Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number

Abstract

The steady axisymmetric flow in and around a deformable drop moving under the action of gravity along the axis of a vertical tube at intermediate Reynolds number is studied by solving the nonlinear free-boundary problem using a Galerkin finite-element method. For the case where the drop and suspending liquid have the same viscosity, the ratio of the densities is 6/5 or 5/6, and the radius of the tube is equal to twice the radius of a sphere having the drop volume, four significant results are apparent in the computations. First, we compute drops showing much more deformation, and in particular the development of considerably more non-convexity, than those found in previous calculations for non-zero Reynolds number. The degree of non-convexity typically grows with the Reynolds number. Secondly, external recirculation zones can be attached to or disjoint from the drop. We find when there is a single external recirculation zone, that is disjoint (as found by Dandy & Leal), it can attach to the drop as the Reynolds number is increased. As the Reynolds number further increases, this is immediately followed by division of the drop into two adjacent recirculating regions. Thirdly, we sometimes find two recirculation zones in the suspending liquid. Finally, the drag coefficient, axis ratio, and normalized interfacial and frontal areas of the drop can vary non-monotonically with the Weber number, exhibiting as many as four local extrema. The results are compared to previous theoretical and experimental work, and implications for drop motion and heat and mass transfer are discussed.