 Viscous fluid flow

Viscous fluid flow includes the transfer of momentum by internal viscous stresses. For an incompressible Newtonian fluid with constant dynamic viscosity $\mu$, the extra stress is proportional to the rate-of-strain tensor:Combining this relation with momentum conservation gives the Navier-Stokes equation. Other constitutive equations, including those for non-Newtonian fluids, give different stress responses.

A simple example is Couette flow between plates separated by $h$, with one fixed and the other moving at speed $U$. A steady linear profile $u_x=Uy/h$ produces uniform shear stress ${\tau }_{xy}=\mu U/h$. The moving plate supplies power per area $\mu U^2/h$, which becomes viscous dissipation. ^[ref]

The Reynolds number $\mathrm{Re}=\rho UL/\mu$ compares advective momentum transport with viscous transport. Small values motivate Stokes flow; large values can still require thin viscous boundary layers near solid walls. Viscous flow therefore includes both smooth laminar flow and turbulent flow.