= Solution
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.
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