Low-order models for thick-film flows
Alexander Wray (University of Strathclyde, Glasgow)
Thursday 14th February, 2019 14:00-15:00 Maths 311B
A common approach to understanding a fluid-dynamical system is that of a low-order model: a mathematically rigorous simplification of the governing equations (typically an asymptotic reduction of the Navier-Stokes equations) to a system of PDEs that are more analytically tractable. This usually relies on the identification of some disparity in length scales within the problem, such as the “aspect ratio” of a wave’s length to its height, to allow analytic simplification of the governing equations.
In non-planar situations, the curvature of the underlying substrate complicates the situation by incorporation of two additional length scales: the radii of curvature. In such situations, modelling has typically relied on some potentially prohibitive symmetry assumption, or the assumption that the substrate shape is slowly varying. We have shown in recent years that a combination of a novel set of scalings, together with the application of the method of weighted residuals, can relax these assumptions to accurately model situations where the fluid thickness is in excess of the radius of curvature.
We discuss briefly the methodology used, before examining its application to a variety of different physical situations. By comparison with Direct Numerical Simulations (DNS), we show that good accuracy can be maintained even for thick films at moderate levels of inertia. We show that the rapid speed of computation can allow for the full exploration of parameter space in seconds or minutes, rather than the weeks or months as required for DNS. We also show that, at least for the Moffatt Problem discussed in detail herein, the resultant equations are analytically tractable, allowing for a detailed understanding of the phenomena at play.