Solutions of a two-dimensional grade-two fluid model with a tangential boundary condition on non-smooth domains

Abstract

In this Note, we construct a solution in H 1 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. In addition, we establish that, if the domain is a curvilinear polygon with curved segments of class C 1.1, each solution of the grade-two fluid tends to a solution of the Navier-Stokes equations when the material modulus α tends to zero. To our knowledge, these results are new. When the domain is a polygon, we show that the regularity of the solution corresponds to that of a Stokes problem. Uniqueness is established in a convex polygon, for sufficiently small data.