On the selection of viscosity to suppress the Saffman-Taylor instability in a radially spreading annulus

We examine the stability of a system with two radially spreading fronts in a He le-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.