Three-dimensional Navier-Stokes simulations of gravitationally and viscously unstable, miscible displacements in Hele-Shaw cells are discussed. In neutrally bouyant displacements, quasisteady fingers are observed whose tip velocity increases with the Peclet number and the unfavourable viscosity ratio. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole along the length of the finger. This quadrupole leads to the emergence of a longitudinal, inner splitting phenomenon behind the tip. The generation of those vorticity quadrupoles is related to the viscous fingering instability acting along the streamwise direction. By looking at different aspect ratios we assess the competition between multiple fingers, and the interaction between inner splitting and tip-splitting; and by analysing displacements with nonmonotonic viscosity profiles, we present a natural way to control finger growth.

In vertical Hele-Shaw cells, a destabilizing density difference generally increases the velocity of the two-dimensional base flow displacement front. These two-dimensional simulations reveal several possible configurations for the streamline patterns, from single stagnation points in gravitationally unstable displacements to multiple stagnation points and recirculation zones, and spike formation in stable configurations. A two-dimensional pinch-off governed by dispersion is observed some distance behind the displacement front. Three-dimensional simulations show that the inner splitting instability in vertical displacements is delayed for increasing unfavorable density differences, even though the streamwise vorticity quadrupole's strength increases for larger gravity numbers. For large unstable density differences the formation of a secondary, downward propagating front develops an anchor-like shape as a result of the flow induced by quadrupoles.

In horizontal displacements, we identify a number of mechanisms concerning the interaction of viscous fingering with a spanwise Rayleigh-Taylor instability. The dominant wavelength of the Rayleigh-Taylor instability generally is shorter than that of the fingering instability, which results in the formation of plumes of the more viscous fluid not only in between neighboring viscous fingers, but also along the center of fingers, thereby destroying their shoulders and splitting them longitudinally. We identify a transition from viscously dominated to gravitationally dominated displacements, and correlate this behavior to the magnitudes of the streamwise and gapwise vorticity components. Gap-averaged, time-dependent concentration profiles show that variable density displacement fronts propagate more slowly than their constant density counterparts. This indicates that the gravitational mixing results in a more complete expulsion of the resident fluid from the Hele-Shaw cell. This observation may be of interest in the context of enhanced oil recovery or carbon sequestration applications.