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Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications.

Scholle, M. and Gaskell, P. H. and Marner, F. (2018) 'Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications.', Journal of mathematical physics., 59 (4). 043101.

Abstract

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.

Item Type:Article
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Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1063/1.5031119
Publisher statement:© 2018 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Scholle, M. and Gaskell, P. H. and Marner, F. (2018) 'Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications.', Journal of mathematical physics., 59 (4). 043101 and may be found at https://doi.org/10.1063/1.5031119
Date accepted:25 March 2018
Date deposited:04 May 2018
Date of first online publication:18 April 2018
Date first made open access:18 April 2019

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