(353h) Scaling in the Transition From Selective Withdrawal to Entrainment

In selective withdrawal (SW), fluid is withdrawn through a tube that has its tip suspended a distance S above a flat interface separating two fluids. When the withdrawal rate Q is low, the interface forms a steady-state hump and only the upper fluid is withdrawn. When Q is increased (or S decreased), the interface undergoes a topological transition so that the lower fluid is entrained with the upper one, forming a steady-state spout. Here, this discontinuous transition is analyzed computationally when both fluids are incompressible and Newtonian. The numerical method employed is an implicit method of lines ALE algorithm which uses finite elements with elliptic mesh generation.  In particular, it is shown that the critical withdrawal rate at which the aforementioned transition occurs scales with the nozzle separation raised to some power n.  Moreover, two distinct types of transition from SW to entrainment are identified.  The first, which occurs at low flow rates, involves a transition from SW to viscous entrainment (VE) in which normal viscous stresses overcome capillary pressure to cause the transition.  Here, the value of the scaling exponent is found to equal to three (n=3).  This finding agrees with previous experiments and resolves a long standing controversy involving previous simulations.  The second transition, which has heretofore not been known, involves one from SW to inertial entrainment (IE) in which viscous stresses play no role.  Here, the value of the scaling exponent is found to equal to two (n=2).  Scaling arguments are then used to rationalize the simulation results, the existence of these two distinct regimes, and the transition from the first regime, where n=3, to the other, where n=2.