Projects per year
Abstract
The paper is devoted to the studies of viscous flows caused by a vibrating boundary. The fluid domain is a halfspace, 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 VishikLyusternik approach, combined with twotiming 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 nonoscillating components. It is shown that the nonoscillating outer part of the solution is governed either by the full NavierStokes 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 VishikLyusternik method to the development of the general theory of vibrational boundary layers.
Original language  English 

Pages (fromto)  337367 
Number of pages  31 
Journal  Studies in Applied Mathematics 
Volume  121 
Issue number  4 
DOIs  
Publication status  Published  Nov 2008 
Keywords
 SURFACE
 WALLS
Projects
 2 Finished


Stability of Inviscid flows through a given domain
1/03/06 → 29/02/08
Project: Research project (funded) › Research