Onset of motion of a three-dimensional droplet on a wall in shear flow at moderate Reynolds numbers

Abstract

We investigate the critical conditions for the onset of motion of a three-dimensional droplet on a wall in shear flows at moderate Reynolds number. A diffuse-interface method is used for this purpose, which also circumvents the stress singularity at the moving contact line, and the method allows for a density and viscosity contrast between the fluids. Contact-angle hysteresis is represented by the prescription of a receding contact angle θ_Rand an advancing contact angle value θ_A. Critical conditions are determined by tracking the motion and deformation of a droplet (initially a spherical cap with a uniform contact angle θ₀). At sufficiently low values of a Weber number,We(based on the applied shear rate and the drop volume), the drop deforms and translates for some time, but subsequently reaches a stationary position and attains a steady-state shape. At sufficiently large values ofWeno such steady state is found. We present results for the critical value ofWeas a function of Reynolds numberRefor cases with the initial value of the contact angle θ₀=θ_Ras well as for θ₀=θ_A. A scaling argument based on a force balance on the drop is shown to represent the results very accurately. Results are also presented for the static shape, transient motion and flow structure at criticality. It is shown that at lowReour results agree (with some qualifications) with those of Dimitrakopoulos & Higdon (1998,J. Fluid Mech. vol. 377, p. 189). Overall, the results indicate that the critical value ofWeis affected significantly by inertial effects at moderate Reynolds numbers, whereas the steady shape of droplets still shows some resemblance to that obtained previously for creeping flow conditions. The paper concludes with an investigation into the complex structure of a steady wake behind the droplet and the occurrence of a stagnation point at the upstream side of the droplet.