Geodesic
The Miles Theorem and the first boundary value problem for the Taylor--Goldstein equation
Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 3, pp. 24-38

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We study the problem of the linear stability of stationary plane-parallel shear flows of an inviscid stratified incompressible fluid in the gravity field between two fixed impermeable solid parallel infinite plates with respect to plane perturbations in the Boussinesq approximation and without it. For both cases, we construct some analytical examples of steady plane-parallel shear flows of an ideal density-heterogeneous incompressible fluid and small plane perturbations in the form of normal waves imposed on them, whose asymptotic behavior proves that these perturbations grow in time regardless of whether the well-known result of spectral stability theory (the Miles Theorem) is valid or not.
Keywords: stratified fluid, stationary flow, instability, Taylor–Goldstein equation, Miles Theorem, analytical solution, asymptotic expansion.
Mots-clés : small perturbation 
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     author = {A. A. Gavril'eva and Yu. G. Gubarev and M. P. Lebedev},
     title = {The {Miles} {Theorem} and the first boundary value problem for the {Taylor--Goldstein} equation},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
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     publisher = {mathdoc},
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     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2019_22_3_a2/}
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A. A. Gavril'eva; Yu. G. Gubarev; M. P. Lebedev. The Miles Theorem and the first boundary value problem for the Taylor--Goldstein equation. Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 3, pp. 24-38. http://geodesic.mathdoc.fr/item/SJIM_2019_22_3_a2/