Shear fronts in shear-thickening suspensions

We study the fronts that appear when a shear-thickening suspension is submitted to a sudden driving force at a boundary. Using a quasi-one-dimensional experimental geometry, we extract the front shape and the propagation speed from the suspension flow field and map out their dependence on applied shear. We find that the relation between stress and velocity is quadratic, as is generally true for inertial effects in liquids, but with a prefactor that can be much larger than the material density. We show that these experimental findings can be explained by an extension of a phenomenological model originally developed to describe steady-state shear thickening. This is achieved by introducing a sole additional parameter: the characteristic strain scale that controls the crossover from startup response to steady-state behavior. The theoretical framework we obtain points out a linkage between transient and steady-state properties of shear-thickening materials.

Figure 1

Quasi-1D shear experiment. (a) Illustration of the experimental setup, consisting of a layer of cornstarch suspension (yellow) that floats on oil (blue). Dashed black lines represent rubber sheets confining the suspension. An acrylic plate with roughened surface was inserted in the middle of the container (at $x=0$ m) and moved with speed $U_0$. The dashed red box indicates the area used for data analysis. The shaded area in the side view illustrates the cross sectional area $S$. (b) Exemplary velocity profiles of a shear front for $\varphi =0.532$ and $U_0=0.46\pm 0.02$ m/s, propagating transversely to either side of the plate (dashed blue line). [(c), (d)] Front position $x_{\text{f}}$ and accumulated strain $\gamma$ as functions of time $t$. In panel (c), red lines show linear fits. In panel (d), time $t=0$ ms represents the time when $x=x_{\text{f}}$. Red lines show fits to a power law. Black dashed lines indicate the asymptotic accumulated strain ${\gamma }_{\infty }$. (e) Local shear rate calculated from the mean velocity profiles. Blue and pink represent the left and right branches, respectively, in panel (b).

Figure 2

Characteristics of propagating jamming fronts as function of pushing speed $U_0$. Experimentally obtained data for different packing fractions $\varphi$ are shown by solid symbols. In panels (a)–(c), dashed lines are from model calculations. (a) Dimensionless front propagation speed $k$. (b) Asymptotic accumulated strain ${\gamma }_{\infty }$. (c) Stress at the boundary $\mathrm{\Sigma }$. Solid circles are calculated by plugging experimentally measured $U_0$ and $k$ into Eq. (2). Dashed lines show the stress at the boundary obtained from the numerical calculations, which satisfy $\mathrm{\Sigma }\sim U_0^2$. (d) Maximum shear rate ${\dot{\gamma }}_{\text{max}}$. Open circles are from the model (the same color scheme is used as for data from the experiments). Black lines are power law fits.

Figure 3

Dimensionless front propagation speed $k$ (black) and asymptotic accumulated strain ${\gamma }_{\infty }$ (blue) as functions of packing fraction $\varphi$. The blue curve shows Eq. (11), and the black curve shows its reciprocal, both with ${\gamma }^{*}=0.197$. Data at $\varphi =0.556$ and 0.544 are represented by open circles.

Figure 4

The 1D model system for numerical calculation. For all data shown, the parameters are ${\varphi }_0=0.593,{\varphi }_{\text{m}}=0.452,{\eta }_0=13.6\mathrm{mPa}\mathrm{s},{\mathrm{\Sigma }}^{*}=20.4\mathrm{Pa}$, and ${\gamma }^{*}=0.197$. [(a), (b)] Velocity profiles at different times for $\varphi =0.521,U_0=0.01$ m/s (a) and 0.1 m/s (b). (a) Fluidlike regime. (b) Unstable regime. [(c)–(g)] Results in the jamming regime for $\varphi =0.532$ and $U_0=0.456$ m/s. The results can be directly compared with the experiment shown in Fig. 1. [(c), (d)] Velocity profiles and accumulated strain $\gamma$ at different times. (e) Front position $x_{\text{f}}$ as a function of time. The red line shows a linear fit. (f) Accumulated strain $\gamma$ as a function of time in element $n=80$. The red curve is a fit to a power law. The dashed black line indicates the asymptotic strain. (g) Local shear rate calculated from the mean velocity profile.

Figure 5

Flow profiles in the fluidlike regime and “front” position $x_{\text{f}}$ at $\varphi =0.521$ and $U_0=0.01$ m/s. (a) Numerical calculation based on the model. (b) Experimental data. (c) Position of the front, defined as the position where $u_{\text{y}}=0.45U_0$.

Figure 6

Front position as a function of time for different preshear. The fast plate speed was $U_0=360$ mm/s, and the slow preshear speed $U_{\text{pre}}$ varied as labeled in the plot. Positive $U_{\text{pre}}$ represents pre-shear in the same direction as $U_0$, and negative $U_{\text{pre}}$ was in the opposite direction.

Figure 7

Schematic illustration of the model system used for the numerical calculations. The black boxes represent fluid elements, the blue arrows represent the local velocities and the red arrows show the shear stress applied on the left and right boundaries of the $n\mathrm{th}$ element. The boundary conditions are $u_1=U_0$ and $u_{\text{N}}=0$. The width of an element is $\mathrm{\Delta }l$.

Figure 8

Evolution of $\mathrm{\Sigma }\text{−}\dot{\gamma }$ at $\varphi =0.521$. The blue curves show the $\mathrm{\Sigma }\text{−}\dot{\gamma }$ relation at different $\gamma$ (starting from zero, with strain increments of 0.0105 between adjacent curves), as predicted by the generalized Wyart-Cates model. The dashed black line corresponds to the relation at steady state $\left(\gamma \to +\infty \right)$. The thick black, green, and red lines show the relation between stress and shear rate in element no. 2, calculated numerically for different $U_0$ as indicated.

Figure 9

(a) Viscosity $\eta$ at different shear rates $\dot{\gamma }$ and packing fractions $\varphi$. Squares connected by thin solid lines represent stress-controlled experiments; circles connected by thin dashed lines represent shear-rate-controlled experiments. The predictions of the Wyart-Cates model are shown by the thick curves with the same color as the experiments. The dashed black line indicates a constant stress $\mathrm{\Sigma }=500\mathrm{Pa}$, which is provided by surface tension and corresponds to the upper limit of stress in steady-state experiments using our shear cell geometry. (b) The lower Newtonian viscosity ${\eta }_{\text{N},1}$ (solid circles) and higher Newtonian viscosity ${\eta }_{\text{N},2}$ (open circles) at different $\varphi$. The two curves show the best fit of ${\eta }_{\text{N},1}$ and ${\eta }_{\text{N},2}$ with Eq. (F2). The vertical dashed lines label ${\varphi }_{\text{m}}$ (left) and ${\varphi }_0$ (right) obtained from the fitting. (c) Relation between rescaled packing fraction $\mathrm{\Phi }$ and onset stress ${\mathrm{\Sigma }}_{\text{DST}}$. For each $\mathrm{\Phi }$, the corresponding $\varphi$ is labeled on the right. The solid black points are experimental data. The red curve is the prediction of the model for ${\mathrm{\Sigma }}^{*}=20.4\mathrm{Pa}$.

Figure 10

Dimensionless front propagation speed $k$ and asymptotic accumulated strain ${\gamma }_{\infty }$ at different packing fraction $\varphi$ obtained numerically at ${\gamma }^{*}=0.197$ and $U_0=1\mathrm{m}/\mathrm{s}$. The solid curves show Eq. (11) and its reciprocal at the same ${\gamma }^{*}$.

Figure 11

Comparison of $R\left(\varphi \right)$ obtained from the numerical calculation (open circles) with the prediction of Eq. (H1) (blue line). The dashed black lines show ${\varphi }_{\text{m}}$ and ${\varphi }_0$.