Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support
type: Item
instance of: scholarly article
title: Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support
zbMATH Open document ID: 1204.35133
DOI: 10.5052/RACSAM.2010.13
author: Roland Glowinski
author: J. I. Díaz
author: Giovanna Guidoboni
author: Taebeom Kim
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2010-07-09T00:00:00Z
review text: The isothermal and unsteady flow of an incompressible Bingham visco-plastic medium, during the time interval \((0,T)\), is modeled by the following system of equations (clearly of the Navier- Stokes system type): \[ \rho(\partial _t \mathbf{u}+ (\mathbf{u}\cdot \bigtriangledown\nabla)\mathbf{u})= \nabla \cdot \pmb{\sigma}+\mathbf{f}(t,x) \quad\text{in }(0,T)\times \widetilde{\Omega}, \] \[ \nabla \cdot \mathbf{u}=0 \quad\text{in }\quad (0,T)\times \widetilde{\Omega}, \] \[ \pmb{\sigma}=-p\mathbf{I}+\sqrt{2}g\frac{\mathbf{D(u)}}{|\mathbf{D(u)}|}+2\mu\mathbf{D(u)}, \] \[ \mathbf{u}(0)=\mathbf{u}_0 \quad (\text{with }\nabla \cdot\mathbf{u}_0)=0. \] Here \(\mathbf{u}\) and \(p\) represent velocity and pressure, respectively. The positive constants \(\rho, \mu\) and \(g\) represent density, viscosity and plasticity yield of the Bingham medium, respectively. Moreover, \(\mathbf{f}(t,x)\) is a given density of external forces, \(\mathbf{D(v)}=[\nabla \mathbf{v}+(\nabla \mathbf{v})^t]/2(=D_{ij}(\mathbf{v})_{1\leq i,j\leq d}, \forall \mathbf{v} \in (H^1(\widetilde{\Omega}))^d\), and \(|\mathbf{D(v)}|\) is the Frobenius norm of tensor \(\mathbf{D(v)}\), i.e., \[ |\mathbf{D(v)}|=\left( \sum_{1\leq i,j\leq d} |D_{ij}(\mathbf{v})|^2\right)^{1/2}. \] The domain \(\widetilde{\Omega}\) is an open and connected subset of \(\mathbb{R}^d\) (\(d=2\) or \(3\) in most of the applications), \(\widetilde{\Gamma}\) is the boundary of \(\widetilde{\Omega}\) and \(T>0\) is arbitrarily fixed (and possibly \(T=+\infty\)). Only homogeneous Dirichlet boundary conditions are considered. For the two-dimensional flow in a bounded domain, it is shown the extinction in a finite time even under suitable nonzero external forces. Also, the special case of three-dimensional domain given as an infinitely long cylinder of bounded cross section is considered. Sufficient conditions leading to a scalar formulation on the cross section are given. The stabilization of the solutions is proved, when \(t\) goes to …
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 76A10
zbMATH DE Number: 5735154
zbMATH Keywords: Bingham visco-plastic medium flows
zbMATH Keywords: approximation of solutions
zbMATH Keywords: stabilization for large time
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
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full work available at URL: racsam.2010.13
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description: scientific article; zbMATH DE number 5735154