T h e one-dimensional m otion o f com pressible, viscous a n d heat-conductive fluids is described by th e equations in th e L a g ra n g ia n m a s s coordin ate (the suffix denotes th e partial differentiation with respect to th e v a ria b le t or x) { Pt+ p 2 u.v=0 ut+ P .=(110u .).r (e + 1.172) t ( p u ) x =-c D O z ) + (g p u u x ) , w h ere p is th e density, u is th e velocity, 0 is th e absolute temperature, g is th e viscosity coefficient and i s th e co efficien t o f heat conduction, and the pressure p a n d th e internal energy e a r e rela ted to p , 0 b y t h e equation of state of th e fluids.H e r e w e assum e that th e equation of state is one for the ideal polytropic gas:
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Kawashima et al. (1981) studied this question.