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Three problems about of the shock waves
Three problems about of the shock waves
for the curious students
(hydrodynamics)
The first problem
As it informs in your text-books, for an ideal gas the Hugoniot equation has the form
p[(h+1)V-(h-1)V0]=p0[(h+1)V0-(h-1)V] , h=cp/cv ;
here p- the pressure, V- the specific volume, cp and cV - the specific heat capacities with constant p and V . Hence it follows
[(h-1)V0-(h+1)V](dp/dV)=[(h+1)p+(h-1)p0] (1)
As you know, the energy conserwation law is
de=dQ-pdV (2)
where e - the specific inner energy, dQ - the summary contribution in de from all nonmechanical influences (so called "the heat flow"). The Hugoniot equation bases on the idea about of adiabatic deforming in a shock wave:dQ=0 , and from (2):
de=-pdV (3)
For the function e(p,V) with any form, de=(de/dp)Vdp+(de/dV)pdV; the substitution it in (3) gives
(dp/dV)=-[p+(de/dV)]/(de/dp) ;
for an ideal gas it will be
(dp/dV)=-hp/V (4)
As you see, the formula (1) turns into (4) only at p=p0 and V=V0: in an ideal gas the shock waves can be described by Hugoniot equation only if its amplitudes near to nought.
Why it is so?
- Leonid G. Philippenko's blog
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