Viscous flow and lubrication theory in slowly-varying channels

Understanding the flow through non-uniform channels remains a major technological goal. Lubrication theory is a well-known approach to analyzing such flows. In a recent study, Tavakol, Froehlicher, Holmes & Stone (2018) considered pressure-driven viscous flows in channels with both smooth and sharp changes using extended lubrication theory to analyze the problems. They proposed an approximation up to O(δ⁴), with δ being the height-to-length ratio, to surpass the limitations of standard lubrication theory. The aim of our study is to extend the lubrication theory model by generating terms of much higher orders and to understand the behavior of the flow in this limit. We examine a Domb-Sykes plot for piecewise affine channels. Discontinuities in channel slope lead to matching regions that transition between lubrication theory away from the locations of discontinuity.