Non-Newtonian fluid

Remarks

In a simple shear flow $u = (\dot\gamma y, 0, 0)$ with shear rate $\dot\gamma > 0$, the shear stress is $\tau_{xy}$, and the apparent viscosity is $\eta(\dot\gamma) = \tau_{xy} / \dot\gamma$. A Newtonian fluid has $\eta(\dot\gamma) = \mu$ for all $\dot\gamma$.

Example: a power-law fluid with $\tau_{xy} = K \dot\gamma^{1/2}$ and $K > 0$ has $\eta(\dot\gamma) = K \dot\gamma^{-1/2}$. This is not constant, so the fluid is non-Newtonian; it is shear-thinning.

Non-example: a fluid with $\tau = 2 \mu D$ and $\mu = 10^{-3}$ (in SI units, roughly water) is Newtonian, because the ratio is the same constant in every flow.

Common classes are shear-thinning and shear-thickening fluids, viscoplastic fluids with a yield stress such as the Bingham plastic, and viscoelastic fluids such as the Oldroyd-B model.