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.
