Immiscible Newtonian displacement by a viscoplastic material in a capillary plane channel

The immiscible displacement in a capillary plane channel of a Newtonian liquid by a viscoplastic one that obeys a Papanastasiou’s constitutive equation is numerically analyzed. An elliptic mesh generation technique, coupled with the Galerkin finite element method is used to determine the velocity field and the configuration of the interface between the two materials. We investigate the displacement efficiency and the flow patterns of the problem as functions of the dimensionless parameters that govern the problem: the capillary number (Ca), the viscosity ratio of the two fluids (N _η ) and the yield number, (τ′₀). The numerical results showed that for a fixed viscosity ratio, the fraction of mass attached to the wall is a decreasing function of τ′₀. We constructed maps of streamlines in the Cartesian space defined by τ′₀ and Ca for fixed viscosity ratios in order to capture the rough location of bypass and recirculating flow regimes. Higher yield number values induce bypass flow regimes, especially for high Ca. The dimensionless forms of the momentum conservation equation and the force balance at the interface were essential for the understanding of the role played by the dimensionless numbers that govern the problem.