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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-70/1/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 70 / 1 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Write

v = u_{x}

and

D = \partial_{t} + v \partial_{x}

. The Euler equations and conservation of mass give

D ρ = - ρ v_{x}

and

ρ D u = - p_{x} e_{x}

. The internal-energy density of the ideal gas is

e = p / (γ - 1)

. The adiabatic pressure equation therefore gives

\partial_{t} e + \partial_{x} (v e) = - p \partial_{x} v .

Multiply the momentum equation by

u

and use continuity to obtain the kinetic-energy balance

\partial_{t} (ρ u^{2} /2) + \partial_{x} (v ρ u^{2} /2) = - v \partial_{x} p .

Adding the two equations moves

\partial_{x} (p v)

into the flux. Thus the entire system is

\partial_{t} Q + \partial_{x} F = 0

, where

Q = ρ ρ v ρ u_{y} ρ u_{z} E, F = ρ v ρ v^{2} + p ρ v u_{y} ρ v u_{z} v (E + p), E = \frac{1}{2} ρ (v^{2} + u_{y}^{2} + u_{z}^{2}) + \frac{p}{γ - 1} .

This derives the conservative energy flux of a polytropic ideal gas. In particular one-dimensional spatial dependence does not remove the transverse velocities from the total energy. The energy flux can equally be written

ρ v (u^{2} /2 + w)

with specific enthalpy

w = γ p / [(γ - 1) ρ]

.

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