Coriolis effects on fingering patterns under rotation

Rotationally induced fingering patterns in a two-dimensional heterogeneous porous medium

Rotating fluid flows under two-dimensional homogeneous porous media conditions (or in a rotating Hele-Shaw cell) reveal the development of complex interfacial fingering patterns. These pattern-forming structures are characterized by the occurrence of finger competition events, finger pinch-off episodes, as well as by the production of satellite droplets. In this work, we use intensive numerical simulations to investigate how these fully nonlinear pattern growth phenomena are altered by the presence of permeability heterogeneities in the rotating porous medium. This is done by employing a diffuse-interface Darcy-Cahn-Hilliard description of the problem and considering a permeability field presenting a log-Gaussian distribution, characterized by a variance s and a correlation length l. We study how the heterogeneity measures s and l couple to the governing hydrodynamic dimensionless parameters of the problem and introduce important changes on the pattern formation dynamics of the system.

Diffuse-interface approach to rotating Hele-Shaw flows

When two fluids of different densities move in a rotating Hele-Shaw cell, the interface between them becomes centrifugally unstable and deforms. Depending on the viscosity contrast of the system, distinct types of complex patterns arise at the fluid-fluid boundary. Deformations can also induce the emergence of interfacial singularities and topological changes such as droplet pinch-off and self-intersection. We present numerical simulations based on a diffuse-interface model for this particular two-phase displacement that capture a variety of pattern-forming behaviors. This is implemented by employing a Boussinesq Hele-Shaw-Cahn-Hilliard approach, considering the whole range of possible values for the viscosity contrast, and by including inertial effects due to the Coriolis force. The role played by these two physical contributions on the development of interface singularities is illustrated and discussed.

Inertial effects on rotating Hele-Shaw flows

We examine the finite surface tension radial viscous fingering problem in a rotating Hele-Shaw cell, and focus on the action of inertial effects on the stability and morphology of the emerging patterns. To study such a flow we use an alternative version of the usual Darcy's law derived by gap-averaging the Navier-Stokes equation, but retaining its inertial terms. The importance of inertial forces is pertinently characterized by a rotational Reynolds number. Linear and weakly nonlinear stages of the dynamics are described analytically through a mode coupling approach. The linear stability results indicate that inertia has a stabilizing role. However, the characteristic number of fingers and the width of the band of linearly unstable modes are not altered by inertia. In the early nonlinear regime we find that inertia acts to favor the development of fingers with narrower tips than those obtained in its absence. We have also verified that finger competition events are affected by inertial effects which tend to restrain finger length variability among inward-moving fingers.

Coriolis effects on rotating Hele-Shaw flows: a conformal-mapping approach

The zero surface tension fluid-fluid interface dynamics in a radial Hele-Shaw cell driven by both injection and rotation is studied by a conformal-mapping approach. The situation in which one of the fluids is inviscid and has negligible density is analyzed. When Coriolis force effects are ignored, exact solutions of the zero surface tension rotating Hele-Shaw problem with injection reveal suppression of cusp singularities for sufficiently high rotation rates. We study how the Coriolis force affects the time-dependent solutions of the problem, and the development of finite time singularities. By employing Richardson's harmonic moments approach we obtain conformal maps which describe the time evolution of the fluid boundary. Our results demonstrate that the inertial Coriolis contribution plays an important role in determining the time for cusp formation. Moreover, it introduces a phase drift that makes the evolving patterns rotate. The Coriolis force acts against centrifugal effects, promoting (inhibiting) cusp breakdown if the more viscous and dense fluid lies outside (inside) the interface. Despite the presence of Coriolis effects, the occurrence of finger bending events has not been detected in the exact solutions.