Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 352 / 3 / b

LetAxial momentum balance gives shear-stress magnitude $|{\tau }_{rz}|=Gr/2$. The wall stress must exceed the pressure-dependent yield stress, so motion requiresWhen this holds, define the plug radiusThe no-slip solid velocity isIn the sheared annulus,so

The volume flux carried with the particle skeleton isBy Darcy law, the fluid moves relative to the solids at $kG/\eta$. The total mixture-volume flux is thereforeThis is explicit because $\varphi (r)$ is given above. For example, ifthen

At fixed $G=\Delta P/L$, increasing $p_s$ increases the plug radius and lowers both the particle velocity and porosity. The total flux decreases from a nearly particle-free Poiseuille value toward a nonzero Darcy seepage flux through a jammed packing. Particles stop when $a=R$, namely at

Put $\delta =1-p_s/p_s^{\ast }=1-a/R$. As $\delta \downarrow 0$, almost the whole cross-section is a plug at concentration ${\varphi }_m$ and speed $G(R-a{)}^2/(4\eta )$. The particle flux is thereforeThe jammed seepage flux is approximatelyIt appreciably changes the total flux when $Q_p\lesssim Q_{\mathrm{seep}}^{\ast }$, orBecause $k\ll R^2$, seepage matters principally in a narrow pressure interval just below jamming.

At fixed $Q_T$, the pressure gradient is no longer fixed. As $p_s\to 0$, the constitutive law gives a dilute particle fraction and hence $Q_p\to 0$. Raising $p_s$ first adds particles and increases their flux. At high $p_s$, the packing becomes dense and nearly jammed; maintaining $Q_T$ then requires a large pressure gradient that sends most of the liquid through the packing by Darcy seepage, while $Q_p$ falls back toward zero. Thus $Q_p(p_s)$ has an interior maximum, and every particle flux below that maximum occurs at two solid pressures.

On the low-$p_s$ branch the suspension is dilute, shear is distributed broadly, particle motion carries much of the total volume, and the required $\Delta P$ is relatively small. On the high-$p_s$ branch the suspension contains a large nearly jammed plug, particles move slowly, seepage carries a substantial fraction of $Q_T$, and a much larger $\Delta P$ is required. Equal particle flux therefore does not imply equal concentration, flow structure, or pumping cost.