Stretching liquid bridges with moving contact lines: The role of inertia

Liquid bridges with moving contact lines are found in a variety of settings such as capillary feeders and high-speed printing. Although it is often assumed that the length scale for these flows is small enough that inertial effects can be neglected, this is not the case in certain applications. To address this issue, we solve the Navier-Stokes equations with the finite element method for the stretching of a liquid drop between two surfaces for non-zero Reynolds numbers. We consider an axisymmetric liquid bridge between a moving flat plate and either a stationary flat plate or a cavity. The contact lines are allowed to slip, and we evaluate the effect of the Reynolds number and contact angles on the transfer of liquid to the moving plate. In the case of two flat plates, we find that inertia forces the interface to map onto a similarity solution in a manner that shifts the breakup point toward the more wettable surface. Inertia and wettability are thus competing effects, with inertia driving fluid toward the surface with the higher contact angle and wettability driving fluid toward the surface with the lower contact angle. When a cavity is present, contact line pinning on the cavity wall biases breakup toward the cavity as the Reynolds number is increased, leading to improved cavity emptying. As the flat plate is made more wettable, a second pinch-off point can form near that plate, leading to a satellite drop and a reduction in liquid transfer to the plate. Therefore, higher liquid transfer is not always obtained with a more wettable substrate when inertia is present, in contrast to Stokes flow. We also compare our results to those obtained using a model based on the long-wave approximation and find good qualitative agreement, with the long-wave model overpredicting the amount of liquid transferred after breakup. (C) 2011 American Institute of Physics. [doi:10.1063/1.3623427]