The thickness of cylindrical shocks and the PLK method
Author:
H. C. Levey
Journal:
Quart. Appl. Math. 17 (1959), 77-93
MSC:
Primary 76.00
DOI:
https://doi.org/10.1090/qam/105980
MathSciNet review:
105980
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Abstract: Cylindrical shocks occur when a viscous heat-conducting gas flows radially in a plane. This is a singular perturbation problem in which the perturbing parameter is the reciprocal of ${R_e}$ , the Reynolds number of the flow. It is shown that for both inward facing shocks (source flow) and outward facing shocks (sink flow) the shock thickness is of order $R_e^{ - 1}{S^{ - 1}}\log \left ( {{R_e}{S^3}} \right )$ where $S$ is the shock strength. This is contrary to results for sink flow which have been obtained by the use of Lighthill’s technique for rendering approximations uniformly valid—the $PLK$ method. It is shown that this method fails when applied to singular perturbation problems of the type discussed here in which the small parameter multiplies the highest derivatives.
- Akira Sakurai, On the theory of cylindrical shock wave, J. Phys. Soc. Japan 4 (1949), 199–202. MR 38798, DOI https://doi.org/10.1143/JPSJ.4.199
- H. C. Levey, Two dimensional source flow of a viscous fluid, Quart. Appl. Math. 12 (1954), 25–48. MR 63859, DOI https://doi.org/10.1090/S0033-569X-1954-63859-0
- T. Yao-Tsu Wu, Two dimensional sink flow of a viscous, heat-conducting, compressible fluid; cylindrical shock waves, Quart. Appl. Math. 13 (1956), 393–418. MR 74225, DOI https://doi.org/10.1090/S0033-569X-1956-74225-X
- H. S. Tsien, The Poincaré-Lighthill-Kuo method, Advances in applied mechanics, vol. IV, Academic Press Inc., New York, N.Y., 1956, pp. 281–349. MR 0079929
- M. J. Lighthill, A technique for rendering approximate solutions to physical problems uniformly valid, Philos. Mag. (7) 40 (1949), 1179–1201. MR 33941
A. Sakurai, On the theory of cylindrical shock wave, J. Phys. Soc. Japan 4, 199-202 (1949)
H. C. Levey, Two dimensional source flow of a viscous fluid, Quart. Appl. Math. 7, 25-48 (1954)
Y. T. Wu, Two dimensional sink flow of a viscous, heat-conducting compressible fluid; cylindrical shock waves, Quart. Appl. Math. 13, 393-418 (1956)
H. S. Tsien, The Poincare-Lighthill-Kuo method, Advances in Applied Mechanics, IV, Academic Press Inc., N. Y., 1956, pp. 281-349
M. J. Lighthill, A technique for rendering approximate solutions to physical problems uniformly valid, Phil. Mag. (7) 40, 1179-1201 (1949)
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Article copyright:
© Copyright 1959
American Mathematical Society