Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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.

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DOI: https://doi.org/10.1090/qam/105980
Article copyright: © Copyright 1959 American Mathematical Society

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