Numerical simulation of the motion of a single drop in a shear flow at finite Reynolds numbers

Three-dimensional numerical simulation of the cross-stream migration of a drop in simple shear flow at finite Reynolds numbers neglecting the gravity influence is presented. In this study the full Navier-Stokes equations are solved by a finite difference/front tracking method. A drop is shown to migrate to the centreline of the channel in a shear flow. In other words, the centre of the channel is a global attractor of trajectories of a drop, regardless of the initial position and velocity. The migration velocity of the drop depends on surface tension and fluid velocity. Increasing the Weber number and decreasing the Reynolds number decrease the time of the migration to the centreline. The study showed that after an initial transient period the drop leads the local undisturbed velocity, for all cases. While time progresses, the velocity of the drop along the flow direction (x) decreases, whereas the velocity of drop along the flow velocity gradient increases. When the x-velocity reaches the local undisturbed velocity, the slip velocity tends to zero, except during a short initial transient. The slip velocity is the difference between the drop velocity and the ambient fluid velocity at the centre of the drop for the undisturbed flow. To validate the present calculations, some typical results are compared with the available experimental and theoretical data, which confirm that the present approach is qualitatively reliable in predicting the drop migration.