The thickness of cylindrical shocks and the PLK method

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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 {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 ⁡ ( R e S 3 ) R_e^{ - 1}{S^{ - 1}}\log \left ( {{R_e}{S^3}} \right ) where S 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 P L K 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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