The thickness of cylindrical shocks and the PLK method

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.