Fragility in sheared dense granular suspensions
Rheology of granular suspensions
Hard particles, size $10\mu m - 1mm$, density-matched, in Newtonian fluid
- Rate-independant rheology
- Steady-state $\eta(\phi)$
- However, rich transient rheology
Fragility
Mechanical response depends strongly on the compatibility of the instantly applied load with respect to the load history.
Measuring fragility
1. Shear at constant stress in the $x$ direction
(with gradient along $y$), up to steady-state
2. Shear in a new direction making an angle $\theta$ with $x$
(keeping the gradient along $y$)
Simulation technique
[Mari et al, JOR 2014]
[Mari et al, PRE 2015]
Imposed velocity profile $U_{\infty}$, deformation rate tensor $E_{\infty} = \frac{1}{2}(\nabla U_{\infty} + \nabla U_{\infty}^{\mathrm{T}}) = \dot\gamma \hat{E}_{\infty}$
Eq. of motion (force balance)
$$\require{color}
\definecolor{mblue}{RGB}{0,40,200}
\definecolor{mred}{RGB}{200,40,0}
\definecolor{mgreen}{RGB}{30,140,30}
\definecolor{mpurple}{RGB}{100,0,100}
0=
\textcolor{mred}{- R_{\mathrm{FU}} \cdot (U-U_{\infty}) + \dot\gamma R_{\mathrm{FE}} : \hat{E}_{\infty}}
+ \textcolor{mblue}{F_{\mathrm{C}}}
$$
hydrodynamics
contacts
+ Imposed stress
$$\require{color}
\definecolor{mblue}{RGB}{0,40,200}
\definecolor{mred}{RGB}{200,40,0}
\definecolor{mgreen}{RGB}{30,140,30}
\definecolor{mpurple}{RGB}{100,0,100}
\sigma =
\textcolor{mred}{\dot\gamma (R_{\mathrm{SE}} + R_{\mathrm{SU}}\cdot R_{\mathrm{FU}}^{-1} \cdot R_{\mathrm{FE}}):\hat{E}_{\infty}}
+
\textcolor{mblue}{(R_{\mathrm{SU}}\cdot R_{\mathrm{FU}}^{-1} + xI)\cdot F_{\mathrm{C}}}
$$
Solve for $U$ and $\dot\gamma$.
$$\require{color} \definecolor{mblue}{RGB}{0,40,200} |\textcolor{mblue}{F^{\mathrm{t}}_{\mathrm{C}}}|<\mu |\textcolor{mblue}{F^{\mathrm{n}}_{\mathrm{C}}}| $$
Shear-rate maps
Rate maxima
Max. rate location
Rate along the $x$ direction
Primary flows with oscillatory cross shear
"Unthickening" of a shear-thickening suspension (oobleck/cornstarch)
Speeding sedimentation of a large intruder
Simple shear + oscillatory cross shear
$\dot\gamma_x = \dot \gamma_0$
$\dot\gamma_z = \dot \gamma_{\perp} \cos(\omega t)$
$\gamma_x = \dot \gamma_0 t$
$\gamma_z = \dot \omega^{-1} \gamma_{\perp} \sin(\omega t)$
2 dimensionless numbers:
- amplitude $\gamma^{\max}_z = \dot \omega^{-1} \gamma_{\perp}$,
- rate ratio $\dot \gamma_{\perp}/\dot \gamma_0$.
Simple shear + oscillatory cross shear
When $\gamma^{\max}_z \approx 1\%$ and $\dot \gamma_{\perp}/\dot \gamma_0\gg 1$,
viscosity drop actually very generic.
Simple shear + oscillatory cross shear
Viscosity drop due to contact number drop.
"Echo" experiment
"Echo" experiment
$\phi < \phi_{\mathrm{c}}$
$\phi > \phi_{\mathrm{c}}$
Simple shear + oscillatory cross shear
Viscosity drop for $\dot \gamma_{\perp}/\dot \gamma_0\gg 1$.
Viscosity drop when "echo" experiment + simple shear drift
