Abstract
The one-dimensional isothermal motion of a compressible elastic fluid or solid can be described in Lagrangian coordinates by the coupled system
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. D. James, The propagation of phase boundaries in elastic bars, Archive for Rational Mechanics and Analysis.
M. Shearer, The Riemann problem for a class of conservation laws of mixed type, J. Differential Equations46 (1982), 426–443.
M. Shearer, Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type, Archive for Rational Mechanics and Analysis93 (1986), 45–59.
M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Archive for Rational Mechanics and Analysis81 (1983), 301–315.
R. Hagan & M. Slemrod, The viscosity-capillarity criterion for shocks and phase transitions, Archive for Rational Mechanics and Analysis83 (1984), 333–361.
M. Slemrod, Dynamics of first order phase transition, in Phase Transformations and Material Instabilities in Solids, ed. M. Gurtin, Academic Press: New York (1984), 163–203.
C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Archive for Rational Mechanics and Analysis52 (1973), 1–9.
C. M. Dafermos & R. J. Di Perna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations20 (1976), 90–114.
C. S. Morawetz, On a weak solution for a transonic flow, Comm. Pure and Applied Math.38 (1985), 797–818.
J. Mawhin, Topological degree methods in nonlinear boundary value problems, Conference Board of Mathematical Sciences Regional Conference Series in Mathematics, No. 40, American Mathematical Society (1979).
I. P. Natanson, Theory of functions of a real variable, Vol. 1, F. Ungar Publishing Co., New York (1955).
M. Shearer, Dynamic phase transitions in a van der Waals gas, to appear Quarterly of Applied Math.
A. S. Kalasnikov, Construction of generalized solutions of quasilinear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter, Dokl. Akad. Nauk. SSR127 (1959), 27–30 (Russian).
V. A. Tupciev, The asymptotic behavior of the solution of the Cauchy problem for the equation \({\varepsilon ^{2}}t{u_{{xx}}} = {u_{t}}\left[ {\varphi \left( u \right)} \right]x\) that degenerates for ξ = 0 into the problem of the decay of an arbitrary discontinuity for the case of a rarefraction wave. Z. Vycisl. Mat. Fiz.12 (1972), 770–775; English translation in USSR Comput. Math. and Phys.12.
V. A. Tupciev, On the method of introducing viscosity in the study of problems involving decay of a discontinuity, Dokl. Akad. Nauk. SSR211 (1973), 55–58; English translation in Soviet Math. Dokl.14.
C. M. Dafermos, Structure of solutions of the Riemann problem for hyperbolic systems of conservation laws, Arch. for Rational Mechanics and Analysis53 (1974), 203–217.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Bernard Coleman on the occasion of his sixtieth birthday
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Slemrod, M. (1991). A Limiting “Viscosity” Approach to the Riemann Problem for Materials Exhibiting Change of Phase. In: Markovitz, H., Mizel, V.J., Owen, D.R. (eds) Mechanics and Thermodynamics of Continua. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75975-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-75975-8_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52999-6
Online ISBN: 978-3-642-75975-8
eBook Packages: Springer Book Archive