Multidimensional similarity models for exploding foils

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Abstract
A general derivation of the Gaussian, isothermal similarity equations for multidimensional hydrodynamic expansions is presented. Analytical solutions for the cases of constant heating and adiabatic flow for planar, cylindrical, and spherical expansions are given. An energy integral is derived for general multidimensional adiabatic flows. For two‐dimensional adiabatic expansions, a second invariant of the motion is demonstrated. It is shown that this invariant gives the asymptotic shape of a Gaussian object in terms of its initial shape. A simple model for laser heating of an exploding foil is described and the asymptotic form of the corresponding solutions are given for two and three dimensions. The similarity solutions are compared with two‐dimensional numerical hydrodynamic simulations for an example of an exploding foil, designed to produce a recombination x‐ray laser.