Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Formal equivalence of the nonlinear string and one-dimensional fluid flow


Author: Gerald Rosen
Journal: Quart. Appl. Math. 23 (1965), 286-287
MSC: Primary 76.35; Secondary 73.00
DOI: https://doi.org/10.1090/qam/187538
MathSciNet review: 187538
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Abstract: By applying transformations to the dynamical equation for the longitudinal vibrations of a nonlinear model string, we obtain a set of equations which describes the one-dimensional flow of an ideal compressible polytropic fluid. Thus it is shown that the nonlinear string problem is formally equivalent to the classical problem of fluid flow analyzed by Riemann and by others.


References [Enhancements On Off] (What's this?)

  • [1] Norman J. Zabusky, Exact solution for the vibrations of a nonlinear continuous model string, J. Mathematical Phys. 3 (1962), 1028–1039. MR 0146545, https://doi.org/10.1063/1.1724290
  • [2] For example, see: R. von Mises, Mathematical theory of compressible fluid flow, Academic Press Inc., New York, 1958, Article 12, p. 155
  • [3] A more general form of Eq. (7), a form that takes account of real fluid viscosity, has been derived and used as a starting point for non-Riemannian fluid flow theory; see: G. Rosen, Phys. of Fluids 2, 517 (1959); 3, 188 (1960); 3, 191 (1960)
  • [4] A modern analysis of this classical problem is presented by: P. A. Fox, J. Math, and Phys. 34, 133 (1955)

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Additional Information

DOI: https://doi.org/10.1090/qam/187538
Article copyright: © Copyright 1965 American Mathematical Society


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