Numerical study of miscible fingering in a time-dependent gap Hele-Shaw cell

We perform a detailed numerical study of the evolution of a miscible fluid droplet in a time-dependent gap Hele-Shaw cell. The development of the emerging fingering instabilities is systematically analyzed by intensive and highly accurate numerical simulations. We focus on the influence of three relevant physical parameters on the interface dynamics: the Peclet number Pe, the viscosity contrast A, and the Korteweg stress parameter delta. Consistently with conventional miscible Saffman-Taylor studies in constant-gap Hele-Shaw cells, our results demonstrate that more vigorous fingering is observed at higher Pe and larger A. Concerning the specific role of Pe and A, we deduce two general results: higher Peclet number favors branching around a nearly circular region (which leads to longer interfacial lengths); while larger viscosity contrast results in more significant finger penetrations (which is quantitatively expressed by larger diameter of gyration). We have also verified that the Korteweg stress parameter delta does act as an effective interfacial tension: it stabilizes the miscible interface, leading to fingering patterns that present a greater resemblance with the structures obtained in similar immiscible situations. Finally, we have identified the development of a visually striking phenomenon in the limit of high Pe, large A, and relatively small delta: some outward fingers pinch, and subsequent droplet detachment is observed. We show that such a droplet detachment process can be prevented by the action of stronger interfacial stresses. This last finding provides additional evidence for the claim that the Korteweg stresses can be treated as an ersatz interfacial tension in diffusing fluids.