Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity
A class of nonlinear variational problems describing incompressible fluids and solids by stationary Stokes equations given in a planar domain with a crack (infinitely thin flat plate in fluids) is considered. Based on the Fourier asymptotic analysis, general analytical solutions are obtained in pola...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Taylor & Francis Group
2022-12-01
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| Series: | Applied Mathematics in Science and Engineering |
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| Online Access: | http://dx.doi.org/10.1080/27690911.2022.2084542 |
| _version_ | 1797641024478117888 |
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| author | Victor A. Kovtunenko Kohji Ohtsuka |
| author_facet | Victor A. Kovtunenko Kohji Ohtsuka |
| author_sort | Victor A. Kovtunenko |
| collection | DOAJ |
| description | A class of nonlinear variational problems describing incompressible fluids and solids by stationary Stokes equations given in a planar domain with a crack (infinitely thin flat plate in fluids) is considered. Based on the Fourier asymptotic analysis, general analytical solutions are obtained in polar coordinates as the power series with respect to the distance to the crack tip. The logarithm terms and angular functions are accounted in the asymptotic expansion using recurrence relations. Then boundary conditions imposed between the opposite crack faces in the sector of angle $ 2\pi $ determine admissible exponents and parameters in the power series. For the specific conditions of Dirichlet, Neumann, impermeability, non-penetration and shear crack, the principal asymptotic terms are derived, which verify the singular behaviour. In particular, the analytical solution answers the questions of a square-root singularity at the crack tip and the presence of log-oscillations of variational solutions for the Stokes problems. |
| first_indexed | 2024-03-11T13:39:39Z |
| format | Article |
| id | doaj.art-a6b8b09b7e574c9e96b47553646484bf |
| institution | Directory Open Access Journal |
| issn | 2769-0911 |
| language | English |
| last_indexed | 2024-03-11T13:39:39Z |
| publishDate | 2022-12-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | Applied Mathematics in Science and Engineering |
| spelling | doaj.art-a6b8b09b7e574c9e96b47553646484bf2023-11-02T13:48:31ZengTaylor & Francis GroupApplied Mathematics in Science and Engineering2769-09112022-12-0130144847910.1080/27690911.2022.20845422084542Asymptotic series solution of variational stokes problems in planar domain with crack-like singularityVictor A. Kovtunenko0Kohji Ohtsuka1University of GrazHiroshima Kokusai Gakuin UniversityA class of nonlinear variational problems describing incompressible fluids and solids by stationary Stokes equations given in a planar domain with a crack (infinitely thin flat plate in fluids) is considered. Based on the Fourier asymptotic analysis, general analytical solutions are obtained in polar coordinates as the power series with respect to the distance to the crack tip. The logarithm terms and angular functions are accounted in the asymptotic expansion using recurrence relations. Then boundary conditions imposed between the opposite crack faces in the sector of angle $ 2\pi $ determine admissible exponents and parameters in the power series. For the specific conditions of Dirichlet, Neumann, impermeability, non-penetration and shear crack, the principal asymptotic terms are derived, which verify the singular behaviour. In particular, the analytical solution answers the questions of a square-root singularity at the crack tip and the presence of log-oscillations of variational solutions for the Stokes problems.http://dx.doi.org/10.1080/27690911.2022.2084542stokes systemincompressibilitymixed variational problemasymptotic theoryfourier analysispower seriesrecurrence relationcracknon-penetrationcrack-tip singularity |
| spellingShingle | Victor A. Kovtunenko Kohji Ohtsuka Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity Applied Mathematics in Science and Engineering stokes system incompressibility mixed variational problem asymptotic theory fourier analysis power series recurrence relation crack non-penetration crack-tip singularity |
| title | Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity |
| title_full | Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity |
| title_fullStr | Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity |
| title_full_unstemmed | Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity |
| title_short | Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity |
| title_sort | asymptotic series solution of variational stokes problems in planar domain with crack like singularity |
| topic | stokes system incompressibility mixed variational problem asymptotic theory fourier analysis power series recurrence relation crack non-penetration crack-tip singularity |
| url | http://dx.doi.org/10.1080/27690911.2022.2084542 |
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