Rabinowitsch Shear-Rate Correction

Weissenberg and Rabinowitsch worked out in 1929 how to get the true wall shear rate of a non-Newtonian fluid out of a measurement that only gives you flow rate and radius. The answer for a power-law fluid is a single tidy factor: multiply the apparent rate by $(3n+1)/4n$.

The physical picture is worth carrying. A Newtonian fluid in a round pipe flows in a parabola: fastest at the centre, zero at the wall, and the velocity gradient rises smoothly across the radius. A shear-thinning melt does something different. Where the shear is highest — near the wall — the viscosity is lowest, so that material moves comparatively easily; near the centre the shear is low, the viscosity is high, and the melt drifts along nearly as a solid plug. The result is a blunted profile: flat across the middle, and all the velocity change crammed into a thin layer at the wall. Crammed into a thinner layer means a steeper gradient, which means a higher true wall shear rate than the parabola-derived formula predicts.

Check the factor at the ends. At $n = 1$ the fluid is Newtonian and $(3+1)/4 = 1$ — no correction, exactly as it must be. That identity is the quickest way to confirm you have the expression the right way up. At $n = 0.3$, a common figure for a filled polyolefin at moulding rates, the factor is $1.9/1.2 = 1.583$: the apparent rate understates the true one by nearly 60 %. At $n = 0.2$ it is 1.6 and still climbing. Since the correction is always $\geq 1$, the true rate is never below the apparent rate, and if your arithmetic says otherwise the two are swapped.

What it assumes, and where those assumptions fail. Fully developed laminar flow in a round channel; no slip at the wall; uniform temperature across the section. Take them one at a time.

No slip is the one filled and highly filled melts break routinely. A rigid PVC, a heavily mineral-filled compound, a metal-injection feedstock — these can and do slide along the die wall rather than sticking to it, and the plug motion that results is flow the correction has no way of seeing. Worse, filler particles migrate away from the wall in a shear field, leaving a thin lubricating layer of nearly pure matrix at exactly the place the measurement is most sensitive to. What comes out looks like extreme shear thinning and is not. The way to separate the two is a Mooney analysis: run the same material at the same wall shear stress through capillaries of several different radii, and if slip is present the apparent rate depends on radius in a way that pure viscosity cannot explain.

Uniform temperature fails at high shear because the melt heats itself. All the work done shearing it turns into heat, the material is a poor conductor so the heat has nowhere to go in the time available, and the melt near the wall runs measurably hotter than the melt at the centre. The viscosity there falls for a reason that has nothing to do with $n$, and it contaminates a high-rate flow curve in the same direction as slip. At the rates a real injection gate sees, viscous heating is not a small effect.

And the one that catches careful people: $n$ is a local slope, not a constant. It is the tangent to the flow curve at the shear rate you are working at, and the flow curve bends — it flattens toward $n = 1$ at low rate where the melt goes Newtonian. Using an $n$ fitted at 100 s⁻¹ to correct a measurement at 10,000 s⁻¹ is applying the wrong slope. Fit $n$ from points that bracket the rate you care about.