An Accurate and Comprehensive Model of Thin Fluid Flows with Inertia on Curved Substrates

Consider the three dimensional flow of a viscous Newtonian fluidupon a curved two dimensional substrate when the fluid film is thinas occurs in many draining, coating and biological flows. We derivea comprehensive model of the dynamics of the film, the model beingexpressed in terms of the film thicknen and the average lateral velocity Pu. Based upon centre manifold theory, we are assured thatthe model accurately and systematically includes the effects of thecurvature of substrate, gravitational body force, fluid inertia and dissipation.The model resolves wave-like phenomena in the dynamics ofviscous fluid flows over arbitrarily curved substrates such as cylinders,tubes and spheres. We briefly illustrate its use in simulating dropformation on cylindrical fibres, wave transitions, three dimensionalinstabilities, Faraday waves, viscous hydraulic jumps, flow vortices ina compound channel and flow down and up a step. These modelsare the most complete models for thin film flow of a Newtonian fluid;many other thin film models can be obtained by different restrictionsand truncations of the model derived here.