The Miles Theorem and New Particular Solutions to the Taylor--Goldstein Equation
The linear stability problem of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid in the gravity field between two immovable impermeable solid planes is studied in and without the Boussinesq approximation. Using the Lyapunov direct method, it is prove...
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Kazan Federal University
2016-06-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | http://kpfu.ru/portal/docs/F1759423227/158_2_phys_mat_1.pdf |
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| author | A.A. Gavrilieva Yu.G. Gubarev M.P. Lebedev |
| author_facet | A.A. Gavrilieva Yu.G. Gubarev M.P. Lebedev |
| author_sort | A.A. Gavrilieva |
| collection | DOAJ |
| description | The linear stability problem of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid in the gravity field between two immovable impermeable solid planes is studied in and without the Boussinesq approximation. Using the Lyapunov direct method, it is proved that these flows are absolutely unstable in the theoretical sense with respect to small plane perturbations. The applicability domain boundaries of the known necessary condition of the linear instability of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid in the gravity field is strictly determined in the Boussinesq approximation and without it (Miles theorem). It is found that this theorem is, by its character, both sufficient and necessary statement with respect to some uncompleted unclosed subclasses of studied perturbations. The analytical examples are constructed with the view of illustrations of the mentioned stationary flows and small plane perturbations imposed on these flows. These perturbations are not under the Miles theorem and they increase with time irrespective of the validity of the theoretical linear stability criterion in and without the Boussinesq approximation. Therefore, the results derived earlier by other authors with the help of the method of integral relations for the linear stability problems of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid demand strict description for the studied partial classes of small plane perturbations as otherwise they can be mistaken. |
| format | Article |
| id | doaj-art-bad4e73708fb4be08338f9ddfa9c9cfc |
| institution | DOAJ |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2016-06-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-bad4e73708fb4be08338f9ddfa9c9cfc2025-08-20T02:56:39ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982016-06-011582156171The Miles Theorem and New Particular Solutions to the Taylor--Goldstein EquationA.A. Gavrilieva0Yu.G. Gubarev1M.P. Lebedev2Larionov Institute of Physical and Technical Problems of the North, Siberian Branch, Russian Academy of Sciences, Yakutsk, 677891 RussiaLavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia; Novosibirsk National Research State University, Novosibirsk, 630090 RussiaLarionov Institute of Physical and Technical Problems of the North, Siberian Branch, Russian Academy of Sciences, Yakutsk, 677891 Russia; Ammosov North-Eastern Federal University, Yakutsk, 677000 RussiaThe linear stability problem of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid in the gravity field between two immovable impermeable solid planes is studied in and without the Boussinesq approximation. Using the Lyapunov direct method, it is proved that these flows are absolutely unstable in the theoretical sense with respect to small plane perturbations. The applicability domain boundaries of the known necessary condition of the linear instability of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid in the gravity field is strictly determined in the Boussinesq approximation and without it (Miles theorem). It is found that this theorem is, by its character, both sufficient and necessary statement with respect to some uncompleted unclosed subclasses of studied perturbations. The analytical examples are constructed with the view of illustrations of the mentioned stationary flows and small plane perturbations imposed on these flows. These perturbations are not under the Miles theorem and they increase with time irrespective of the validity of the theoretical linear stability criterion in and without the Boussinesq approximation. Therefore, the results derived earlier by other authors with the help of the method of integral relations for the linear stability problems of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid demand strict description for the studied partial classes of small plane perturbations as otherwise they can be mistaken.http://kpfu.ru/portal/docs/F1759423227/158_2_phys_mat_1.pdfideal stratified fluidBoussinesq approximationstationary plane-parallel shear flowsstabilityLyapunov direct methodinstabilitysmall plane perturbationsa priori estimateMiles theoremanalytical solutionsBessel functionsWhittaker functions |
| spellingShingle | A.A. Gavrilieva Yu.G. Gubarev M.P. Lebedev The Miles Theorem and New Particular Solutions to the Taylor--Goldstein Equation Учёные записки Казанского университета: Серия Физико-математические науки ideal stratified fluid Boussinesq approximation stationary plane-parallel shear flows stability Lyapunov direct method instability small plane perturbations a priori estimate Miles theorem analytical solutions Bessel functions Whittaker functions |
| title | The Miles Theorem and New Particular Solutions to the Taylor--Goldstein Equation |
| title_full | The Miles Theorem and New Particular Solutions to the Taylor--Goldstein Equation |
| title_fullStr | The Miles Theorem and New Particular Solutions to the Taylor--Goldstein Equation |
| title_full_unstemmed | The Miles Theorem and New Particular Solutions to the Taylor--Goldstein Equation |
| title_short | The Miles Theorem and New Particular Solutions to the Taylor--Goldstein Equation |
| title_sort | miles theorem and new particular solutions to the taylor goldstein equation |
| topic | ideal stratified fluid Boussinesq approximation stationary plane-parallel shear flows stability Lyapunov direct method instability small plane perturbations a priori estimate Miles theorem analytical solutions Bessel functions Whittaker functions |
| url | http://kpfu.ru/portal/docs/F1759423227/158_2_phys_mat_1.pdf |
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