Instability in the evolution equations describing incompressible granular flow
type: Item
instance of: scholarly article
title: Instability in the evolution equations describing incompressible granular flow
zbMATH Open document ID: 0647.35037
DOI: 10.1016/0022-0396(87)90038-6
author: D. G. Schaeffer
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: Constitutive equations for incompressible granular materials are derived and analyzed. Raw materials are often stored and moved in granular form, so there is considerable interest in mathematical theories that can be applied to engineering design. The author gives a detailed derivation of the equations governing incompressible rigid-perfectly plastic Coulomb powders, using a von Mises type yield condition. The equations formally resemble the Navier-Stokes equations, but are homogeneous of degree zero in the strain rate tensor. The equations then are linearized, exponentially unstable solutions are identified, and the steady state equations are analyzed (which need not be of elliptic type). These results are then applied to a well-known similarity solution (Jenike's radial solution), which is shown to lead to ill-posed problems in most practical applications. The paper concludes with a discussion of other possible constitutive laws, including compressible materials, and of oscillatory motions which are observed in industrial silos.
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 74C99
Mathematics Subject Classification ID: 74L10
zbMATH DE Number: 4056161
zbMATH Keywords: Constitutive equations
zbMATH Keywords: incompressible granular materials
zbMATH Keywords: incompressible rigid-perfectly plastic Coulomb powders
zbMATH Keywords: von Mises type yield condition
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: linearized
zbMATH Keywords: exponentially unstable
zbMATH Keywords: steady state equations
zbMATH Keywords: similarity solution
zbMATH Keywords: Jenike's radial solution
zbMATH Keywords: ill-posed problems
zbMATH Keywords: oscillatory motions
reviewed by: Hans Engler
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90038-6
OpenAlex ID: W2011660773
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II
cites work: Q4097611
cites work: Gravity Flow of Frictional-Cohesive Solids—Convergence to Radial Stress Fields
cites work: Q3275501
cites work: Compressible fluid flow and systems of conservation laws in several space variables
cites work: Q3270352
cites work: A theory of the kinematics of ideal soils under plane strain conditions
cites work: Q3933886
description: scientific article; zbMATH DE number 4056161