Root
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Conference Proceedings
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6th EUROMECH Nonlinear Dynamics Conference (ENOC 2008)
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Viscous Flows in a Half Space caused by
Tangential Vibrations on its Boundary.
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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 non-deformable 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 plane stretchable

membrane. In contrast to the classical boundary layer theory we aim to build a global solution. In order 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 contain both

oscillating and non-oscillating components. It is shown that the

non-oscillating `outer' part of the solution is governed by the

either 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 adapting of the Vishik-Lyusternik method to

the development of the general theory of vibrational boundary

layers.

vibrating boundary. The fluid domain is a half space, its boundary

is a non-deformable 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 plane stretchable

membrane. In contrast to the classical boundary layer theory we aim to build a global solution. In order 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 contain both

oscillating and non-oscillating components. It is shown that the

non-oscillating `outer' part of the solution is governed by the

either 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 adapting of the Vishik-Lyusternik method to

the development of the general theory of vibrational boundary

layers.