= Specific internal energy
{title2=$e$}
Specific <internal energy> $e=U/M$ is <internal energy> per unit <mass>, measured in $\mathrm{J\,kg^{-1}}$. For a <perfect gas> with constant <specific-heat ratio> $\gamma$, choosing the constant reference as zero gives $e=c_vT=p/[(\gamma-1)\rho]$, where $c_v$ is the specific <heat capacity at constant volume>, $T$ the <temperature>, $p$ the <pressure> and $\rho$ the <mass density>.
For a single-component <fluid>, the local <first law of thermodynamics> can be written
$$
\rho\frac{De}{Dt}=-p\nabla\cdot\mathbf u+\boldsymbol\tau:\nabla\mathbf u-\nabla\cdot\mathbf q+r.
$$
The <material derivative> follows <velocity> $\mathbf u$; $\boldsymbol\tau$ is the <viscous stress tensor>, $\mathbf q$ the <heat flux density>, and $r$ an external heating rate per unit <volume>. Compression supplies <pressure> <work>, the stress contraction accounts for <viscous dissipation>, and convergence of <heat flux> raises <internal energy>. Writing only $-p\nabla\cdot\mathbf u+Q$ assumes that viscous heating is negligible or included in $Q$. An insulated sheared <fluid> can warm even when no <heat> crosses its boundary.https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node14.html[Richard Fitzpatrick, Energy Conservation, equations 1.72–1.74: local internal-energy balance and viscous heating]{ref}
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