Residual mass and flow regimes for the immiscible liquid-liquid displacement in a plane channel

The motion of two immiscible liquids in a plane channel is analyzed for the case in which the flow conditions and the interactions between the liquids and the solid surface maintain the displaced fluid attached to the wall. The Galerkin Finite Element Method is used to compute the velocity field and the configuration of the interface between the two fluids. We compare the residual mass fraction left on the wall with its two counterparts in capillary tubes, namely residual mass fraction and dimensionless layer thickness of the displaced fluid. The main result of this comparison was that although there is a qualitative similarity concerning the layer thickness between the two cases, the residual fraction of mass presented an important difference, showing that when the aspect ratio of the capillary passage is large there is an increase in the displacement efficiency. The thickness of the displaced liquid film attached to the channel walls is a function of the capillary number (Ca) and the viscosity ratio (Nμ). A map of streamlines in the Cartesian space (Ca, Nμ) with the different flow regimes of the problem is presented. We also showed that we can adapt the available analytical results obtained for gas-displacement in capillary tubes to the plane channel case, for low values of Ca.