Tsvelodub O.Yu.

Institute of Thermophysics, Siberian Branch of Russian Academy of Sciences,

Prospect Lavrentyev 1, Novosibirsk 630090, Russia

Abstract

As well known, in the case of small flow rates the investigation of nonlinear perturbations on a thin layer of a viscous liquid, vertically flowing by gravity, can be reduced to the consideration of a single equation for the film thickness. One from the cases when this redaction can be done shall be considered in this report. The film is flowing on a vertical cylinder. The next assumptions are used : the Reynolds numbers are small, the film thickness is much less than the wave length of perturbation, cylinder has enough large radius. For this case an equation involving our interest was obtained in /1/ .

In this report two special classes of waves regimes with the use of this model equation are considered: a) the spiral waves and b) the periodic space waves that travelling along cylinder and are symmetric with respect to angle coordinate.

The problem to obtain the spiral waves are solved by special transformations of the equation to well-known Kuramoto-Sivashinsky one.

The space periodic solutions branch from the trivial one when the wave numbers of small perturbations are neutral. The cylinder film have a discrete set of neutral wave numbers. Appropriate travelling solutions of small but final amplitude exist in vicinities of these neutral points. In the general cases these solutions can be easily constructed. The exception is presented by vicinities of resonant points. In difference from general case. In this case it is necessary to take at once the sum of two resonant harmonics. From a condition of solvability of equations the second order we come to a system of two equations for determination of these harmonics. From its analysis follows that a picture of branches in a vicinity of resonant wave points is essentially richer, than in general case. In particular, the steady-state periodic solution with wave numbers in the neighbourhood of the neutral wave numbers were constructed and their stability was studied.

If the wave numbers are far away from the neutral wave numbers, here using the analytical results several families of spatially periodic steady-state travelling solutions are numerically constructed.

As a result are obtained the nonlinear space solutions for equations describing the evolution of spatial long-wave perturbations on the surfaces of liquid film flowing down a vertical cylinder.

Reference

1. Shlang T, Sivashinsky G.I. J. Phys. Paris, v. 43, 459 (1982).