Boundary effects on the onset of miscible viscous fingering in a Hele-Shaw flow

Abstract

Previous studies on the onset of miscible viscous fingering (VF) have been modeled with two semi-infinite fluid regions, where the more viscous fluid from one region is displaced by the less viscous fluid of the other region. However, most of the experiments were performed by filling the Hele-Shaw cell with a highly viscous fluid and then displacing it by a less viscous fluid through the inlet boundary. This causes the difference in the boundary condition at the inlet, where the reflective boundary arises for the former, and the absorbing boundary is for the latter case, and they correspond to a Neumann- and Dirichlet-type of boundary condition, respectively, in the species transport convection-diffusion equation. The literature on the linear stability analysis (LSA) of miscible VF for the reflective boundary case is largely available. Here we investigate the effects, and we compare both boundary conditions for the onset of miscible VF. In either of the configurations, the base-state is found to be unsteady, which leads to a nonautonomous dynamical system, and perturbations are not localized around the interface in the original coordinate system. To overcome these difficulties, we used the initial value problem (IVP) in a similarity domain, and we found that the onset of VF is delayed in the absorbing boundary case as compared to the reflective one. Interestingly, it is observed from the LSA that there exists a threshold mobility ratio for which the boundary conditions can affect the maximum at the onset of instability. Furthermore, fully nonlinear simulations are carried out using COMSOL MULTIPHYSICS, and the results are found to be in good agreement with the LSA.