Shearing Liquids of Unequal Viscosity: Can Mixing be Independent of Stirring Rate at a Large Viscosity Ratio?

Mixing liquids with unequal viscosity involves complex flows and a nontrivial diffusive feedback. As an insightful reference case, we study the deformation of a spherical blob with viscosity λη, sheared in a miscible bath with viscosity η. Experiments resolve the transition from unbounded stretching, at low λ, to rollinglike motion, at large λ. In between the blob is transiently substretched before destabilizing into swirling structures. Exact solutions of the flow recover all three regimes and predict a transition ratio λ_{c}≃3.975. We discuss the dramatic impact of the flow transition on mixing and propose scaling laws for the blob mixing time in the limit of large Péclet numbers, suggesting mixing could be independent of shear rate at large λ.