Thin-film flow in helically wound shallow channels of arbitrary cross-sectional shape

Abstract

We consider the steady, gravity-driven flow of a thin film of viscous fluid down a helically wound shallow channel of arbitrary cross-sectional shape with arbitrary torsion and curvature. This extends our previous work [D. J. Arnold et al., “Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015)] on channels of rectangular cross section. The Navier-Stokes equations are expressed in a novel, non-orthogonal coordinate system fitted to the channel bottom. By assuming that the channel depth is small compared to its width and that the fluid depth in the vertical direction is also small compared to its typical horizontal extent, we are able to solve for the velocity components and pressure analytically. Using these results, a differential equation for the free surface shape is obtained, which must in general be solved numerically. Motivated by the aim of understanding flows in static spiral particle separators used in mineral processing, we investigate the effect of cross-sectional shape on the secondary flow in the channel cross section. We show that the competition between gravity and inertia in non-rectangular channels is qualitatively similar to that in rectangular channels, but that the cross-sectional shape has a strong influence on the breakup of the secondary flow into multiple clockwise-rotating cells. This may be triggered by small changes to the channel geometry, such as one or more bumps in the channel bottom that are small relative to the fluid depth. In contrast to the secondary flow which is quite sensitive to small bumps in the channel bottom, the free-surface profile is relatively insensitive to these. The sensitivity of the flow to the channel geometry may have important implications for the design of efficient spiral particle separators.