Apparent Wall Shear Rate

Shear rate is how fast one layer of melt slides past the next, measured as a velocity gradient — metres per second of velocity difference per metre of gap, which reduces to reciprocal seconds. In a round channel with a Newtonian fluid the gradient at the wall works out to exactly $4Q/\pi R^{3}$, and that expression is the x-axis of every capillary-rheometer flow curve ever plotted.

The cube on the radius is what makes this worth knowing. Halve the radius at the same flow rate and the shear rate goes up eightfold. That is why a pin gate shears a melt so much harder than the runner feeding it — the runner might be 5 mm across and the gate 1 mm, and the melt goes through the gate at over a hundred times the shear rate of the runner at the same flow. Degradation at the gate, brown streaks in a clear part, a measurable drop in molecular weight, splay and burn marks all start there. When a part shows a processing defect and the machine settings look reasonable, the gate is the first place to look, and this equation is why.

$R$ is a radius. Entering a diameter gives an answer eight times too small, and the mistake is quiet because the number that comes back is still perfectly plausible. Half the errors on this page are that one. The other half are unit slips between cubic centimetres per second and per minute, which is a factor of sixty.

Why it is called APPARENT. The $4Q/\pi R^{3}$ expression is derived from the Newtonian velocity profile — the parabola. A polymer melt is not Newtonian. Because it thins where it is sheared hardest, the fluid near the wall gets easier to move and the fluid at the centre effectively moves as a plug, so the real profile is blunter than a parabola: flatter in the middle and steeper at the wall. Steeper at the wall means the true shear rate there is higher than the Newtonian formula says. How much higher is exactly what the Rabinowitsch correction on the neighbouring page computes, and for a typical melt it is a factor of 1.5 or so — not a rounding error.

So report apparent rates as apparent, always. A flow curve built from one source's corrected data plotted against another's uncorrected data is not a flow curve; it is two different measurements sharing an axis label. The same discipline applies when handing numbers to a filling simulation, whose material model expects one or the other and will not warn you which it got.

One more use for the equation, going the other way. Pick a shear rate the material tolerates, and it gives you the flow rate a channel can carry — which is how a die or a runner gets sized. What rate a material tolerates is not a universal number: it depends on the polymer, on the residence time at temperature, and on what is in it. PVC will degrade at rates polypropylene shrugs off. A glass-filled compound will break its fibres down to a fraction of their length long before the matrix minds, and that fibre attrition is a permanent loss of the mechanical properties you paid for.