The steady behavior of the interface between two immiscible liquid layers flowing in a channel where one wall is flat and the other exhibits isolated or periodic step-changes in surface topography is considered in this work. A one-dimensional model based on the lubrication approximation is used to describe the interfacial shape and explore its behavior over a wide range of problem parameters. The density and thickness ratios, as well as the periodicity of the topography, strongly influence interfacial shape, whereas the viscosity ratio has a relatively minor effect. In some cases, interfacial features that are present in the flow of single liquid layers over surfaces with topography can be completely suppressed by large pressure gradients that arise due to the bounded geometry. A scaling analysis is applied to explain variations in the capillary length scale. The results presented in this work are potentially relevant to a range of applications including microfluidics, oil recovery, lubricated piping, and lithographic printing.
Bibliographical noteFunding Information:
This work was supported through the Industrial Partnership for Research in Interfacial and Materials Engineering of the University of Minnesota. We are especially grateful for a gift from the Dai Nippon Printing Company. We also thank 3M for their support of a graduate fellowship to R.D.L.