Weissenberg–Rabinowitsch equation

The Weissenberg-Rabinowitsch equation recovers the wall shear rate of a generalized Newtonian fluid from a measured pressure-drop versus flow-rate curve. For steady fully developed flow in a horizontal circular pipe with radius $R$, length $L$ and no slip, define positive wall shear stressIf $Q({\tau }_R)$ is the volumetric flow rate, the wall shear rate isThe derivative samples the local slope of the measured flow curve.

To derive it, momentum balance gives $\tau (r)={\tau }_{R}r/R$. With $\dot{\gamma }(r)=-du/dr$ and $u(R)=0$, integrating the velocity profile givesDifferentiating ${\tau }_R^{3}Q$ proves the formula. This assumes a local stress/shear-rate relation at the specified measurement conditions; elastic memory or wall slip requires additional modeling.

For a power-law fluid with $\tau =K{\dot{\gamma }}^n$, $n>0$, one has $Q\propto {\tau }_R^{1/n}$ andThe Newtonian case $n=1$ recovers $4Q/(\pi R^3)$. The correction is useful because a non-Newtonian wall shear rate cannot generally be inferred from the Newtonian expression alone.