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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 number measures deformation accumulated over a viscoelastic relaxation time, often as the product of that time and a shear rate or an extension rate. Its precise rate convention should be stated. Unlike a Deborah number based on externally imposed temporal variation, it can remain nonzero in a steady flow. A weak-flow constitutive expansion generally requires both slow temporal change and small deformation over the memory time.

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 stressIf $Q({\tau }_R)$ is the volumetric flow rate, the wall shear rate isThe derivative samples the local slope of the measured flow curve.

To derive it, momentum balance gives $\tau (r)={\tau }_{R}r/R$. With $\dot{\gamma }(r)=-du/dr$ and $u(R)=0$, integrating the velocity profile givesDifferentiating ${\tau }_R^{3}Q$ 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 $Q\propto {\tau }_R^{1/n}$ andThe 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.

A yield-stress fluid can sustain a finite shear stress without steady flow, but flows when the applied loading exceeds its yield stress. The ideal Bingham plastic treats the unyielded region as rigid and gives, after yield,Here $\dot{\gamma }$ is the shear rate, ${\tau }_y$ the yield stress and ${\eta }_p$ the post-yield dynamic viscosity. Real materials can undergo small elastic deformations below the flow threshold; an ideal rigid branch does not describe every part of their response.

A pressure-driven experiment makes the spatial distinction visible. Where the local shear stress is below threshold, a plug flow of a yield-stress fluid can move without internal shearing, while adjacent yielded layers deform. If every point remains below threshold and the walls have no wall slip, the ideal material does not flow. Slowly evolving microstructure and thixotropy make the apparent threshold depend on preparation, waiting time and the loading protocol.