$$ v(r) = V_{max} \left[ 1 - \left(\frac{r}{R}\right)^2 \right] $$
Exact parabolic (Hagen–Poiseuille) profile for laminar flow.
$$ v(r) = V_{max} \left( 1 - \frac{r}{R} \right)^{\frac{1}{n}} $$
Power-law profile where $n$ is derived from the $V_{avg}/V_{max}$ ratio.
$$ \dot{\gamma} = \frac{4Q}{\pi R^3} = \frac{8V}{D} $$
Nominal wall shear rate — the value returned by most commercial viscometers;
equals the true wall shear rate for a Newtonian fluid in laminar pipe flow.
⚠ Valid only for Newtonian laminar pipe flow. For turbulent flow use the friction factor method.