Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 2 / a

Solution

E = \frac{p}{γ - 1} + \frac{1}{2} ρ u_{x}^{2} .

The conservative mass, momentum, and energy equations are

\partial_{t} ρ + \partial_{x} (ρ u_{x}) = 0,

\partial_{t} (ρ u_{x}) + \partial_{x} (ρ u_{x}^{2} + p) = 0,

\partial_{t} E + \partial_{x} [(E + p) u_{x}] = 0 .

A monatomic nonrelativistic perfect gas has

γ = 5/3

, an ultrarelativistic gas or radiation-dominated fluid has

γ = 4/3

, and a rotationally active diatomic gas has

γ = 7/5

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