Viscous fluid injection into a confined channel

Abstract

We analyze the injection of a viscous fluid into a two-dimensional horizontal confined channel initially filled with another viscous fluid of different density and viscosity. We study the flow using the lubrication approximation and assume that the mixing between the fluids and their interfacial tension are negligible. When the injection rate is maintained constant, the evolution of the fluid-fluid interface can be described by a nonlinear advection-diffusion equation dependent only on the viscosity ratio between the two fluids. In the early time period, the advection-diffusion equation reduces to a well-known nonlinear diffusion equation, and a self-similar solution is obtained. In the late time period, the advection-diffusion equation is approximated by a nonlinear hyperbolic equation, and a compound wave solution is constructed to describe the time evolution of the fluid-fluid interface. Numerical solutions of the full equation show good agreement with the analytical solutions in both the early and late time periods. Finally, a regime diagram is obtained to summarize the flow behaviours with regard to two dimensionless groups: the viscosity ratio of the two fluids and the dimensionless time; three different dynamical behaviours are identified in the regime diagram: a nonlinear diffusion regime, a hyperbolic regime, and a transition regime. This problem is analogous to the corresponding injection flow problem into a confined porous medium.