Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime

Abstract

This is the second paper in a two part series in which we analyze two diffuse interface models to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. Diffusion between the components is limited if the components are macroscopically immiscible. In one of the systems (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we focus on buoyancy driven flow and the Rayleigh–Taylor instability. In the fully nonlinear regime before pinchoff, results from the HSCH and BHSCH models are compared to highly accurate boundary-integral simulations of the classical sharp interface system. In this case, we find that the diffuse interface models yield nearly identical results and we demonstrate convergence to the boundary-integral solutions as the interface thickness vanishes. We find that the break-up of an unstably stratified fluid layer is smoothly captured by both models. The HSCH model seems to be more diffusive than the BHSCH model and predicts an earlier pinchoff time which causes subtle differences between the two in the pinchoff region. Further, in the limit of zero interface thickness, we find that the effect of compressibility does not vanish at pinchoff. This distinguishes the HSCH model from all others in which compressibility effects are neglected. It may turn out, for example, that characterizing the limiting effect of compressibility at pinchoff may suggest a physically based selection mechanism for cutting and reconnecting sharp interfaces. Varying the gravitational force and viscosities of the fluids yields different pinchoff times and numbers of satellite drops. Moreover, using the analysis of the linear growth rates from our first paper [paper I, Phys. Fluids 14, 492 (2002)], we confirm that the modified HSCH/BHSCH parameters suggested in that work lead to improved agreement with sharp interface results at finite interface thicknesses. Last, we also consider a case in which the fluid components are miscible. We find competition between buoyancy, viscous, diffusional and, at very early times, surface tension-like forces.