Three-dimensional Navier–Stokes simulations of viscously unstable, miscible Hele-Shaw displacements are discussed. Quasisteady fingers are observed whose tip velocity increases with the Péclet 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 Péclet number in miscible displacements differs in some ways from that of the capillary number in immiscible flows. Specifically, larger Péclet 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 Péclet numbers.

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Direct numerical simulations of the variable density and viscosity Navier–Stokes equations are employed, in order to explore three-dimensional effects within miscible displacements in horizontal Hele-Shaw cells. These simulations 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 along the upper, gravitationally unstable side of the interface generally is shorter than that of the fingering instability. This results in the formation of plumes of the more viscous resident fluid not only in between neighbouring viscous fingers, but also along the centre of fingers, thereby destroying their shoulders and splitting them longitudinally. The streamwise vorticity dipoles forming as a result of the spanwise Rayleigh–Taylor instability place viscous resident fluid in between regions of less viscous, injected fluid, thereby resulting in the formation of gapwise vorticity via the traditional, gap-averaged viscous fingering mechanism. This leads to a strong spatial correlation of both vorticity components. For stronger density contrasts, the streamwise vorticity component increases, while the gapwise component is reduced, thus indicating a transition from viscously dominated to gravitationally dominated displacements. 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.

One efficient way of determining the bond strength of adhesives is to measure the force or the work required to separate two surfaces bonded by a thin adhesive film. We consider the case in which the thin film is not a conventional adhesive material but a high viscosity ferrofluid confined between two narrowly spaced parallel flat plates subjected to an external magnetic field. Our theoretical results demonstrate that both the peak adhesive force and the separation energy are significantly influenced by the action and symmetry properties of the applied field. Specifically, we show that the adhesive strength of a ferrofluid is reduced if the applied magnetic field is perpendicular to the plates or if the applied field is in plane and exhibits azimuthal symmetry. Conversely, the adhesive strength can be either enhanced or reduced if the applied field is in plane and is directed radially outward. This establishes an interesting connection between adhesion and ferrohydrodynamic phenomena, allowing the control of important adhesive properties by magnetic means.

We consider the flow of a ferrofluid droplet in a Hele-Shaw cell with a time-dependent gap width. When the surface tension and applied magnetic field are zero, interfacial instabilities develop and the droplet breaks. We execute a mode-coupling approach to the problem and focus on understanding how the development of singularities is affected by the action of an external field. Our analytical results indicate that the introduction of an azimuthal magnetic field profoundly modifies pattern formation, allowing the inhibition of interfacial singularities. We suggest the magnetic field can be used as a controllable parameter to discipline singular behavior.