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Constitutive surfaces in fluid mechanics

Published online by Cambridge University Press:  24 October 2008

M. J. Sewell  and
D. Porter
M. J. Sewell
Affiliation:
University of Reading
D. Porter
Affiliation:
University of Reading
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Abstract

The new concept of a constitutive surface is introduced into inviscid fluid mechanics, with special reference to compressible gas dynamics and to shallow water theory. The detailed shape of such surfaces is calculated, including the manner of their transition across singularities where the Mach or Froude number passes through unity. The calculation draws upon recent work describing the transition of a Legendre transformation through its singularity. For example, mass flow Q, total energy h and flow stress P are always related by part of a ‘swallowtail’ surface, regardless of the particular motion.

The addition of dynamical conditions defines particle history tracks which always lie on constitutive surfaces even for unsteady flow, except that they may jump from one part to another of such a surface when shock waves or hydraulic jumps are passed through.

Illustrations given include the steady flow of a general gas through a standing normal shock, general shallow water theory, and flow along a sloping-sided channel. Connections with existing literature are described.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 3 , November 1980 , pp. 517 - 546
Copyright
Copyright © Cambridge Philosophical Society 1980

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