Specific internal energy ($e$)

Specific internal energy $e=U/M$ is internal energy per unit mass, measured in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$. For a perfect gas with constant specific-heat ratio $\gamma$, choosing the constant reference as zero gives $e=c_{v}T=p/[(\gamma -1)\rho ]$, 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 writtenThe material derivative follows velocity $\mathbf{u}$; $\mathbf{\tau }$ 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.^[ref]