Shear Thinning Viscous Fluids in Cylindrical Domains. Regularity up to the Boundary

Abstract

We consider the motion of a non-Newtonian fluid with shear dependent viscosity between two cylinders. We prove regularity results for the second derivatives of the velocity and the first derivatives of the pressure up to the boundary. A similar problem is studied in reference [2] in the case of a flat boundary. Here we extend the techniques applied in [2] to cylindrical coordinates.

On the regularity of shear thickening viscous fluids

“…Finally, we remark that the same improvements also hold in the cases considered in [10,11], where, in the shear thinning fluids case, we deal with the regularity problem for cylindrical boundary domains. Concerning general nonflat boundaries, the extension of the above kind of results presents new obstacles, in comparison with the classical case p = 2.…”

Global regularity of a class of p-fluid flows in cylinders

“…In the recent paper [9] we were concerned with the regularity problem for a certain type of non-flat boundary domain, namely the open domain between two coaxial cylinders. This case is easier to treat than the one of a generic curvilinear boundary, which requires much more elaborate techniques of localization and "flattening the boundary."…”

“…There we avoid such techniques, by appealing to the natural change of coordinates associated with a cylindrical surface. Actually, the choice for the domain in [9] enabled us to pass to cylindrical coordinates, by a one-to-one mapping, and then to extend to cylindrical coordinates the technique used in [4] for Cartesian coordinates. Indeed, changing from a Cartesian coordinate system (x 1 , x 2 , x 3 ) to a cylindrical one, (r, θ, x 3 ) ≡ (y 1 , y 2 , y 3 ), we return to a situation similar to the one of a flat boundary, where the cylindrical coordinates play now a formal role similar to that of the Cartesian coordinates in the flat boundary case.…”

Boundary Regularity of Shear Thickening Flows

“…Concerning the shear thinning case, strongly related W 2, q regularity results up to the boundary, under the boundary condition (1.2), are proved, for flat boundaries in [4,5,10], for cylindrical domains in [20,21], and for smooth arbitrary boundaries in [7]. Appeal to Troisi's anisotropic embedding theorems (instead of classical, isotropic, Sobolev embedding theorems), also used below, was introduced in [10].…”