Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 352 / 1

Letbe the rate-of-strain tensor and let $\dot{\mathbf{\gamma }}=2\mathbf{D}$. The invariant magnitude convention that reproduces the ordinary scalar rate in simple shear flow isThe tensor contraction makes this definition independent of a rigid rotation of coordinates.

A generalized Newtonian fluid responds instantaneously with stress determined only by the current scalar shear rate. It therefore cannot represent viscoelasticity, including stress relaxation, elastic recoil, memory, and rate-dependent normal stresses. It also cannot represent thixotropy, in which microstructure and viscosity evolve with the duration and history of shearing. Other omitted effects include hysteresis and genuine finite-time yielding.

Put $G=\Delta p/L>0$, with flow in the positive $z$ direction, so $dp/dz=-G$. Balancing pressure forces and wall shear on a cylindrical fluid volume of radius $R$ and length $L$ givesThus the wall stress is fixed before any constitutive equation is specified.

For fully developed pipe flow, the axial Cauchy equation reduces toRegularity at the axis removes the $1/r$ integration constant, soIts magnitude rises linearly from zero at the axis to ${\tau }_R$ at the wall.

The axial velocity $u(r)$ decreases toward the wall, so $\dot{\gamma }=-du/dr\ge 0$. For a power-law fluid, the stress magnitude isHenceIntegrating inward from the no-slip boundary condition $u(R)=0$ givesFor $n=1$ this reduces to the parabolic Hagen--Poiseuille profile.

The volumetric flow rate is $Q=2\pi {\int }_0^{R}u(r)rdr$. Integration by parts, using finite $u(0)$ and $u(R)=0$, givesBecause the stress magnitude is $\tau ={\tau }_{R}r/R$, change variables from $r$ to $\tau$:Therefore the required power is $n=3$. The fundamental theorem of calculus gives the pipe form of the Weissenberg–Rabinowitsch equation:Finally $\eta ({\dot{\gamma }}_R)={\tau }_R/{\dot{\gamma }}_R$, soA measured pressure-drop--flow-rate curve therefore recovers the wall viscosity without assuming a constitutive form.