Newtonian fluid

Remarks

The form is the general isotropic stress that is linear in $E$. For an incompressible flow, $\tr E = 0$ and the law becomes $\sigma = -p I + 2 \mu E$. Example: in the simple shear flow $\mathbf{u} = (\dot\gamma y, 0, 0)$, $E_{12} = \dot\gamma / 2$, so the shear stress is $\sigma_{12} = \mu \dot\gamma$, proportional to the shear rate. Non-example: a power-law fluid with shear stress $\tau = K \dot\gamma^{m}$ for a constant $m \ne 1$ is not Newtonian, since its stress is not linear in the shear rate. Substituting the Newtonian law into the balance of momentum gives the Navier–Stokes equations.