Control of viscous instability by variation of injection rate in a fluid with time-dependent rheology

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}