On the stability of viscous free-surface flow supported by a rotating cylinder

Using an adaptive finite-element (FE) scheme developed recently by the authors, we shed new light on the long-standing fundamental problem of the unsteady free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. For supportable loads, we observe that the steady-state is more readily attained for near-maximal fluid loads on the cylinder than for significantly sub-maximal loads. For the latter, we investigate large-time dynamics by means of a finite-difference approximation to the thin-film equations, which is also used to validate the adaptive FE simulations (applied to the full Stokes equations) for these significantly sub-maximal loads. Conversely, by comparing results of the two methods, we assess the validity of the thin-film approximation as either the load is increased or the rotation rate of the cylinder is decreased. Results are presented on the independent effects of gravity, surface tension and initial film thickness on the decay to steady-state. Finally, new numerical simulations of load shedding are presented

Differential constraints and exact solutions of nonlinear diffusion equations

The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries

New classes of exact solutions of three-dimensional Navier-Stokes equations

New classes of exact solutions of the three-dimensional unsteady Navier-Stokes equations containing arbitrary functions and parameters are described. Various periodic and other solutions, which are expressed through elementary functions are obtained. The general physical interpretation and classification of solutions is given.Comment: 11 page

Mrštík, Vilém

(1863 - 1912), Schriftstelle

The Problem of Filling a Spherical Cavity in an Aqueous Solution of Polymers

The problem of filling a spherical cavity in a liquid has attracted the attention of many authors. The study of bubble behavior in liquid allows to estimate the consequences of cavitation processes, which can lead to the intensive destruction of the material surface. Regarding this connection, it becomes necessary to study the influence of impurities, including polymeric additives on the strengthening or suppression of cavitation. In this paper, this problem is considered in three models of a relaxing fluid. It is shown that for all models, the cavity filling time is finite if the surface tension is not equal to zero. This result was previously established for the cases of ideal and viscous fluids. However, the relaxation factor can significantly change the flow pattern by slowing down the filling process and lowering the level of energy accumulation during the bubble collapse

Analysis of the Models of Motion of Aqueous Solutions of Polymers on the Basis of Their Exact Solutions

The qualitative properties of solutions of a hereditary model of motion of aqueous solutions of polymers, its modification in the limiting case of short relaxation times, and a similar second grade fluid model are studied. Unsteady shear flows are considered. In the first case, their properties are similar to those of motion of a usual viscous fluid. Other models can include weak discontinuities, which are retained in the course of fluid motion. Exact solutions are found by using the group analysis of the examined systems of equations. These solutions describe the fluid motion in a gap between coaxial rotating cylinders, the stagnation point flow, and the motion in a half-space induced by plane rotation (analog of the Karman vortex). The problem of motion of an aqueous solution of a polymer in a cylindrical tube under the action of a streamwise pressure gradient is considered. In this case, a flow with straight-line trajectories is possible (analog of the Hagen-Poiseuille flow). In contrast to the latter, however, the pressure in the flow considered here depends on all three spatial variables

Exact Solutions of Boundary Layer Equations in Polymer Solutions

The paper presents new exact solutions of equations derived earlier. Three of them describe unsteady motions of a polymer solution near the stagnation point. A class of partially invariant solutions with a wide functional arbitrariness is found. An invariant solution of the stationary problem in which the solid boundary is a logarithmic curve is constructed