Creeping films with vortices over strongly undulated bottoms
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We study the influence of an undulated bottom profile on steady two–dimensional gravity driven film flows of a Newtonian fluid. Traditional approaches towards this topic are based on lubrication approximation, on special perturbation methods or on numerics. However, lubrication approximation and perturbation methods deliver acceptable results only within their range of validity. Especially if the bottom is strongly undulated, conventional analytical methods fail. Neither can the classical separation solution of the biharmonic equation in terms of an infinite series be applied because of massive convergence problems if the waviness exceeds a limit. In this paper we present an analytical method based on a representation of the solution of Stokes equations in terms of holomorphic functions. Applying the complex function theory, convergence problems are avoided and the problem is reduced to solving ordinary differential equations and integral equations at the boundaries only. Our calculations show the creation, formation and evolution of vortices if waviness and film thickness exceed critical values. A detailed parameter study on size and strength of the vortices is shown. Moreover, we present a quantitative study on the effect of the vortices on the flow rate. Our calculations show very good agreement with experimental results.
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