DOI: 10.17586/2226-1494-2018-18-3-529-534


STABILITY OF VISCOUS FILM ON SURFACE OF SLIGHTLY INCLINED ROTATING VERTICAL CYLINDER

I. F. Melikhov


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For citation: Melikhov I.F. Stability of viscous film on surface of slightly inclined rotating vertical cylinder. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2018, vol. 18, no. 3, pp. 529–534 (in Russian). doi: 10.17586/2226-1494-2018-18-3-529-534

Abstract
 Subject of Research.The paper deals with the problem of viscous film evolution on a surface of rotating cylinder. The objective of the work is stability analysis of such system in the cases when the cylinder axis is nearly vertical.Stability of uniform film on a strictly vertical cylinder is analyzed, and the system dynamics  under small time-periodic deviations of cylinder axis is studied.Methods. Initial Navier-Stokes equations, which describe dynamics of viscous fluids, are considered with introduced small parameter – the ratio of fluid layer thickness to cylinder radius. This fact allows applying of asymptotic methods and helps to derive a simplified equation on the film thickness. The obtained equation is a non-linear partial-derivative equation and  cannot be solved analytically in general form. Perturbation theory is applied for the subsequent analysis. Linear stability analysis for uniform film on the vertical cylinder surface is carried out. Response to time-periodic perturbation of the cylinder tilt is studied. Main Results. In the case of strictly vertical cylinder, capillary forces imply instability of axisymmetric perturbations with a wavelength exceeding critical value. The obtained critical wavelength value is proportional to the cylinder radius. It is shown that when a cylinder axis oscillates around the vertical, non-uniform fluid layer appears with diametrically opposite thickening and thinning. In the case when the axis oscillation frequency is equal to the rotation frequency, a resonance appears.Practical Relevance. The obtained results might be used for better understanding and further improvement of existing manufacturing processes.

Keywords: viscous films, lubrication theory, fluid on cylinder surface, flow stability

Acknowledgements. The work is partially supported by the Government of the Russian Federation (grant 074-U01).

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