Effective slip due to Stokes flow through channels with patterned stick-slip walls

Abstract

This thesis comprises studies of two problems on Stokes flows through (i) a plane microchannel, and (ii) a microscale circular capillary tube, of which the walls are superhydrophobic surfaces featuring micro- or nano-patterns. In the first problem, an analytical study on longitudinal and transverse flows through a plane microchannel, which is made up of a periodic array of ribs and grooves on the upper and lower walls, is performed. This study focuses not only on the Cassie state, but also on the state whereby the liquid is allowed to penetrate the grooves between the ribs. These grooves are filled with inviscid gas and this periodic array gives rise to heterogeneous boundary conditions for the flow. Partial-slip and shear-free conditions are applied on the solid–liquid interface and the liquid–gas interface, respectively. Using the methods of eigenfunction expansions and domain decomposition, semi-analytical models are developed for four different settings. Two of them correspond to longitudinal flow with in-phase and 180-out-of-phase alignments of ribs between the upper and lower walls. The other two are for transverse flow with the in-phase and out-of-phase wall alignments. These models enable the effective slip lengths, normalized by half the pitch of the pattern, to be deduced as functions of the channel height, the microscopic or intrinsic partial slip length, the depth of liquid penetration, and the width of the grooves or the shear-free area fraction of the liquid-gas interface. Numerical calculations are performed to examine effects of these parameters on the effective slip length. The effect of the phase of alignment of ribs is appreciable when the surface is in the Cassie state and the channel height is sufficiently small. In-phase alignment yields a larger effective slip length in longitudinal flow. In sharp contrast, out-of-phase alignment is preferable in transverse flow. In the case involving penetration, a larger liquid penetration can give rise to a larger slip length in a thin channel.

In the second problem, an analytical study on flow through a microscale circular tube, of which the wall is patterned with a periodic array of spots or holes, is performed. Void region is filled with inviscid gas and patterns of circular and square shapes are considered. For simplicity, liquid penetration into the cavities is not considered, and hence the focus is on the Cassie state. No-slip and shear-free conditions are applied on the solid–liquid interface and the liquid–gas interface, respectively. By the methods of eigenfunction expansions and point collocation, a semi-analytical model is established. The effective slip length, normalized by the tube radius, is found as a function of the pitch of the pattern in the streamwise direction, the number of periodic units in the circumferential direction, and the solid fraction. Comparisons with some proposed scaling laws, varying pitch, solid fraction and tube size are performed. Large slip length is produced by arranging small circular no-slip spots with large separation in the streamwise direction. In some situations, spots and holes can be replaced by parallel stripes to increase the slip length.