The drag on a flattened bubble moving across a plane substrate

Abstract

AbstractThe equilibrium shape of an axisymmetric liquid drop or gas bubble in an immiscible supporting liquid held under gravity against a horizontal plane rigid surface is derived. A thin film of supporting liquid remains between the drop/bubble surface and the rigid substrate at equilibrium, of thickness ${h}_{0} $ determined by the balance of the disjoining pressure $\Pi (h)$ between the drop and the substrate and the internal Laplace pressure of the drop/bubble. The interface is macroscopically flat around the drop axis out to a radius $a$ but matches smoothly into the outer shape of radius $A$ through a boundary layer region of width $a{ H}_{0}^{1/ 2} $ where ${H}_{0} = 2{h}_{0} A/ {a}^{2} $ is a small parameter. The outer drop shape is determined by a balance of buoyancy forces and local Laplace pressure and is roughly spherical if $A/ \lambda \ll 1$, where $\lambda = \sqrt{\gamma / \mrm{\Delta} \rho g} $ is the capillary length in the interface with a logarithmic correction due to the action of the disjoining pressure across the flattened region. With the shape determined, we calculate the drag force on this flattened bubble to lowest order in the velocity as it moves across the rigid substrate using a lubrication approximation valid to terms of $O( \mathop{ ({h}_{0} / a)}\nolimits ^{2} )$ as an integral over the flattened bubble surface of the hydrodynamic pressure. The lubrication theory of itself is not sufficient to determine the drag due to the divergence of that integral if the outer flow field properties are neglected. By using the known exact result for the drag force on an undistorted bubble, the drag on the flattened bubble can be computed as an integral over the lubrication region alone. We derive the drag as a series expansion in the small parameter ${H}_{0} $ by means of a fairly intricate boundary layer analysis. The logarithmic divergence of the translational drag with film thickness ${h}_{0} $ for the undistorted bubble is replaced by the stronger divergence ${ h}_{0}^{\ensuremath{-} 1/ 2} $ to leading order for the flattened bubble case. We present explicit numerical results for the first few terms in the expansion for the case of an exponentially repulsive disjoining pressure, and analytic expressions for these terms in the limit of very short-range disjoining pressure forces. The results of this calculation are compared with recent work (Hodges, Jensen & Rallison, J. Fluid Mech., vol. 512, 2004, p. 95) where disjoining pressure is neglected and hydrodynamic pressure is balanced against buoyancy and Laplace forces. The limits of validity of this linear drag theory are also presented.