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A free boundary-value problem for a nonisothermal flow of two heavy viscous incompressible capillary liquids descibing a slot coating process is mathematically studied. The problem is stationary and two-dimensional and the Boussinesq approximation for the Navier-Stokes equations is used. The existence and uniqueness of a weak solution is proved in weighted Holder spaces for small data.
Analysis of a nonisothermal viscous flow problem with free boundaries and a dynamic contact line
(1995)
Two problems that appear in the linearization of certain free boundary value problems of the hydrodynamics of two viscous fluids are studied in the strip-like domain
Π = {x = (x1, x2) ∈ ℝ2 : x1 ∈ ℝ1, (0 < x2 < h*) ∨ (h* < x2 < 1)}.
The first problem arises in the linearization of a two-layer flow down a geometrically perturbed inclined plane. The second one appears after the linearization of a two-layer flow in a geometrically perturbed inclined channel with one moving (smooth) wall. For this purpose the unknown flow domain was mapped onto the double strip Π.
The arising linear elliptic problems contain additional unknown functions in the boundary conditions. The paper is devoted to the investigation of these boundary problems by studying the asymptotics of the eigenvalues of corresponding operator pencils. It can be proved that the boundary value problems are uniquely solvable in weighted Sobolev spaces with exponential weight. The study of the full (nonlinear) free boundary value problems will be the topic of a forthcoming paper.
Two stationary plane free boundary value problems for the Navier-Stokes equations are studied. The first problem models the viscous two-fluid flow down a perturbed or slightly distorted inclined plane. The second one describes the viscous two-fluid flow in a perturbed or slightly distorted channel. For sufficiently small data and under certain conditions on parameters the solvability and uniqueness results are proved for both problems. The asymptotic behaviour of the solutions is investigated. For the second problem an example of nonuniqueness is constructed. Computational results of flow problems that are very close to the above problems are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)