The Lateral Migration of a Drop under Gravity between Two Parallel Plates at Finite Reynolds Numbers

Abstract

A finite difference / front tracking method is used to examine the lateral migration of a three-dimensional deformable drop in plane Poiseuille flow at a finite-Reynolds-number. The computations are based on an improved implementation of the front tracking method at finite Reynolds numbers that include convective terms. The elliptic pressure equation is solved by a multigrid method. Both neutrally buoyant and non-neutrally buoyant drop are studied. The computation is performed within a unit cell which is periodic in the direction along the channel. A neutrally buoyant drop lags the fluid slightly, and the wall effect balances the effect of the curvature of the velocity profile, giving rise to an equilibrium lateral position about halfway between the wall and the centerline (the SegreSilberberg effect). Results are presented over a range of density ratios. In the non-neutrally buoyant case, the gravity force is imposed along the flow direction. Non-neutrally buoyant drops have more complicated patterns of migration, depending upon the magnitude of the buoyancy force. When the density difference is small, the equilibrium position is either near the wall or near the centerline, depending on whether the drop leads or lags the local fluid. When the density difference is large enough, the equilibrium position shifts towards the centerline, irrespective of whether the drop is lighter or heavier than the fluid. The effect of Reynolds number and capillary number on the non-neutrally buoyant drops is investigated. The accuracy of the method is assessed by comparison with the other simulations and experiments.