The velocity equations for dilatant granular flow and a new exact solution
type: Item
instance of: scholarly article
title: The velocity equations for dilatant granular flow and a new exact solution
zbMATH Open document ID: 0896.73005
DOI: 10.1007/PL00001461
author: Jim Hill
author: Tui L. Katoanga
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1998-10-06T00:00:00Z
review text: A new exact solution is presented to kinematically determined velocity equations for axially symmetric flow of dilatant granular materials. Two sets of three first-order partial differential equations are obtained from the stress equations in the limits of certain plastic regimes, assuming a Coulomb-Mohr plastic potential in the first case and dilatant double shearing in the second one. All known solutions of these partial differential equations are classified into four known functional forms. Using standard algebraic packages for the determination of Lie symmetries of main equation set, it is shown that the velocity equations admit a four-dimensional symmetry algebra. Then all above-mentioned functional forms of existing solutions are systematically obtained by identification of the so-called ``optimal systems'' of the invariant Lie algebra. This result indicates that there are no further solutions which can be generated from the classical symmetries. Any new families of solutions arise only from the so-called ``non-classical'' symmetries. Then a simple new solution of the velocity equations is obtained in closed form for a special case which has not been given in the literature. For this particular case, the integral curves (streamlines) and simple analytical approximations for particle paths are found and shown graphically.
Mathematics Subject Classification ID: 74A99
Mathematics Subject Classification ID: 74L10
Mathematics Subject Classification ID: 74C99
zbMATH DE Number: 1059761
zbMATH Keywords: optimal systems of invariant Lie algebra
zbMATH Keywords: nonclassical symmetries
zbMATH Keywords: axially symmetric flow
zbMATH Keywords: plastic regimes
zbMATH Keywords: Coulomb-Mohr plastic potential
zbMATH Keywords: dilatant double shearing
zbMATH Keywords: Lie symmetries
zbMATH Keywords: four-dimensional symmetry algebra
zbMATH Keywords: classical symmetries
zbMATH Keywords: integral curves
reviewed by: Ivan A. Parinov
MaRDI profile type: Publication
full work available at URL: pl00001461
OpenAlex ID: W2033245562
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description: scientific article; zbMATH DE number 1059761