Viscous flows in a half space caused by tangential vibrations on its boundary

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

The paper is devoted to the studies of viscous flows caused by a vibrating boundary. The fluid domain is a half-space, its boundary is a nondeformable plane that exhibits purely tangential vibrations. Such a simple geometrical setting allows us to study general boundary velocity fields and to obtain general results. From a practical viewpoint, such boundary conditions may be seen as the tangential vibrations of the material points of a stretchable plane membrane. In contrast to the classical boundary layer theory, we aim to build a global solution. To achieve this goal we employ the Vishik-Lyusternik approach, combined with two-timing and averaging methods. Our main result is: we obtain a uniformly valid in the whole fluid domain approximation to the global solutions. This solution corresponds to general boundary conditions and to three different settings of the main small parameter. Our solution always include the inner part and outer part that both contain oscillating and non-oscillating components. It is shown that the nonoscillating outer part of the solution is governed either by the full Navier-Stokes equations or the Stokes equations (both with the unit viscosity) and can be interpreted as a steady or unsteady streaming. In contrast to the existing theories of a steady streaming, our solutions do not contain any secular (infinitely growing with the inner normal coordinate) terms. The examples of the spatially periodic vibrations of the boundary and the angular torsional vibrations of an infinite rigid disc are considered. These examples are still brief and illustrative, while the core of the paper is devoted to the adaptation of the Vishik-Lyusternik method to the development of the general theory of vibrational boundary layers.

Original languageEnglish
Pages (from-to)337-367
Number of pages31
JournalStudies in Applied Mathematics
Volume121
Issue number4
DOIs
Publication statusPublished - Nov 2008

Fingerprint

Viscous flow
Viscous Flow
Half-space
Boundary layers
Vibration
Boundary conditions
Fluids
Streaming
Global Solution
Navier Stokes equations
Boundary Layer
Viscosity
Torsional Vibration
Membranes
Fluid
General Boundary Conditions
Averaging Method
Stokes Equations
Small Parameter
Velocity Field

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Viscous flows in a half space caused by tangential vibrations on its boundary. / Vladimirov, V. A.

In: Studies in Applied Mathematics, Vol. 121, No. 4, 11.2008, p. 337-367.

Research output: Contribution to journalArticle

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