Generalized Newtonian fluid

A generalized Newtonian fluid has instantaneous scalar constitutive law $\tau =\eta (\dot{\gamma })\dot{\gamma }$, with viscosity depending on the current shear rate but carrying no memory.

Differential shear viscosity measures the incremental shear-stress response around a given positive base shear rate. It differs from the secant shear viscosity mu. A regular locally stable stress-to-rate inversion needs positive differential viscosity. Zero differential viscosity makes a first-order stress perturbation singular, and negative differential viscosity identifies a decreasing constitutive branch.

Perturb the Couette flow between walls moving at +-U by a small pressure gradient -G. The mean shear rate remains U/h, and the stress variation across the gap is -Gy. Linearizing the monotone generalized Newtonian fluid law divides this variation by differential viscosity, giving the displayed velocity and flux per unit width. Require $Gh\ll (U/h){\eta }_d(U/h)$ and a smooth constitutive law on the perturbed rates. For a power-law fluid, the denominator is $nk(U/h{)}^{n-1}$.

The Weissenberg--Rabinowitsch equation recovers the wall shear rate of a generalized Newtonian fluid from a measured pressure-drop--flow-rate curve. For a circular pipe,