Define the <viscous number>
$$
I_v=\frac{\eta|\dot\gamma|}{p_s}.
$$
The pressure relation gives
$$
\frac\phi{\phi_m-\phi}=I_v^{-1/2},
\qquad
\boxed{\phi=\frac{\phi_m}{1+\sqrt{I_v}}}.
$$
Using $\eta\widehat\mu\phi^2|\dot\gamma|/(\phi_m-\phi)^2=\widehat\mu p_s$, the shear stress at imposed pressure is
$$
\boxed{\tau=\eta\dot\gamma
+\widehat\mu p_s\operatorname{sgn}\dot\gamma}.
$$
The material therefore behaves as a pressure-dependent <Bingham plastic> with yield stress $\widehat\mu p_s$, 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.

Solution

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

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Shear-induced dilation is the tendency of a dense granular or suspended-particle packing to increase its volume under shear. In a saturated confined system it requires pore-fluid migration and can delay stress adjustment.


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 Incoming links: Shear-induced dilation