Power-Law Viscosity Ratio

Plot the viscosity of a polymer melt against shear rate on logarithmic axes and, over the middle of the range, you get a straight line sloping down. Ostwald in 1925 and de Waele in 1923 wrote that line down, and the result is the most-used constitutive equation in plastics processing. In the textbooks it appears as $\tau = K \dot{\gamma}^{\,n}$: shear stress equals a consistency index times shear rate to the power $n$. Dividing through by $\dot{\gamma}$ gives the viscosity form $\eta = K \dot{\gamma}^{\,n-1}$, and $n$ — the power-law index — is the slope of the flow curve. For polymer melts $n$ runs from about 0.15 for a heavily filled compound to nearly 1 for a nylon at low rate. At $n = 1$ the fluid is Newtonian.

This page does not use that form, and the reason is worth explaining properly, because it is a lesson about units rather than a limitation of the site.

Look at what $K$ has to be dimensionally. Stress is pascals; shear rate is reciprocal seconds; so for $\tau = K\dot{\gamma}^{\,n}$ to balance, $K$ must carry the dimension $\mathrm{Pa}\cdot\mathrm{s}^{n}$. The exponent in that unit is another variable in the same equation. A melt with $n = 0.3$ has a $K$ in Pa·s⁰·³; a melt with $n = 0.5$ has a $K$ in Pa·s⁰·⁵. These are not the same kind of quantity, and no conversion between them exists.

A units engine converts dimensions. It can express Pa·s, and it can express Pa·√m — that is the unit of fracture toughness, and this site does carry it, because that half-power is fixed, deliberately built for, and the same for every material anyone will ever type in. What no units engine can express is a unit whose dimension depends on a value the reader has not entered yet. The tempting shortcut — type $K$ as dimensionless and print "Pa·sⁿ" beside it as a cosmetic label — is precisely the pattern this site calls the units engine failing quietly. It would let a reader convert a number that cannot be converted, and it would offer no warning at all when they did. So we do not do it.

Instead the page carries the power law as a ratio between two points on the same flow curve: $\eta/\eta_{ref} = (\dot{\gamma}/\dot{\gamma}_{ref})^{\,n-1}$. Both viscosities are pascal-seconds, both shear rates are reciprocal seconds, $n$ is dimensionless, and every one of them converts honestly. It says exactly the same physics — it is the same equation with $K$ eliminated between two points — and it is how a rheologist reads a flow curve anyway: not as an intercept and a slope, but as a measured point and a slope you walk along.

If you arrived here holding a $K$ from a datasheet, here is what you have and how to use it. $K$ is the shear stress at a shear rate of exactly one reciprocal second — set $\dot{\gamma} = 1$ in $\tau = K\dot{\gamma}^{\,n}$ and there it is. Since $\eta = \tau/\dot{\gamma}$, the viscosity at 1 s⁻¹ is numerically $K$, in pascal-seconds. So enter $\dot{\gamma}_{ref} = 1$ s⁻¹ and $\eta_{ref} = K$ in Pa·s, and this page reproduces the textbook curve exactly. If the source quoted its stress in something other than pascals, convert the stress first and then use it. That is the whole bridge.

And here is the classic mistake, which the ratio form makes impossible: carrying a $K$ between two sources with different $n$. Two grades, two papers, two datasheets, each fitted independently — one reports $K = 12{,}000$ at $n = 0.28$ and the other $K = 9{,}000$ at $n = 0.41$, and somebody averages them or substitutes one into a calculation set up for the other. The numbers are not comparable, because they are not even the same unit. The error can run to an order of magnitude at working shear rates, and there is nothing in the arithmetic to flag it. Whenever you must compare two power-law fits, do it by evaluating both at a common shear rate you care about — which is exactly what this page does.

Finally, the straight line is only straight in the middle. Below roughly 0.1 to 1 s⁻¹, most melts flatten into a Newtonian plateau at the zero-shear viscosity $\eta_0$, and the power law extrapolated down there predicts a viscosity running away to infinity — which matters for anything gravity-driven, for sag in a parison, for slow flows in thick sections. At the top end, viscous heating and slip bend the curve the other way. The Cross and Carreau-Yasuda models exist to cover the plateau and the transition; the power law is the high-shear asymptote of both. Fortunately, high shear is where injection moulding and extrusion actually live, which is why so simple a model has survived a century of better ones.