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Kinetic-energy balance for an incompressible Newtonian fluid

For a fixed domain

D

, body-force density

f

, and stress tensor

σ

, the kinetic energy satisfies

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

The last term is the nonnegative rate at which viscosity converts mechanical energy into heat.

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