Grigori M. Sisoev

Axisymmetric wave regimes in viscous liquid film flow over a spinning disk

Finite-amplitude capillary waves, which can accompany the axisymmetric flow of a thin viscous film over a rotating disk, are considered. A system of approximate evolution equations for the film thickness and volumetric flow rates in the radial and azimuthal directions is derived, which contains two similarity parameters. In order to inspire confidence in this model, its steady solutions and their linear stability characteristics are compared to those of the full Navier–Stokes equations. Localized equations, which account for the presence of inertial, capillary, centrifugal and Coriolis forces, are obtained via truncation of the approximate system. Periodic solutions of these equations are then determined and found to be similar to those observed experimentally. Our results suggest that the steady quasi-periodic waves with largest amplitude compare well with experimentally observed wave profiles.

Development of dominating waves from small disturbances in falling viscous-liquid films

For wavy liquid films, the principle of selection of the periodic solutions realized experimentally as regular waves is justified. By means of numerical methods, the bifurcations of the families of steady periodic waves and the attractors of the corresponding nonstationary problem are systematically studied. A comparison of the bifurcations and the attractors shows that, when several periodic solutions exist for a given wave number, the solution with the maximum wave amplitude and the maximum phase velocity develops from small initial disturbances (the dominating wave regime). With wave number variation, near the bifurcation points the attractor passes discontinuously from one family to another. This passage is accompanied by the appearance of two-periodic solutions in small neighborhoods of these points. The relations between the calculated parameters of the dominating waves are in a good agreement with all the available experimental data.

The flow of thin liquid films over spinning disks: Hydrodynamics and mass transfer

We study the hydrodynamics and mass transfer associated with gas absorption into a thin liquid film flowing over a spinning disk. We use the thin-layer approximation in conjunction with the Karman–Polhausen method to derive evolution equations for the film thickness and the volumetric flow rates in the radial and azimuthal directions. We also use the integral balance method to derive evolution equations for the thickness of the diffusion boundary layer as well as the concentration of solute at the disk surface. Numerical solutions of these partial differential equations, which govern the hydrodynamics and the associated mass transfer, reveal the formation of large finite-amplitude waves and elucidate their significant effect on the mass-transfer characteristics. We illustrate this dependence quantitatively by examining the effect of system parameters on the time-averaged and spatially averaged Sherwood numbers. The results are assessed by comparison with computations of the parabolized convective diffusio...

Evolution scales for wave regimes in liquid film flow over a spinning disk

We study the spatiotemporal development of a thin viscous film flowing over a spinning disk. A coupled system of evolution equations for the film thickness and volumetric flow rates in the radial and azimuthal directions is derived using the Karman–Polhausen method, assuming a parabolic profile for the film velocity. In the limit of large Eckman number, this system reduces to equations previously used to study the falling film problem. Numerical solutions of the system are obtained starting from initially waveless profiles, which correspond to the Nusselt solution for the case of a spinning disk. Results from these simulations reveal the development of finite-amplitude waves, which, locally, approximate closely to the shape of quasisteady periodic traveling waves. These waves are found to be in good agreement with the predictions of the localized version of the model.

Instability of a two-layer capillary jet

Abstract An axisymmetrical flow of a two-layer capillary jet is studied. The most general formulation of the jet instability problem is considered. The influence of the governing parameters on the amplification factor of the disturbances is investigated taking into account the variation along the jet of the main flow velocity profile. The boundary layer approximation to full Navier-Stokes formulation is applied for calculation of a stationary mean flow. Stability analysis is carried out for various cross sections along the jet on the assumption of a locally parallel main flow. Numerical calculations of the corresponding eigenvalue problem reveal two types of unstable perturbations connected with the presence of the free surface and interface accordingly. The predominant type of instability is defined by the main flow properties. Numerous solutions of the multiparameter stability problem are presented.

Flow stability of a film of viscous liquid on the surface of a rotating disk

The stability of a steady axisymmetric flow of a film is studied using the assumption of local plane parallelity. We present the results of numerical calculation.

Helical waves in a liquid film on a rotating disk

The stability of steady-state axisymmetric flow against non-axisymmetric perturbations is considered.

Flow of a viscous liquid film on the surface of a rotating disk

Results are presented from numerical calculations of steady-state axisymmetric flow of a film of viscous incompressible liquid over the surface of a plane rotating disk.

Stationary spiral waves in film flow over a spinning disk

Stationary spiral waves in liquid film flowing over a spinning disk have been observed in earlier experiments [H. Espig and R. Hoyle, “Waves in a thin liquid layer on a rotating disk,” J. Fluid Mech. 22, 671 (1965); A. F. Charwat et al., “The flow and stability of thin liquid films on a rotating disk,” J. Fluid Mech. 53, 227 (1972); G. Leneweit et al., “Surface instabilities of thin liquid film flow on a rotating disk,” Exp. Fluids 26, 75 (1999)]. In the framework of a mathematical model derived by the integral method, it is shown that the waves develop due to nonaxisymmetric liquid feeding onto the spinning disk, and the wave shapes are approximated by the Archimedean spirals, whose coefficients depend on the Eckman number. The dependence has been confirmed by experimental data from recorded videos.

Towards the Problem of Instability of a Falling Viscous-Liquid Film with a Dissolved Surfactant

The linear stability of a viscous-liquid film with a dissolved surfactant, flowing down a vertical surface, is investigated. The model employed takes into account the surfactant diffusion, adsorption-desorption, and evaporation. Three kinds of unstable disturbances are calculated and the limits of their existence in the space of the governing parameters determining the mathematical model of the flow are studied.