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In a parallel-plate rheometer, a point at radius $r$ on the upper plate moves at speed $\Omega r$. The thin-gap approximation therefore gives the local shear rateDefine the rim valueAn annulus of radius $r$ and width $dr$ has area $2\pi rdr$; its tangential force is $\tau (r)2\pi rdr$, and its moment arm is $r$. The measured torque is consequently

For a generalized Newtonian fluid, $\tau =\eta (\dot{\gamma })\dot{\gamma }$. Changing the integration variable from $r$ to $\dot{\gamma }$ givesThus the requested kernel isMultiplying by ${\dot{\gamma }}_R^3$ and taking a derivative with respect to ${\dot{\gamma }}_R$ turns the upper-limit contribution of the integral into the rim viscosity:ThereforeA measured torque curve and its slope hence determine $\eta$ over the range of imposed rim shear rates.

The experimental law impliesSubstitution givesThe graph is a decreasing rectangular hyperbola with a positive high-rate plateau $2/(\pi \alpha R^3)$ and a $1/\dot{\gamma }$ divergence at the origin. Equivalently,so the inferred material is a Bingham plastic with yield stress $3T_c/(2\pi R^3)$.

Let $z\in [0,h]$ measure distance through the thin Bingham plastic layer, with a stationary wall at $z=0$ and large-scale velocity $u_b$ at $z=h$. To leading lubrication order the shear stress ${\tau }_b$ is uniform across the layer. If $|{\tau }_b|\le {\tau }_c$, the material is unyielded and the no-slip wall makes the entire layer stationary. After yield,Integrating across the depth gives the Bingham sliding lawEquivalently, for nonzero sliding,

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.