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A family of systems including the Herschel-Bulkley fluid equations

Nikolai V. Chemetov, Marcelo M. Santos

Abstract

We analyze the Navier-Stokes equations for incompressible fluids with the {\lq\lq}viscous stress tensor{\rq\rq} $\mathbb{S}$ in a family which includes the Bingham model for viscoplastic fluids (more generally, the Herschel-Bulkley model). $\mathbb{S}$ is the subgradient of a convex potential $V=V(x,t,X)$, allowing that $V$ can depend on the space-time variables $(x,t)$. The potential has its one-sided directional derivatives $V'(X,X)$ uniformly bounded from below and above by a $p$-power function of the matrices $X$. For $p\geqslant 2.2$ we solve an initial boundary value problem for those fluid systems, in a bounded region in $\mathbb{R}^3$. We take a nonlinear boundary condition, which encompasses the Navier friction/slip boundary condition.

A family of systems including the Herschel-Bulkley fluid equations

Abstract

We analyze the Navier-Stokes equations for incompressible fluids with the {\lq\lq}viscous stress tensor{\rq\rq} in a family which includes the Bingham model for viscoplastic fluids (more generally, the Herschel-Bulkley model). is the subgradient of a convex potential , allowing that can depend on the space-time variables . The potential has its one-sided directional derivatives uniformly bounded from below and above by a -power function of the matrices . For we solve an initial boundary value problem for those fluid systems, in a bounded region in . We take a nonlinear boundary condition, which encompasses the Navier friction/slip boundary condition.
Paper Structure (5 sections, 16 theorems, 157 equations)

This paper contains 5 sections, 16 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. The systems \ref{['SR explicit']} and the modified version \ref{['SR01g']}
  3. Auxiliary results
  4. Construction of the approximation problem
  5. Limit transition

Key Result

Theorem 1.1

Let $V$ be the potential (function) given in pot-potential, being $\Omega\equiv\Omega(x,t)$ a matrix function in $L^p(\mathcal{O}_T;\mathbb{R}^{d\times d})$, and $g$ as above (a Caratheodory function from $\Gamma_T\times\mathbb{R}^3$ to $\mathbb{R}$, differentiable and convex with respect to the sec where $p^{\prime }=\frac{p}{p-1}$ and $V_{p}^{\ast }$ is the dual space of $V_{p}$. The pair $(\mat

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 3.1
  • Remark 3.2
  • Definition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • Corollary 3.7
  • Corollary 3.8
  • ...and 9 more