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

Define the viscous numberThe pressure relation givesUsing $\eta \hat{\mu }{\varphi }^2|\dot{\gamma }|/({\varphi }_m-\varphi {)}^2=\hat{\mu }{p}_s$, the shear stress at imposed pressure isThe material therefore behaves as a pressure-dependent Bingham plastic with yield stress $\hat{\mu }{p}_s$, while its steady concentration dilates as shear rate increases.

After a step in shear rate, the particles must rearrange and undergo shear-induced dilation or compaction. Because the sample is saturated, that volume change requires pore fluid to migrate through the packing, so pore pressure and effective particle pressure relax over a finite poro-viscous time. Measuring the transient stress can therefore constrain the permeability, and with an independently known permeability can constrain the suspension's compressibility or dilatancy law.