Pressure-Driven Suspension Flow near Jamming

We report here magnetic resonance imaging measurements performed on suspensions with a bulk solid volume fraction (${\varphi }_0$) up to 0.55 flowing in a pipe. We visualize and quantify spatial distributions of $\varphi$ and velocity across the pipe at different axial positions. For dense suspensions (${\varphi }_0>0.5$), we found a different behavior compared to the known cases of lower ${\varphi }_0$. Our experimental results demonstrate compaction within the jammed region (characterized by a zero macroscopic shear rate) from the jamming limit ${\varphi }_m\approx 0.58$ at its outer boundary to the random close packing limit ${\varphi }_{\text{rcp}}\approx 0.64$ at the center. Additionally, we show that $\varphi$ and velocity profiles can be fairly well captured by a frictional rheology accounting for both further compaction of jammed regions as well as normal stress differences.

Figure 1

Evolution of the friction coefficient ($\mu =\tau /{\sigma }_n^{\prime }$) and suspension viscous number $I={\eta }_f\dot{\gamma }/{\sigma }_n^{\prime }$ as a function of the solid volume fraction $\varphi$ for a neutrally buoyant dense suspension of monodisperse spheres in a Newtonian fluid. Experimental results (symbols) obtained in annular shear cell [2] and a frictional rheology (lines) [14] extended to the jammed state ($I=0$, $\varphi \ge {\varphi }_m$). Figures adapted from [2, 14]. Details can be found in the Supplemental Material [16].

Figure 2

(a) Diagram of our experimental setup. The pump induces hydraulic pressure in the water tank, driving the suspension out of the fluid bag (red pattern) at a constant rate into the pipe toward the exit reservoir. The distance of the measurement from the pipe inlet, $L$, is measured from the bag to the coil which is located in the middle of the magnet. For long distance measurements, a circularly wound pipe was attached appropriately with a winding radius of 0.5 m (resulting in a Dean number of $\sim 8\times {10}^{-5}$, small enough not to disturb the 1D nature of the flow). (b) Diagram of the MRI pulse sequence. A shaped $\pi /2$ pulse with a slice selective gradient ($G_z$) and a broadband $\pi$ pulse were used. The slice thickness was 20 mm. For velocity measurements along the $z$ direction bipolar gradients were used, separated by 15 ms. rf represents radio frequency currents in the coil and ACQ stands for MRI signal acquisition. $\delta t$ ($\mathrm{\Delta }T$) represents the width of (distance between) the velocity gradient fields.

Figure 3

Axial development of the velocity and $\varphi (r)$ for ${\varphi }_0=0.35$ where the entrance length, $L_{\mathrm{ENT}}=2.7\mathrm{m}$. The velocity is normalized by the averaged velocity, $V_{\text{avg}}$, and the radial distance, $r$, is normalized by the pipe radius, $R$. The identical velocities for different $L$ indicate that the velocity has a much shorter entrance length compared to the $L_{\mathrm{ENT}}$. However, we have observed a gradual change in $\varphi (r)$ as a function of $L$. At $L=2\mathrm{m}$, $\varphi$ is indistinguishable from $\varphi$ at $L=3\mathrm{m}$, and thus, the flow is fully developed. Theoretical prediction using a frictional rheology [14], neglecting any normal stress difference, is also displayed.

Figure 4

The fully developed velocity and $\varphi$ profiles are shown for ${\varphi }_0=0.52$ and 0.55 (experimental results as circles). A blunted velocity profile is clearly observed in both cases and covers more than half of the pipe radius as shown in (a) and (b). The limits of the “jammed” regions, where the velocity is within 1% of its maximum value and where $\varphi$ is at ${\varphi }_{\text{rcp}}\approx 0.64\pm 0.01$, are indicated by downward arrows on the velocity and the solid volume fraction profiles, respectively. Theoretical predictions for a frictional rheology, extended for compaction above ${\varphi }_m$ or not, are also displayed, either neglecting any normal stress difference (${\omega }_2=0$) or assuming a normal stress difference dependence after [23] (${\omega }_2=0.5$). The relative particle size is also displayed.

Figure 5

2D MRI images of the fully developed $\varphi (r)$ distribution across the pipe cross section is shown for each ${\varphi }_0$.