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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-2/39c/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 39C / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

The Navier-Stokes equation for constant mass density

ρ

, dynamic viscosity

μ

, and body force per unit mass

f

is

ρ (\partial_{t} u + u \cdot \nabla u) = \nabla \cdot σ + ρ f, \nabla \cdot u = 0,

where the Newtonian fluid stress tensor is

σ = - p I + 2 μ e, e = \frac{1}{2} (\nabla u + (\nabla u)^{T}) .

Taking the dot product with

u

gives

ρ u \cdot (\partial_{t} u + u \cdot \nabla u) = \nabla \cdot (σ u) - σ : \nabla u + ρ f \cdot u .

Incompressibility implies

σ : \nabla u = 2 μ e : e,

and converts the left side into a local time derivative plus the divergence of kinetic-energy flux. The divergence theorem therefore gives the kinetic-energy balance for an incompressible Newtonian fluid

\frac{d}{d t} \int_{Ω} \frac{1}{2} ρ ∣ u ∣^{2} d V = - \int_{\partial Ω} \frac{1}{2} ρ ∣ u ∣^{2} u \cdot n d S + \int_{\partial Ω} u \cdot σ n d S + \int_{Ω} ρ f \cdot u d V - 2 μ \int_{Ω} e : e d V .

The four terms are respectively outward advective transport of kinetic energy, mechanical power supplied by surface traction, power supplied by the body force, and irreversible viscous dissipation into heat.

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