Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition |
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Authors: | V Girault L R Scott |
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Institution: | a Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 75252 Paris cedex 05, France;b Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1581, USA |
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Abstract: | This article studies the solutions in H1 of a two-dimensional grade-two fluid model with a non-homogeneous Dirichlet tangential boundary condition, on a Lipschitz-continuous domain. Existence is proven by splitting the problem into a generalized Stokes problem and a transport equation, without restricting the size of the data and the constant parameters of the fluid. A substantial part of the article is devoted to a sharp analysis of this transport equation, under weak regularity assumptions. By means of this analysis, it is established that each solution of the grade-two fluid model satisfies energy equalities and converges strongly to a solution of the Navier–Stokes equations when the normal stress modulus α tends to zero. When the domain is a polygon, it is shown that the regularity of the solution is related to that of a Stokes problem. Uniqueness is established in a convex polygon, with adequate restrictions on the size of the data and parameters. |
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Keywords: | Grade-two fluid Generalized Stokes problem Transport equation Generalized Friedrichs' Lemma |
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