We examine the stability of a system with two radially spreading fronts in a Hele-Shaw cell in which the viscosity increases monotonically from the innermost to the outermost fluid. The critical parameters are identified as the viscosity ratio of the inner and outer fluids and the viscosity difference between the intermediate and outer fluids as a fraction of the viscosity difference between the inner and outer fluids. There is a minimum viscosity ratio of the inner and outer fluids above which, for each azimuthal mode, the system is stable to perturbations of that mode at any flow rate. This condition is directly analogous to the result for a single interface. Below this minimum ratio, the system may be stable at any flow rate early in the flow. However, once the inner radius reaches a critical fraction of the outer radius, this absolute stability ceases to apply owing to the coupling of the inner and outer interfaces. We determine the maximum flow rate, as a function of time, in order that all modes remain stable due to the effects of interfacial tension. These criteria for stability are then used to select the viscosity of the intermediate fluid so that a fixed volume of the intermediate and then inner fluid can be added to the system in the minimum time with the system remaining stable throughout. The optimal viscosity for this intermediate fluid depends on the relative volume of the inner and intermediate fluid and also on the overall viscosity ratio of the innermost fluid and the original fluid in the cell, with the balance being to suppress the early time instability of the outer interface and the late time instability of the inner interface. We discuss application of this approach to a problem of injection of treatment fluid in an oil well.

Using variational calculus, we investigate the time-dependent injection rate that minimises the growth of the Saffman–Taylor instability when a finite volume of fluid is injected in a finite time, $t_{f}$, into a Hele-Shaw cell. We first consider a planar interface, and show that, with a constant viscosity ratio, the constant injection rate is optimal. When the viscosity of the displacing fluid, $\unicode[STIX]{x1D707}_{1}(t)$, gradually increases over time, as may occur with a slowly gelling polymer solution, the optimal injection rate, $U^{\ast }(t)$, involves a gradual increase in the flow rate with time. This leads to a smaller initial value of flow rate than the constant injection rate, finishing with a larger value. Such optimisation can lead to a substantial suppression of the instability as compared to the constant injection case if the characteristic gelling time is comparable to $t_{f}$. In contrast, for either relatively slow or fast gelling, there is much less benefit in selecting the optimal injection rate, $U^{\ast }(t)$, as compared to the constant injection rate. In the case of a constant injection rate from a point source, $Q$, then with a constant viscosity ratio the fastest-growing perturbation on the radially spreading front involves axisymmetric modes whose wavenumber increases with time. Approximating the discrete azimuthal modes by a continuous distribution, we find the injection rate that minimises growth, $Q^{\ast }(t)$. We find that there is a critical time for injection, $t_{f}^{\dagger }$, such that if $t_{f}>t_{f}^{\dagger }$ then $Q^{\ast }(t)$ can be chosen so that the interface is always stable. This critical time emerges from the case with an injection rate given by $Q^{\ast }\sim t^{-1/3}$. As the total injection time is reduced to values $t_{f}<t_{f}^{\dagger }$, the system becomes progressively more unstable, and the optimal injection rate for an idealised continuous distribution of azimuthal modes asymptotes to a flow rate that increases linearly with time. As for the one-dimensional case, if the viscosity of the injection fluid gradually increases over time, then the optimal injection rate has a smaller initial value but gradually increases to larger values than for the analogous constant viscosity problem. If the displacing fluid features shear-thinning rheology, then the optimal injection rate involves a smaller flow rate at early times, although not as large a reduction as in the Newtonian case, and a larger flow rate at late times, although not as large an increase as in the Newtonian case.

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 ≫ π r b 2 , we find r f = 0.4 r b ( V / r b 2 ) 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.