aggregation (DLA), to show that for viscosity ratios in the range 300-10,000, (i) the finger pattern has a fractal dimension of approximately 1.7 and (ii) the azimuthally-averaged fraction of the area occupied by the fingers, S(r,t), is organised into three regions: an inner region of fixed radius, r < r_b, which is fully saturated with injection fluid, S = 1; a frozen finger region, r_b < r < r_f (t), in which the saturation is independent of time, S(r) = (r/r_b)^-0.3; and an outer growing finger region, r_f(t) < r < 1.44 r_f(t), in which the saturation decreases linearly to zero from the value (r_f/r_b)^-0.3 at r_f(t). For a given injected volume per unit thickness of the cell, V ≫ πr_b², we find r_f = 0.4r_b (V/r_b²)^1/1.7. This apparent universality of the saturation profile of non-linear fingers in terms of the inner region radius, r_b, and the injected volume V, demonstrates extraordinary order in such a complex and fractal instability. Furthermore, control strategies designed to suppress viscous fingering through variations in the injection rate, based on linear stability theory, are less effective once the instability becomes fully nonlinear.