Convergence of strong shock waves in an ideal gas with dust particles

In this paper, the authors study the problem of an imploding strong cylindrical/spherical shock wave collapsing at the axis/center of a cylindrical/spherical piston that is filled with a dusty gas of uniform density. The dusty gas is assumed to be a mixture of an ideal gas and a large number of dust particles. The dust particles are of a micrometric size and uniformly distributed in the mixture. A mathematical model using a system of hyperbolic partial differential equations is presented for the considered problem. The perturbation series method is used to solve the implosion problem, providing a global solution and yielding accurately the results of Guderley's local similarity solution, which holds only in the neighborhood of the axis/center of implosion. The values of all possible real similarity exponents and the corresponding amplitudes are determined in the vicinity of the shock collapse by extending the flow variables and shock location in the Taylor series in time t. Furthermore, the obtained values of similarity exponents have been compared with the existing results and numerical results obtained by the other methods. The effects of the adiabatic exponent gamma, the wavefront curvature alpha, and various dusty gas parameters such as sigma, K-p, and G(0) on the shock trajectory and flow variables have been graphically analyzed.