Flow on the surface of sloped rotating cylinder
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Flow on the surface of sloped rotating cylinder Ivan F. Melikhov, Sergey A. Chivilikhin and Igor Y. Popov
Abstract. Viscous flow of fluid film on outer surface of a sloped rotating cylinder in gravitational field is studied. Thickness of the fluid layer is assumed to be small compared to the cylinder radius, which allows asymptotic analysis. Governing equation for the thickness dynamics is derived. The equation accounts for viscous effects, gravity, centrifugal and capillary forces. A criterion for existence of steady flow on the sloped cylinder is obtained. Linear stability of stationary solution for the vertical cylinder is given. Film thickness response to oscillations of the cylinder axis around vertical line is studied. Numerical model is implemented for the case of arbitrary slope angle. Mathematics Subject Classification. 76D08, 35Q35. Keywords. Thin viscous layer, Asymptotics.
1. Introduction Understanding of dynamics of viscous films flowing over a substrate is essential for many natural and manufacturing processes. Fluid properties and process parameters may affect not only coating characteristics, but also its stability. In this paper, we consider the flow of a viscous film over outer surface of a solid rotating cylinder. We describe film dynamics due to gravity, centrifugal and capillary forces. The cylinder can be arbitrarily inclined, so vertical and horizontal orientations are just particular cases of the system. Such kind of systems may occurs in different areas, for example in cable coating, glass manufacturing or chocolate production. Viscous film flow over a substrate is a well-known phenomenon. There are general asymptotic approaches which allow to simplify Navier–Stokes equations for flow with free surface and get a nonlinear equation on film thickness. These approaches assume that the film thickness is small compared to its lateral dimensions and usually neglect inertia effects. Paper [1] gives an overview of thin film dynamics with a lot of examples of different physical phenomena driving evolution of the film (gravity, capillary effects, temperature gradients, evaporation, molecular force and others). More recent review [2] extends it with multilayered flows. Although flow over a plane attracts the most interest, coating of curved surfaces was studied as well. Work [3] considers flow on moving curved surface. Paper [4] suggests a general description of flow over an arbitrary curved substrate, considering flow over torus and cylinder as particular examples. Later, this approach was extended to account for inertial effects [5]. Noticeably, works [4,5] use mathematical framework of center manifold (see [6,7] for more detailed description) and estimate the model errors. As for the flow over a rotating cylinder, Moffatt was the first to study the problem [8] for the case of horizontal cylinder. Almost at the same time, Pukhnachev [9] proved the existence and uniqueness of the stationary solution for sufficiently small Reynolds numbers. After these works, different effects dr
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