Analysis of Two Linearized Problems Modeling Viscous Two-Layer Flows

  • 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. ForTwo 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.show moreshow less

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Metadaten
Author:Konstantinas Pileckas, Jürgen Socolowsky
DOI:https://doi.org/10.1002/1522-2616(200211)245:1<129::AID-MANA129>3.0.CO;2-N
ISSN:1522-2616
ISSN:0025-584X (print)
Parent Title (English):In: Mathematische Nachrichten 245 (2002) 1, 129–166
Document Type:Article
Language:English
Year of Publishing:2002
Date of Publication (online):2014/11/05
Release Date:2014/11/05
First Page:129
Last Page:166
University Bibliography:Hochschulbibliographie
Licence (German):Creative Commons - Namensnennung, Nicht kommerziell, Keine Bearbeitung