The deformation of a liquid drop moving normal to a plane wall

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Abstract
The deformation of a viscous drop moving under the action of gravity normal to a plane solid wall is studied. Under the assumption of creeping flow, the motion is studied as a function of the viscosity ratio between the drop and the suspending fluid, of surface tension, and of the initial drop configuration. Using the boundary integral formulation, the flow inside and outside the drop is represented in terms of a combined distribution of a single-layer and a double-layer potential of Green functions over the drop surface. The densities of these distributions are identified with the discontinuity in the interfacial surface stress, and with the interfacial velocity. The problem is formulated as a Fredholm integral equation of the second kind for the interfacial velocity which is solved by successive iterations. It is found that in the absence of surface tension, a drop moving away from the wall obtains an increasingly prolate shape, eventually ejecting a trailing tail. Depending on the initial drop configuration, ambient fluid may be entrained into the drop along or away from the axis of motion. Surface tension prevents the formation of the tail allowing the drop to maintain a compact shape throughout its evolution. The deformation of the drop has little effect on its speed of rise. A drop moving towards the wall obtains an increasingly oblate shape. In the absence of surface tension, the drop starts spreading in the radial direction reducing into a thinning layer of fluid. This layer is susceptible to the gravitational Rayleigh–Taylor instability. Surface tension restricts spreading, and allows the drop to attain a nearly steady hydrostatic shape. This is quite insensitive to the viscosity ratio and to the initial drop configuration. The evolution of the thin layer of fluid which is trapped between the drop and the wall is examined in detail, and with reference to film-drainage theory. It is shown that the assumptions underlying this theory are accurate when the surface tension is sufficiently large, and when the viscosity of the drop is of the same or lower order of magnitude as the viscosity of the ambient fluid. The numerical results are discussed with reference to film-drainage asymptotic theories.