Hydrodynamic Interaction of Curved Bodies Allowing Slip on Their Surfaces

The hydrodynamics of liquid confined between two approaching curved solids is revisited. Attention is focused on investigating the role of the geometry of curved bodies allowing slip on their surfaces. We derive the equations for pressure and hydrodynamic resistance force to approach of the bodies. We show that they can both be presented as a product of the Reynolds expressions (for pressure and resistance to motion) and corrections for slippage that depend only on the relationships between the slip lengths and the gap. These corrections are shown to be the same for any configuration geometry. The Reynolds parts of the expressions present, in turn, the products of two factors. One of them reflects external conditions, while another one depends only on the curvature of the two surfaces, the relative orientation of their principal radii of curvature, and on the separation. The crucial observation is that this dependence is expressed only through the invariants of the second-order surface. As an application some cases of special colloidal interest are considered. Our analysis provides a method for simple recalculation of pressure or hydrodynamic force from one configuration geometry, or state of the liquid/solid interface, to another.