Title: Converging Shock Flows for a Mie-Grüneisen Equation of State,
Abstract: Previous work has shown that the one-dimensional (1D) inviscid compressible
flow (Euler) equations admit a wide variety of scale-invariant solutions
(including the famous Noh, Sedov, and Guderley shock solutions) when the
included equation of state (EOS) closure model assumes a certain
scale-invariant form. However, this scale-invariant EOS class does not include
even simple models used for shock compression of crystalline solids, including
many broadly applicable representations of Mie-Grüneisen EOS. Intuitively,
this incompatibility naturally arises from the presence of multiple dimensional
scales in the Mie-Grüneisen EOS, which are otherwise absent from
scale-invariant models that feature only dimensionless parameters (such as the
adiabatic index in the ideal gas EOS). The current work extends previous
efforts intended to rectify this inconsistency, by using a scale-invariant EOS
model to approximate a Mie- Grüneisen EOS form. To this end, the adiabatic
bulk modulus for the Mie-Grüneisen EOS is constructed, and its key features
are used to motivate the selection of a scale-invariant approximation form. The
remaining surrogate model parameters are selected through enforcement of the
Rankine-Hugoniot jump conditions for an infinitely strong shock in a
Mie-Grüneisen material. Finally, the approximate EOS is used in conjunction
with the 1D inviscid Euler equations to calculate a semi-analytical,
Guderley-like imploding shock solution in a metal sphere, and to determine if
and when the solution may be valid for the underlying Mie-Grüneisen EOS.