Effective boundary conditions at a rough wall: a high-order homogenization approach
Effective boundary conditions, correct to third order in a small parameter ϵ , are derived by homogenization theory for the motion of an incompressible fluid over a rough wall with periodic micro-indentations. The length scale of the indentations is l , and ϵ = l / L ≪ 1 , with L a characteristic le...
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Published in | Meccanica (Milan) Vol. 55; no. 9; pp. 1781 - 1800 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Dordrecht
Springer Netherlands
01.09.2020
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | Effective boundary conditions, correct to third order in a small parameter
ϵ
, are derived by homogenization theory for the motion of an incompressible fluid over a rough wall with periodic micro-indentations. The length scale of the indentations is
l
, and
ϵ
=
l
/
L
≪
1
, with
L
a characteristic length of the macroscopic problem. A multiple scale expansion of the variables allows to recover, at leading order, the usual Navier slip condition. At next order the slip velocity includes a term arising from the streamwise pressure gradient; furthermore, a transpiration velocity
O
(
ϵ
2
)
appears at the fictitious wall where the effective boundary conditions are enforced. Additional terms appear at third order in both wall-tangent and wall-normal components of the velocity. The application of the effective conditions to a macroscopic problem is carried out for the Hiemenz stagnation point flow over a rough wall, highlighting the differences among the exact results and those obtained using conditions of different asymptotic orders. |
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ISSN: | 0025-6455 1572-9648 |
DOI: | 10.1007/s11012-020-01205-2 |