Numerical study of the effect of nonlinear control on the behavior of a liquid drop in elongational flow with vorticity

Abstract

The problem of controlling a liquid drop suspended in an arbitrary two-dimensional elongational flow with vorticity is revisited. Bentley and Leal (J Fluid Mech 137:219–240, 1986) kept the drop centroid at the stagnation point using a linear proportional control strategy in a four-roll-mill apparatus that projects the drop’s motion onto the stable flow direction of the stagnation point. A nonlinear strategy based on the Poincare–Bendixson theory to ensure a periodic motion of the drop centroid inside a prescribed area around the stagnation point is proposed and studied. In addition, a detailed numerical study is presented to illustrate the effect of the control on the drop motion. The present strategy is effective, allowing for deformation and changes in the drop orientation by less than 1% for extreme flow conditions that cannot be achieved by a four-roll-mill setup.