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Mathematical analysis of a two-phase continuum mixture theory

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Abstract

In this paper, we study the mathematical structure of a continuum reactive mixture model of the combustion of granular energetic materials. We obtain and classify the wave fields associated with this description. This analysis shows that this system of hyperbolic equations becomes degenerate when the relative flow is locally sonic. We derive the corresponding Riemann invariants and construct simple wave solutions. We also discuss special discontinuous solutions of the system of equations. For fixed upstream conditions, different downstream states are possible when the relative velocities exceed the speed of the sound gas.

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Abbreviations

a :

s (solid),g (gas) subscript to indicate the phase

v a :

velocity of phasea

v sg :

v s -v g

P a :

material density

V a :

specific volume

P a :

pressure

T a :

temperature

e a :

internal energy

η a :

entropy

h a :

enthalpy =e a +P a /P a

ψ a :

Helmholtz free energy =e a -T a η a

φ a :

volume fraction

β a :

configuration pressure =\(\phi _a \rho _a \left( {\frac{{\partial \psi _a }}{{\partial \phi _a }}} \right)_{\rho _a ,T_a }\)

c a :

speed of sound =\(\left( {\frac{{\partial p_a }}{{\partial \rho _a }}} \right)_{\eta _a ,\phi _a }^{1/2}\)

Г a :

Grüneisen coefficient =\(\frac{1}{{\rho _a }}\left( {\frac{{\partial p_a }}{{\partial e_a }}} \right)_{\rho _a ,\phi _a }\)

C † a :

rate of mass production

m † a :

rate of momentum production

e † a :

rate of energy production

δ:

drag coefficient

h :

heat transfer coefficient

k a :

thermal conductivity

μ c :

compaction viscosity

F :

φ s φ g [p s −p g −β s ]/μ c

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, 87131, Albuquerque, NM, USA

    P. Embid

  2. Fluid and Thermal Sciences Department, Sandia National Laboratories, 87185, Albuquerque, NM, USA

    M. Baer

Authors
  1. P. Embid
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  2. M. Baer
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Embid, P., Baer, M. Mathematical analysis of a two-phase continuum mixture theory. Continuum Mech. Thermodyn 4, 279–312 (1992). https://doi.org/10.1007/BF01129333

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  • DOI: https://doi.org/10.1007/BF01129333

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