= Weissenberg–Rabinowitsch equation
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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 stress>
$$
\tau_R=\frac{\Delta p\,R}{2L}.
$$
If $Q(\tau_R)$ is the <volumetric flow rate>, the wall <shear rate> is
$$
\dot\gamma_R=\frac{3Q+\tau_R\,dQ/d\tau_R}{\pi R^3}.
$$
The derivative samples the local slope of the measured flow curve.
To derive it, <momentum> balance gives $\tau(r)=\tau_Rr/R$. With $\dot\gamma(r)=-du/dr$ and $u(R)=0$, integrating the <velocity profile> gives
$$
Q=\pi\int_0^R\dot\gamma(r)r^2\,dr
=\frac{\pi R^3}{\tau_R^3}\int_0^{\tau_R}\dot\gamma(\tau)\tau^2\,d\tau.
$$
Differentiating $\tau_R^3Q$ 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 $Q\propto\tau_R^{1/n}$ and
$$
\dot\gamma_R=\left(3+\frac1n\right)\frac{Q}{\pi R^3}.
$$
The 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.
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