Miscible displacements in Hele-Shaw cells: three-dimensional Navier-Stokes simulations

Three-dimensional Navier-Stokes simulations of viscously unstable, miscible Hele-Shaw displacements are discussed. Quasisteady fingers are observed whose tip velocity increases with the Peclet number and the unfavourable viscosity ratio. These fingers are widest near the tip, and become progressively narrower towards the root. The film of resident fluid left behind on the wall decreases in thickness towards the finger tip. The simulations reveal the detailed mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole along the length of the finger. This streamwise vorticity convects resident fluid from the wall towards the centre of the gap in the cross-gap symmetry plane of the finger, while it transports injected fluid laterally away from the finger centre within the mid-gap plane. In this way, it results in the emergence of a longitudinal, inner splitting phenomenon some distance behind the tip that has not been reported previously. This inner splitting mechanism, which leaves the tip largely intact, is fundamentally different from the familiar tip-splitting mechanism. Since the inner splitting owes its existence to the presence of streamwise vorticity and cross-gap velocity, it cannot be captured by gap-averaged equations. It is furthermore observed that the role of the Peclet number in miscible displacements differs in some ways from that of the capillary number in immiscible flows. Specifically, larger Peclet numbers result in wider fingers, while immiscible flows display narrower fingers for larger capillary numbers. Furthermore, while higher capillary numbers are known to promote tip-splitting, inner splitting is delayed for larger Peclet numbers.