Three-dimensional blow-up solutions of the Navier-Stokes equations |
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Authors: | Grundy RE; McLaughlin R |
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Institution: |
A1 School of Mathematical and Computational Sciences, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland, UK A2 School of Computing and Management Sciences, Sheffield Hallam University, Harmer Building, Sheffield S1 1WB, UK
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Abstract: | In this paper we extend the plane blow-up results of Grundy& McLaughlin (1997) to the three-dimensional Navier-Stokes equations.Using a solution structure originally due to Lin we first providenumerical evidence for the existence of blow-up solutions on- < x, z < , 0 y 1 with boundary conditions on y = 0and y = 1 involving derivatives of the velocity components.The formulation enables us to consider plane and radial flowas special cases. Various features of the computations are isolatedand are used to construct a formal asymptotic solution closeto blow-up. We show that the numerical and asymptotic analysesprovide a mutually consistent global picture which supportsthe conclusion that, for the family of problems we considerhere, blow-up in fact can take place in three dimensions butat an inverse linear rate rather than the faster inverse squareof the plane case. |
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