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Vector Eigenfunction Expansions for Plane Channel Flows
Author(s) -
Hennington Dan S.,
Schmid Peter J.
Publication year - 1992
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm199287115
Subject(s) - eigenfunction , vorticity , mathematical analysis , hagen–poiseuille equation , normal mode , mathematics , nonlinear system , degeneracy (biology) , boundary value problem , plane (geometry) , physics , flow (mathematics) , classical mechanics , eigenvalues and eigenvectors , geometry , mechanics , vortex , vibration , quantum mechanics , bioinformatics , biology
The full nonlinear initial‐boundary value problem for the evolution of disturbances in plane Poiseuille flow is considered. The problem is formulated in vector form using the normal velocity and normal vorticity as components. The solution is presented as an expansion in linear eigenmodes. These modes consist of both Orr‐Sommerfeld modes and modes of the normal vorticity (Squire) equation. The case of degenerating eigenmodes is also considered and it is shown that the Benney‐Gustavsson normal velocity‐normal vorticity resonance is a special case of a degeneracy between the vector eigenmodes. The solution to the nonlinear problem is presented as an expansion in the linear eigenmodes as well as in modes of the self‐adjoint part of the linear equation. The full nonlinear solution is further reduced to small systems of coupled amplitude equations using the center manifold theorem.