= Entropy production in a perfect-gas shock
For the <pressure-density Hugoniot relation for a perfect gas>, the change in <specific entropy> is $[s]/c_v=\log P-\gamma\log D(P)$, with $c_v$ the <specific heat capacity> at constant volume. Its derivative is
$$
\frac{d([s]/c_v)}{dP}
=\frac{(\gamma^2-1)(P-1)^2}{P[(\gamma+1)P+\gamma-1][(\gamma-1)P+\gamma+1]}.
$$
It is positive for a compressive <normal shock wave> with $P>1$. For a <weak shock> with $P=1+\delta$, integrating the leading term gives
$$
\frac{[s]}{c_v}=\frac{\gamma^2-1}{12\gamma^2}\delta^3+O(\delta^4).
$$
Thus the <entropy production> is cubic in the small <pressure> jump, even though the <mass density> and <temperature> changes already appear at first order.
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