On non-Newtonian p-fluids. The pseudo-plastic case

Abstract

In the following we study a class of stationary Navier-Stokes equations with shear dependent viscosity, under the non-slip (Dirichlet) boundary condition. We consider pseudo-plastic fluids. A fluid is said pseudo-plastic, or shear thinning, if in Eq. (1.1) below one has p < 2. We are interested in global (i.e., up to the boundary) regularity results, in dimension n = 3, for the second order derivatives of the velocity and the first order derivatives of the pressure. We consider a cubic domain Ω and impose the non-slip boundary condition only on two opposite faces. On the other faces we assume periodicity, as a device to avoid effective boundary conditions. This choice is made so that we work in a bounded domain Ω and simultaneously with a flat boundary.

Hodge-Frobenius equations and the Hodge–Bäcklund transformation

“…Taking the axis of rotation to lie at the origin of coordinates, we write the tangential velocity vector in the form Equating the right-hand sides of equations (3.4) and (3.5), we conclude that η(r) = 2 log r and ω = r 2 du for u(x, y) = arctan(y/x). Then |du| = r −1 , so the singular structure of u in the x, y-plane is analogous to that of the fundamental solution of Laplace's equation in R 3 . In particular, u is singular at the origin of the disc.…”

“…Condition (2) implies, by the Poincaré Lemma, that there exists locally a scalar flow potential ϕ (x) , where x ∈ R 3 denotes the position of a particle in the flow. Perhaps the mildest weakening of the irrotationality condition results from replacing (2) by the integrability condition v · ∇ × v = 0.…”

Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow

“…The existence of weak solutions is then proven for p > 3=2 if n D 2 and for p > 9=5 when n D 3. Finally, the existence of unique regular weak solutions is established, in space dimension n D 2 for p 2, and in space dimension n D 3 for p 11=5 (see, also, [BdV1]). …”

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].…”