Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications
An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead t...
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description | An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns, following which the inviscid limit case is discussed. Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems, two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. Concurrently, the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions, incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface, is established, together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries. |
doi_str_mv | 10.1063/1.5031119 |
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H. ; Marner, F.</creator><creatorcontrib>Scholle, M. ; Gaskell, P. H. ; Marner, F.</creatorcontrib><description>An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns, following which the inviscid limit case is discussed. Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems, two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. Concurrently, the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions, incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface, is established, together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries.</description><identifier>ISSN: 0022-2488</identifier><identifier>EISSN: 1089-7658</identifier><identifier>DOI: 10.1063/1.5031119</identifier><identifier>CODEN: JMAPAQ</identifier><language>eng</language><publisher>New York: American Institute of Physics</publisher><subject>Boundary conditions ; Computational fluid dynamics ; Fluid flow ; Free surfaces ; Integrals ; Navier-Stokes equations ; Quantum physics ; Stokes law (fluid mechanics) ; Three dimensional flow ; Unsteady flow ; Viscosity ; Viscous flow</subject><ispartof>Journal of mathematical physics, 2018-04, Vol.59 (4)</ispartof><rights>Author(s)</rights><rights>Copyright American Institute of Physics Apr 2018</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c327t-6a266e27e063aa43517b135a5734fbff090da6839689edf886c2917aa36c2a583</citedby><cites>FETCH-LOGICAL-c327t-6a266e27e063aa43517b135a5734fbff090da6839689edf886c2917aa36c2a583</cites><orcidid>0000-0001-6965-3454 ; 0000-0001-6945-5247</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://pubs.aip.org/jmp/article-lookup/doi/10.1063/1.5031119$$EHTML$$P50$$Gscitation$$H</linktohtml><link.rule.ids>314,780,784,794,4512,27924,27925,76384</link.rule.ids></links><search><creatorcontrib>Scholle, M.</creatorcontrib><creatorcontrib>Gaskell, P. H.</creatorcontrib><creatorcontrib>Marner, F.</creatorcontrib><title>Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications</title><title>Journal of mathematical physics</title><description>An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns, following which the inviscid limit case is discussed. Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems, two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. Concurrently, the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions, incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface, is established, together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries.</description><subject>Boundary conditions</subject><subject>Computational fluid dynamics</subject><subject>Fluid flow</subject><subject>Free surfaces</subject><subject>Integrals</subject><subject>Navier-Stokes equations</subject><subject>Quantum physics</subject><subject>Stokes law (fluid mechanics)</subject><subject>Three dimensional flow</subject><subject>Unsteady flow</subject><subject>Viscosity</subject><subject>Viscous flow</subject><issn>0022-2488</issn><issn>1089-7658</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp9kEtPwzAQhC0EEqVw4B9Y4gRqih-J4xxRVR5SBQfgHG2ddXFpk9R2EP33BNIzp1lpvt3VDCGXnE05U_KWTzMmOefFERlxposkV5k-JiPGhEhEqvUpOQthzRjnOk1HZD3_BhOpqyOuPETX1LSxNH4g7eoQEap975lm23oMwS03SJ_hy6FPXmPziYHirvvbChO6gm6F1HgX0TuYUKgrCm27cWYgzsmJhU3Ai4OOyfv9_G32mCxeHp5md4vESJHHRIFQCkWOfRyAVGY8X3KZQZbL1C6tZQWrQGlZKF1gZbVWRhQ8B5D9AJmWY3I13G19s-swxHLddL7uX5aCqYKJTCneU9cDZXwTgkdbtt5twe9LzsrfKkteHqrs2ZuBDcbFvzD_wD8Fg3PC</recordid><startdate>201804</startdate><enddate>201804</enddate><creator>Scholle, M.</creator><creator>Gaskell, P. 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Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems, two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. 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source | AIP Journals; Alma/SFX Local Collection |
subjects | Boundary conditions Computational fluid dynamics Fluid flow Free surfaces Integrals Navier-Stokes equations Quantum physics Stokes law (fluid mechanics) Three dimensional flow Unsteady flow Viscosity Viscous flow |
title | Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications |
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