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The dynamics of a viscous drop moving along a substrate under the influence of shear flow in a parallel-walled channel is investigated. A front tracking numerical method is used to simulate a drop with moving contact lines. A Navier slip boundary condition is applied to relax the contact line singularity. Steady state solutions are observed for small Reynolds and capillary number. Unsteady solutions are obtained with increasing Reynolds or capillary number. For large values of the parameters, the interface appears to rupture, but for intermediate parameter values, time periodic drop interface oscillations are possible as the drop is moving along the bottom channel wall. These different states are identified in the Reynolds number–capillary number plane for a specific range of physical parameters. The effects of density and viscosity ratio are also illustrated. [ABSTRACT FROM AUTHOR]

Here we consider the gravity-driven dynamics of two superposed, incompressible, immiscible, viscous fluids flowing in a parallel-wall channel. The fluids are governed by the Navier-Stokes equations with surface tension along the interface. We use the front tracking numerical method to determine the solutions. Both stable and unstable solutions are obtained. The unstable solutions are characterized by a finger of lower fluid rising into the upper fluid. Our results are consistent with the predictions of linear stability theory. In addition, we attempt to characterize the growth of the unstable solutions as a function of the Reynolds and the Bond number. [ABSTRACT FROM AUTHOR]

The motion of a thin viscous layer of fluid on a horizontal solid surface bounded laterally by a dry spot and a vertical solid wall is considered. A lubrication model with contact line motion is studied. We find that for a container of fixed length the axisymmetric equilibrium solutions with small dry spots are unstable to axisymmetric disturbances. As the size of the dry spot increases, the equilibrium solutions become unstable to nonaxisymmetric disturbances. In addition, we present numerical solutions of the nonlinear evolution equations in the axisymmetric and nonaxisymmetric cases for different values of the parameters. The axisymmetric results show good agreement with existing experimental results. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]