Properties of some Hugoniot curves associated with shock instability

Abstract

Combining the Hugoniot equation e e 0  1 2 (p + p 0)(v 0  v) with the (e-p-v) equation of slate gives the Hugoniot curve p h ( p 0, v 0, v) centered at an initial condition (p 0, v 0) in the (p-v) plane. Since the Hugoniot equation satisfies the conservation laws of mass, momentum, and energy, only those Hugoniot curves that are consistent with the second law of thermodynamics define the locus of states connected to (p 0, v 0) by a single compression shock. Hugoniot curves that lie in a domain of the (p-v) plane where ( ∂ 2p ∂v 2 ) 8 > 0 are consistent with the second law of thermodynamics; those that lie in a domain where ( ∂ 2p ∂v 2 ) 8 > 0 , as well as in a domain where ( ∂ 2p ∂v 2 ) 8 < 0 , are inconsistent with the second law and are associated with shock instability. The first law of thermodynamics gives properties of some Hugoniot curves associated with shock instability. The second law of thermodynamics, together with conditions necessary for hydrodynamic stability, gives the locus of states connected to a given initial condition in the domain of the (p-v) plane where a single compression shock is thermodynamically unstable.