On Variational Models for Quasi-static Bingham Fluids

Abstract

We study a quasi‐static incompressible flow of Bingham type with constituent law \[ \begin{array}{ll} T = p\left| {\cal E}u\right| ⁁{p‐2}{\cal E}u+\beta \frac{{\cal E}u}{\left| {\cal E}u\right| } & \text{if }{\cal E}u\neq 0, \\ \left| T\right| \leq \beta & \text{if }{\cal E}u = 0, \end{array} \] T = p∣ℰu∣p‐2ℰu+β ℰu ∣ℰu∣ if ℰu≠0, ∣T∣⩽β if ℰu = 0, where p≥2 and β>0. Here ℰu denotes the strain velocity and T the corresponding stress. The problem admits a variational formulation in the sense that the velocity field u minimizes the energy I(u) = ∫Ω∣ℰu∣p+β∣ℰu∣dx in the space {v∈H1,p(Ω,ℝn): div v = 0} subject to appropriate boundary conditions. We then show smoothness of u on the set {x∈Ω: ℰu≠0}.

Progress in numerical simulation of yield stress fluid flows

“…These authors investigated existence, uniqueness and regularity of the solution for steady and non-stationary flows. Existence and extra regularity results on the problem with Dirichlet boundary conditions for a driven cavity flow was also studied by Seregin (1997, 1998) (see also Fuchs et al (1996); Fuchs and Seregin (2000, chap. 3)). The expected regularity of the solution of the Bingham problem is still an open question, but it seems unreasonable to hope for a high regularity: in specific cases, such as Poiseuille or Couette flows, the velocity is known to have a limited regularity across the yield surface, separating yielded and unyielded regions.…”

On Quasi-static Non=Newtonian Fluids with Power-law

“…The existence of a unique solution of (1.6) is easily verified with the help of standard results on convex variational integrals combined with suitable versions of Korn's inequality in spaces L p (see e.g. [5]) provided we assume uo E H'*"(R, R") and div uo = 0. Next we state our results concerning the regularity of the solution u E K of (1.6).…”

A nonsmooth model for discontinuous shear thickening fluids: Analysis and numerical solution

“…In order to prove partial regularity, the key ingredient is a blow-up estimate, which is verified in the next lemma. Along the proof we make use of results which can be traced back to [12]. In fact, we only give a detailed justification of the main steps, where the proof differs from the one in [12].…”

“…To obtain a contradiction, we will make use of Campanato-type estimates, which follow directly once a blow-up equation is verified at the limit (see [12,Lem. 1.5]).…”