Moving contact lines at non-zero capillary number

Abstract

Consider the unsteady motion of a fluid–fluid interface over and attached to a solid surface with one-dimensional periodic roughness in the limit of small capillary number [Copf ]. We show that the macroscopic behaviour of the interface can be described, to leading order in [Copf ], in terms of well-defined continuum quantities, even though the complicated fluid motion in the neighbourhood of the contact line cannot. The key is that contact-angle hysteresis makes it possible to isolate the viscous stress singularity at the contact line, since for a sufficiently slowly moving fluid—fluid interface all the movement of the contact line occurs in a time much shorter than the macroscopic timescale. The effective ‘slip length’ for the macroscopic description is shown to be velocity dependent and equal to a[Copf ]⁻¹, where a is the wavelength of the roughness on the solid surface. Finally, we consider surfaces with two-dimensional random roughness, and argue that they too would exhibit velocity dependent ‘slip lengths’, though the velocity dependence would be stronger in this case. These results combined with earlier work (Jansons 1985) explain why observed ‘slip lengths’ can be much larger than roughness dimensions, and why the degree of ‘stick-slip’ decreases with increasing speed.