Liquid Migration in Shear Thickening Suspensions Flowing through Constrictions

Dense suspensions often become more dilute as they move downstream through a constriction. We find that as a shear-thickening suspension is extruded through a narrow die and undergoes such liquid migration, the extrudate maintains a steady concentration ${\varphi }_{\mathrm{out}}^{\mathrm{LM}}$, independent of time or initial concentration. At low volumetric flow rate $Q$, ${\varphi }_{\mathrm{out}}^{\mathrm{LM}}$ is a universal function of $Q/r_d^3$, a characteristic shear rate in the die of radius $r_d$, and coincides with the critical input concentration for the onset of LM, ${\varphi }_{\mathrm{in}}^{\mathrm{crit}}$. We predict this function by coupling the Wyart-Cates model for shear thickening and the “suspension balance model” for solvent permeation through particles.

Figure 1

Onset of LM. (a) Extruder schematic. In (b)–(d), $R_b=6.75$ and $r_d=1.7\mathrm{mm}$. (b) Concentration of the extrudate ${\varphi }_{\mathrm{out}}$ and material remaining in the barrel ${\varphi }_{\mathrm{bar}}$ varying ${\varphi }_{\mathrm{in}}$ for $Q=0.048\mathrm{mL}/\mathrm{s}$. (c) ${\varphi }_{\mathrm{out}}$ at varying intervals of plunger displacements at ${\varphi }_{\mathrm{in}}=0.52$, measured from collected extrudates (blue symbols), and measured postextrusion ${\varphi }_{\mathrm{bar}}$ (red symbols). Red lines: calculated ${\varphi }_{\mathrm{bar}}$, assuming constant ${\varphi }_{\mathrm{out}}$. Data for $Q=0.024\mathrm{mL}/\mathrm{s}$ (circles, solid line), $0.048\mathrm{mL}/\mathrm{s}$ (squares, dotted line), and $0.095\mathrm{mL}/\mathrm{s}$ (upside-down triangles, dashed line). (d) Extrusion pressure $P_{\mathrm{ext}}=F/\pi R_b^2$ vs plunger displacement for the same experiments in (c).

Figure 2

LM under varying flow conditions for ${\varphi }_{\mathrm{in}}=0.52$. (a) Extrudate solid mass fraction, ${\varphi }_{\mathrm{out}}(Q)$, for different die radii, $r_d$. (b) Liquid-migrated ${\varphi }_{\mathrm{out}}^{\mathrm{LM}}$ replotted against the scaling variable $Q/r_d^3$. Dotted lines highlight high-$Q$ data beyond the minimum in ${\varphi }_{\mathrm{out}}^{\mathrm{LM}}$.

Figure 3

LM state diagram. Gray diamonds: ${\varphi }_{\mathrm{out}}={\varphi }_{\mathrm{in}}$. Color symbols: $({\varphi }_c,Q/r_d^3)$ for various $({\varphi }_{\mathrm{in}},R_b)$. For $R_b=7.5\mathrm{mm}$, circles: ${\varphi }_{\mathrm{in}}=0.52$, squares: ${\varphi }_{\mathrm{in}}=0.44$, upside-down triangles: ${\varphi }_{\mathrm{in}}=0.46$ and triangles: ${\varphi }_{\mathrm{in}}=0.54$. For ${\varphi }_{\mathrm{in}}=0.52$, “plus” symbols: $R_b=6.75\mathrm{mm}$, open circles: $R_b=10\mathrm{mm}$ and “cross” symbols: $R_b=12.5\mathrm{mm}$. Same $r_d$ color scheme as Fig. 2. Solid lines: predicted LM phase boundary above ${\varphi }_m$. Black: calculated using $\alpha {\dot{\gamma }}_c(\varphi )$ and using SBM coupled with the WC model for any $r_d/a>70$ [see further Fig. 5 and text discussion]; color: calculated using $\mathrm{SBM}+\text{WC}$ below ${\varphi }_m$ for $r_d/a=70$ (yellow), and $r_d/a=140$ (green).

Figure 4

Cornstarch shear rheology. (a) Flow curves ${\eta }_r(\sigma )$ for $\varphi <{\varphi }_m$ (points) with WC fits (lines). (b) Low- and high-$\sigma$ viscosities ${\eta }_1(\varphi )$ (circles) and ${\eta }_2(\varphi )$ (squares), respectively, with ${\eta }_2(\varphi )$ taken from the maximum ${\eta }_r(\sigma )$ from flow curves in (a). Dotted and solid lines: ${\eta }_r=[1-\varphi /{\varphi }_{0,m}{]}^{-2}$, with ${\varphi }_m$ determined by fitting the full set of flow curves. (c) Flow curves for $\varphi >{\varphi }_m$ (points) with steady (filled symbols) and unsteady (open symbols) flow. Predicted backward-bending flow curves (lines), each with a “nose” at ${\dot{\gamma }}_c(\varphi )$. (d) Onset of fluctuations at $\varphi =0.48$ beyond ${\dot{\gamma }}_c$. Open symbols in (c) represent a time average of this unsteady data, and not included in WC fits [30].

Figure 5

A 1D model for LM. (a) Schematic of the “dilation zone” above the die. The resulting stress gradient causes a velocity difference $\mathrm{\Delta }u=u_p-⟨u⟩$ between the particles and mean flow. (b) Normal viscosity ${\eta }_n(\dot{\gamma })$ computed using WC fit parameters. Dotted lines: full backwards bending or $S$ shaped flow curves. Solid lines: profiles used to compute $\mathrm{\Delta }\tilde{u}$ with imposed jumps in the unsteady regime. (c) $\mathrm{\Delta }\tilde{u}(\varphi ,Q/r_d^3)$, computed using $\alpha =4.8$ and $r_d/a=140$. Red dashed line: ${\varphi }_m$. The contour $\mathrm{\Delta }\tilde{u}(\varphi ,Q/r_d^3)={\epsilon }^{†}=0.004$ gives the $\mathrm{SBM}+\mathrm{WC}$ phase boundary in Fig. 3 for $r_d/a=140$.