The method of integral relations is used to derive a nonlinear two-wave equation for long waves on the surface of vertical falling liquid films. This equation is valid within a range of moderate Reynolds numbers and can be reduced in some cases to other well-known equations.The theoretical results for the fastest growing waves are compared with the experimental results concerning velocities, wave numbers, and growth rates of the waves in the inception region. The validity of the theoretical assumptions is also confirmed by direct measurements of instantaneous velocity profiles in a wave liquid film.The results of the experimental investigation concerning nonlinear stationary waves and the evolution of initial solitary disturbances are presented. Vertical falling liquid films are extensively used in interphase heat and mass transfer processes in chemical technology and energetics. It is well known that on the surface of these films there practically always exist waves that can influence the transfer processes. In the case of short test sections and moderate liquid flow rates these waves are ordered and two-dimensional. To estimate the transfer coefficients for the wave flows of films one should be able to calculate the nonlinear wave regimes, because it is the nonlinear waves that are dominating in wave formation on the surface of the above films.Most known publications on nonlinear waves are devoted to calculations of either nonstationary or stationary waves at low and moderate flow rates, respectively. However, there is no general-purpose nonlinear, nonstationary equation which would generalize the available approaches and be valid in a wide range of conditions. The aim of the present study has been to derive a general-purpose model equation for nonlinear, nonstationary waves on the film surface and to substantiate the validity of this approach on the basis of the existing experimental data and theoretical results. CONCLUSIONS AND SIGNIFICANCEA universal model equation to describe nonlinear, nonstationary waves on the surface of liquid films in the range of Re numbers 1 I Re i: tP2 (6 is the long-wave process parameter) is derived by the method of integral relations by using selfsimilar velocity profiles. The equation has a two-wave structure, which implies that at low Re(-I), kinematic waves can be observed, to which the energy is transferred by means of a higher-order wave mechanism. At Re -1/c2 >> 1, the higherorder waves grow at the expense of the kinematic waves. In the limiting cases of low and high Re numbers and in the particular case of stationary waves, the above two-wave equation can be reduced to the well-known equations of Gjevik (1970), Nakoryakov and Shreiber (1973), and Shkadov (1967).In terms of the derived equation, a linear analysis of the stability has been carried out. Analytical expressions to describe neutral disturbances, the fastest growing waves, and capillary ripples observed in front of the large solitary waves have been obtained.An experimental setup has been developed to measure the...

Helical vortices in swirl flow are studied theoretically and experimentally.A theory of helical vortices has been developed. It includes the following results: an analytical solution describing an elementary helical vortex structure – an infinitely thin filament; a solution for axisymmetrical vortices accounting for the helical shape of vortex lines and different laws of vorticity distribution; a formula for calculation of the self-induced velocity of helical vortex rotation (precession) in a cylindrical tube; an explanation of the zone with reverse flow (recirculation zone) arising in swirl flows; and the classification of vortex structures.The experimental study of helical vortices was carried out in a vertical hydrodynamical vortex chamber with a tangential supply of liquid through turning nozzles. Various vortex structures were formed owing to changing boundary conditions on the bottom and at the exit section of the chamber. The hypothesis of helical symmetry is confirmed for various types of swirl flow. The stationary helical vortex structures are described (most of them for the first time) the features of which agree with the results and predictions of the theoretical model developed. They are the following: a rectilinear vortex; a composite columnar vortex; helical vortices screwed on the right or on the left; a vortex with changing helical symmetry; a double helix – two entangled vortex filaments of the same sign.

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