3.3.1. Slow-acceleration phases

The slow-acceleration phases are visco-inertial transients. They can be described simply by considering the inertia of the cylinder, the constant applied torque and the frictionless Newtonian-effective drag of the suspension. Angular momentum theorem applied to the cylinder implies

(3.2)\begin{equation} I\dot{\varOmega} = \varGamma - 2{\rm \pi} hR^2\sigma(R,t),\end{equation}

with $I= 104\pm 1\ \mathrm {\mu }$N m s$^{2}$ the angular moment of inertia of the cylinder plus rotor of the rheometer, which is characterized using an in-built function of the rheometer (when the cylinder is in air). During the acceleration the shear stress is below $\sigma _c$ everywhere in the suspension, momentum has time to diffuse across the flow ($\rho R^2/\eta _0 \sim 10^{-1} {\rm s} \ll \varOmega /\dot {\varOmega }\sim 1\ {\rm s}$, with $\rho \simeq 1240\ {\rm kg}\ {\rm m}^{-3}$ the suspension density and $\eta _0 \simeq 0.38$ Pa s the non-frictional viscosity of the suspension) and the inertia of the flowing part of the liquid ($\sim \rho h R^4 \sim 10^{-7}\ {\rm N}\ {\rm m}\ {\rm s}^{2}$) is much smaller than the cylinder inertia ($I\sim 10^{-4}\ {\rm N}\ {\rm m}\ {\rm s}^{2}$). This means that the shear stress at the cylinder surface is almost equal to the steady, frictionless, Newtonian-effective value, $\sigma (R,t) \simeq \eta _0 \dot {\gamma }(R,t) \simeq 2\eta _0\varOmega (t)$, with $\dot {\gamma }(R,t) \simeq 2\varOmega (t)$ the steady shear rate for a large gap (see (3.1) in the limit $R/R_y\ll 1$). Integrating (3.2) with $\sigma (R,t) = 2\eta _0\varOmega (t)$ gives

(3.3) \begin{equation} \varOmega(t) = \frac{\varGamma}{4{\rm \pi}\eta_0 hR^2} (1-\mathrm{e}^{-\frac{\varGamma_c}{\varGamma}\frac{t-t'}{\tau_{acc}}}),\quad \text{with } \tau_{acc} = \frac{I\sigma_c}{2\eta_0\varGamma},\end{equation}

which is found to agree quantitatively with the measured acceleration phases, as shown in figure 5(a).

Figure 5.

High-torque unsteady regime. (a) Long-time evolution of the cylinder velocity (rheometer measurements, $\varGamma = 3.5 \varGamma _c, h = 35$ mm). Solid-line: (3.3) for the acceleration phases. (b,c) Focus on a deceleration phase resolved with high-speed imaging. Here, $t=0$ stands for the onset of deceleration ($\dot {\varOmega }=0$). (b) Cylinder velocity normalized by the velocity, $\varOmega _c$, at the deceleration onset. (c) Deceleration (left-hand vertical axis) and shear stress at the cylinder surface normalized by the discontinuous shear-thickening stress, $\sigma _c$ (right-hand).

3.3.2. Transitions from slow accelerations to fast decelerations

During the acceleration phase, the rate of acceleration of the cylinder decreases, which means that the stress, $(\varGamma -I\dot {\varOmega })/2{\rm \pi} h R^2$, actually transmitted to the suspension increases, until a value of approximately $\sigma _c$ is reached. Because of the slowness of the acceleration the condition is reached for

(3.4)\begin{equation} \varOmega(t) = \varOmega_c \approx \dot{\gamma}_c/2,\end{equation}

after the typical acceleration time $(\varGamma /\varGamma _c)\ln ({1}/({1-\varGamma _c/\varGamma }))\tau _{acc}$ set by (3.3). Equation (3.4) is valid for a suspension that would be strictly Newtonian up to $\sigma _c$. In practice, there is a continuous shear thickening for stresses just below $\sigma _c$, which augments the actual value of $\varOmega _c$ relative to $\dot {\gamma }_c/2$. For the volume fraction we consider ($\phi = 0.44$), the critical steady cylinder velocity based on the Wyart–Cates rheology (given by $\varOmega _c= \int _R^\infty ({\dot {\gamma }\,\mathrm {d} r}/{r}) = \frac {1}{2}\int _0^{\sigma _c} ({\mathrm {d} \sigma }/{\eta })$) is $\varOmega _c \simeq 1.35 \dot {\gamma }_c/2$. This value is found to be within a factor 2 of the critical velocity, $\varOmega _c \approx 2 \dot {\gamma }_c/2$, observed experimentally, with no trend on the suspension layer thickness (see inset of figure 7c).

Besides this non-trivial relation between the stress level at the cylinder, $\sigma (R)$, and the steady velocity, $\varOmega$, it must be noticed that the condition for unsteadiness is presumably more complex than $\sigma (R)\geq \sigma _c$. As will be discussed in § 4, the criterion is actually expected to depend slightly on the Reynolds number of the flow, as well as on the mode of destabilization. This latter point brings us to the fast-deceleration phases.

3.3.3. Fast-deceleration phases

The deceleration between two acceleration cycles lasts only a fraction of a second, which is shorter than the temporal resolution of the rheometer. To resolve the dynamics we use high-speed imaging of the cylinder and a PIV procedure, which yield both time-resolved measurements of the cylinder velocity $\varOmega (t)$ and the average stress at the cylinder surface, $\sigma (R,t)=(\varGamma -I\dot {\varOmega })/2{\rm \pi} h R^2$, from the measured deceleration rate, $\dot {\varOmega }$. A typical evolution of $\varOmega (t), \dot {\varOmega }(t)$ and $\sigma (R,t)$ is shown in figure 5. The cylinder decelerates from $\varOmega _c\approx 7.5$ to approximately 2 rad s$^{-1}$ within approximately $0.1$ s. This indicates that the drag of the suspension increases steeply. The average stress, $\sigma |_{r=R}$, peaks at almost $10^2 \sigma _c$, which is approximately 30 times as large as the constant stress, $\varGamma /2{\rm \pi} h R^2$, applied by the rheometer and is much larger than the shear stress, $\sigma _J\approx 10$ Pa, at which the suspension jams, according to the fitted Wyart–Cates rheology.

To understand the peak of drag experienced by the cylinder, the flow of the suspension has to be considered. Figure 6(a) presents the evolution of the suspension velocity field measured during the deceleration phase (see also supplementary movie 2). It shows that the beginning of the cylinder deceleration ($\dot {\varOmega }=0$, defining $t=0$) coincides with the emergence of a velocity front, which propagates from the cylinder to the outer wall with a velocity of order 1 m s$^{-1}$. The front is found to nucleate over a small portion of the cylinder surface. After a short transient that will be discussed below, it propagates within an angular sector of approximately 180$^\circ$ around the nucleation direction (the latter being random with respect to both the cylinder and the laboratory). As shown from the velocity profiles along the front propagation direction presented in figure 6(b), this velocity front is also a ‘thickening’ front separating an inner high-stress and low-shear (thickened) region from the outer, unperturbed, low-stress region. The corresponding thickening front position, $R_{f}$, the cylinder angular velocity, $\varOmega$, and the deceleration, $-\dot {\varOmega }$, are plotted vs time in figure 6(c–e), respectively (the front position is defined as the radius, in the front propagation direction, at which $u_\theta = R\varOmega /2$). These analyses confirm that the deceleration phase is triggered by the inception of the thickening front and show that the front propagates at a close to constant speed, $V_{f}$, which is transverse to the suspension velocity and much larger than the latter. This front is of the same nature as those termed jamming or shear fronts, reported previously in imposed-velocity configurations (Peters & Jaeger Reference Peters and Jaeger2014; Peters et al. Reference Peters, Majumdar and Jaeger2016). For high levels of stress these fronts have been rationalized as kinematic shocks with a typical shock width, $\sim 10\eta _0/\rho V_{f}$, limited by viscosity, and a typical propagation velocity, $V_{f} \sim {\rm \Delta} u/\gamma _0$, proportional to the increment of transverse velocity of the suspension across the shock, ${\rm \Delta} u$, and with the inverse of the relaxation strain scale, $\gamma _0$ (Han et al. Reference Han, Wyart, Peters and Jaeger2018). The fronts observed in the present study are found to match these scalings. The typical shock width, $\approx 10$ mm, is of order $10\eta _0/\rho V_{f}\sim 5$ mm, and the relative propagation velocity, $V_{f}/R\varOmega _c\approx 7$, is consistent with reported values (Peters & Jaeger Reference Peters and Jaeger2014; Han et al. Reference Han, Wyart, Peters and Jaeger2018).

Figure 6.

Spatio-temporal analysis of a deceleration phase ($\varGamma = 3.5\varGamma _c, h=35$ mm). (a) Velocity fields at $t= 8$, 28, 48 and 68 ms after the cylinder has started decelerating ($t=0$, see also supplementary movie 2). Colour encodes the relative norm of the velocity, $\sqrt {u_\theta ^2 + u_r^2}/R\varOmega _c$. (b) Radial profiles of the azimuthal velocity component in the direction where the thickening front nucleates and propagates (white arrow in a). (c–e) Time evolution of the thickening front position (c), of the cylinder velocity normalized by the deceleration onset velocity (d) and of the cylinder deceleration rate (e). Red dashed line: arrival time, $t_{out}$, of the front at the outer (vessel) wall. Green-dashed line: release time, $t_{rel}$, at which the next acceleration phase starts.

Figure 7.

Early part of the deceleration phase: maximal deceleration. (a) Normalized maximal deceleration of the cylinder or, equivalently, maximal shear stress in the suspension vs thickness of the suspension layer.(b) Maximal deceleration time normalized by the front arrival time at the outer wall, $t_{out}$, vs $h$. (c) Normalized velocity of the cylinder during a deceleration phase ($h=35$ mm). Insets from top to bottom: cylinder velocity at the onset of deceleration, front velocity and front angular width, ${\rm \Delta} \theta$, vs $h$. (a–c) The pink solid lines show ((3.7)–(3.10a,b)) with $t_{out} = (R_{out}-R)/V_{f}$ for a homogeneous nucleation of the thickening front ($R_{f}(t) /R = 1+V_{f} t$). The black dashed lines show the same equations for a localized nucleation ($R_{f}(t) /R = 1+\sqrt {L^2+(V_{f} t)^2}-L$ with $L^2=R^2+(h/2)^2$, see text).

The important point, now, is to elucidate how the interaction between the front propagation and the cylinder motion sets the oscillation and the effective drag of the suspension. To do so, we model the velocity field and the stresses based on the features of the front observed experimentally. We assume that the velocity of the suspension remains essentially azimuthal ($u_\theta \gg u_r$), that the front is sharp, such that the shear stress immediately upstream of the thickening front vanishes ($\sigma |_{r>R_{f}}\ll \rho V_{f} u_\theta$), that azimuthal stress gradients can be neglected and that the angular momentum imparted by the rheometer during the deceleration is small ($\varGamma \times 0.1\ {\rm s} \ll I\varOmega _c$). In this case, angular momentum conservation of the cylinder and the suspension gives

(3.5)\begin{equation} I\varOmega(t)+S(t)\varOmega(t) = I\varOmega_c,\end{equation}

where $I\varOmega _c$ is the initial angular momentum of the cylinder (before deceleration) and $I\varOmega (t)$ and $S(t)\varOmega (t)$ are the current momenta of the cylinder and of the suspension in the thickened region, respectively. Equation (3.5) simply expresses that the cylinder slows down as it shares momentum with the thickened region. Experimental observations (figure 6a) indicate that the front propagates typically over a half-space, i.e. within an angle span ${\rm \Delta} \theta \approx {\rm \pi}$. Inside the thickened region ($r< R_{f}(t)$) the azimuthal velocity is roughly uniform and close to $R\varOmega (t)$ (figure 6b). Based on these observations the angular momentum of the suspension reads

(3.6)\begin{equation} S(t)\varOmega(t) = \frac{{\rm \Delta}\theta\rho h R}{3} (R_{f}^3(t) - R^3 ) \varOmega(t), \end{equation}

and, from (3.5), the cylinder velocity is expected to follow

(3.7)\begin{equation} \varOmega(t) = \frac{\varOmega_c}{1 + \alpha[R_{f}^3(t)/R^3 - 1]},\quad \text{where } \alpha = \frac{{\rm \Delta}\theta \rho h R^4}{3 I}\end{equation}

represents the characteristic angular moment of inertia of the suspension (at scale $R$) relative to that of the cylinder. Considering $R_{f}(t) = R + V_{f} t$, as a consequence of the approximately constant front speed, $V_{f}$ (figure 6c), gives the following time evolution:

(3.8)\begin{equation} \varOmega(t) = \frac{\varOmega_c}{1 + [(\alpha^{1/3} + t/\tau_{dec})^3 - \alpha]},\quad \text{with } \tau_{dec} = \frac{R}{V_{f}}\alpha^{{-}1/3}.\end{equation}

In the present case, where the cylinder inertia is large ($\alpha ^{-1}\gtrsim 10^{2}$ in the experiments), the maximal deceleration rate is

(3.9)\begin{equation} -\dot{\varOmega}_{max} \simeq \frac{2^{4/3}}{3}\frac{\varOmega_c}{\tau_{dec}} \sim \left(\frac{{\rm \Delta}\theta \rho h R^4}{I}\right)^{1/3} \frac{V_{f} \varOmega_c}{R}, \end{equation}

which is reached at

(3.10a,b)\begin{equation} R_{f,{max}} \simeq (2\alpha)^{{-}1/3} R, \quad \text{and} \quad t_{max} \simeq (2^{{-}1/3} -\alpha^{1/3})\tau_{dec} \sim \tau_{dec},\end{equation}

when the moment of inertia of the thickened region ($\sim {\rm \Delta} \theta \rho h R {R_{f}}^3$) compares with $I$.

Equation (3.9) indicates that the maximal deceleration is expected to scale with the suspension layer thickness as $-\dot {\varOmega }_{max} \propto h^{1/3}$ or, equivalently, that the maximal stress, $\sigma _{max} \propto -\dot {\varOmega }_{max} /h$, follows $h^{-2/3}$. We test this prediction experimentally by varying $h$ systematically from 6 to 35 mm, which is the thickness range accessible with our set-up, while keeping a fixed value of $\varGamma /\varGamma _c = 3.5$. For each thickness, three deceleration events are analysed with high-speed PIV of both the cylinder and the suspension. The latter allows us to verify that the front characteristics remain close to the heuristic features we adopted, i.e. ${\rm \Delta} \theta \approx {\rm \pi}$ and a front velocity, $V_{f} \approx 1.1\ {\rm m}\ {\rm s}^{-1}$, independent of $h$ (insets of figure 7c). We also verify that the cylinder velocity at the onset of deceleration ($\varOmega _c$) is essentially independent of $h$.

Figure 7(a) shows that the maximal deceleration rate obtained experimentally is well captured by (3.9), both in magnitude and trend. It also indicates that the peak stress during the deceleration, $\sigma _{max} = -I\dot {\varOmega }_{max}/({\rm \Delta} \theta h R^2)$, can reach as much as 300 $\sigma _c$ for the thinnest layer. As for the instant of the maximal deceleration, the magnitude of $t_{max}$ is reasonably well captured by (3.10a,b) (see figure 7b), which consistently predicts that, for all thicknesses investigated, the maximal deceleration of the cylinder occurs before the time $t_{out}$ at which the front reaches the outer wall ($t_{max}/t_{out}<1$). However, the model is found to underestimate the exact value, especially for thick layers. We will comment this point in the next paragraph.

Finally, figure 7(c) compares (3.8) with the evolution of the cylinder velocity observed during a whole deceleration event for a thickness $h=35$ mm. The time origin is taken at the onset of deceleration ($\dot {\varOmega }=0$). The match in the maximal deceleration rate is recovered but the deceleration rate at the beginning of the deceleration phase is found to be overestimated by (3.8), which indicates that the initial growth of the thickened region is not as fast as assumed by the model. This suggests that the thickened region nucleates at a single locus on the cylinder wall and has to extend across the layer and azimuthally before propagating radially only. This hypothesis is consistent with the initial growth of the angular span of the front, ${\rm \Delta} \theta$, observed at the layer surface (see figure 6a). It is also in line with the slight underestimation of $t_{max}$ by the model. Indeed, as shown in figure 7(b,c), a better match is found for both the early stage of deceleration and the arrival time, $t_{out}$, by considering heuristically that the average front radius follows $R_{f}(t)/R = 1 + \sqrt {L^2+(V_{f}t)^2}-L$, with $L^2 = R^2+(h/2)^2$, to account for the azimuthal component, $\sim R$, and the vertical component, $\sim h/2$, an initially localized front has to propagate through. Importantly, the maximal deceleration is almost unchanged by the lag, provided $h$ remains small relative to $R_{f,{max}}$ (see figure 7a).

As for the late stage of the deceleration, the model deviates from the measurements close to the time, $t_{out}$, at which the thickening front reaches the outer wall. This is consistent because the leading edge of the front is not stress free anymore. However, the deviation is not towards a stronger deceleration of the cylinder, as could be expected for a suspension fully jammed from the cylinder to the wall, but towards an increasingly milder deceleration until the cylinder eventually re-accelerates approximately $t_{rel} \approx 120$ ms after the onset of deceleration, i.e. $\approx 60$ ms after $t_{out}$ (see figures 6e and 7c). This brings us to the question of the release mechanism terminating the fast-deceleration phases.

3.3.4. Transitions from fast decelerations to slow accelerations

The deceleration of the cylinder results from the extension of the shear-thickened region. On the other hand, shear thickening is conditioned to the shear stress sustained by the suspension. If the latter typically decreases below the critical value $\sigma _c$ at any radius inside the thickened region, the front propagation should stop. This should in turn lead to a large drop in shear stress all over the suspension. Since at this time the deceleration has already reduced the cylinder velocity and the shear rate in the suspension significantly relative to the critical condition for the nucleation of a new thickening front, the cylinder is expected to enter a new acceleration phase (where the suspension is essentially unstressed because most of the applied torque is absorbed by the accelerating cylinder). Therefore, understanding the end of the deceleration phases requires understanding of the stress field in the suspension.

To do so it is insightful to consider, first, the case of a sufficiently large vessel, for which no interaction with the outer wall is expected. Conserving the same assumptions as in § 3.3.3 and applying the angular momentum theorem to the cylinder and thickened suspension up to a fixed radius $r$, the shear stress in the expanding thickened region ($r< R_{f}$) follows

(3.11)\begin{equation} \sigma(r,t) ={-}\frac{(I+S_{{<}r})\dot{\varOmega}}{{\rm \Delta}\theta h r^2} ={-}\frac{1+\alpha(r^3/R^3-1)}{r^2/R^2}\frac{I\dot{\varOmega}}{{\rm \Delta}\theta hR^2}, \end{equation}

with $S_{< r}\varOmega (t)$ the angular moment of the thickened suspension at radii below $r$. The stress is minimal at the same radius, $r_{min} \simeq (2/\alpha )^{1/3}R$, regardless of time, and it evolves as (using 3.8)

(3.12)\begin{equation} \sigma_{min}(t) \simeq \frac{3\rho R \varOmega_c V_{f}}{2^{2/3}}\left(\frac{t}{\tau_{dec}}\right)^{{-}4}.\end{equation}

This means that the release condition, when the shear stress becomes typically lower than $\sigma _c$, is expected at a time and a front position given by

(3.13a,b)\begin{equation} t_{rel,\infty} \simeq \tau_{dec}\left(\frac{3}{2^{2/3}}\frac{V_{f}}{R\varOmega_c}Re\right)^{1/4}, \quad R_{f,rel} \simeq R \left(\frac{3}{2^{2/3}}\frac{V_{f}}{R\varOmega_c}Re \right)^{1/4}\alpha^{{-}1/3},\end{equation}

respectively, where

(3.14)\begin{equation} Re = \frac{\rho R^2 \varOmega_c^2}{\sigma_c} \sim \frac{\rho R^2 \dot{\gamma}_c}{\eta_0},\end{equation}

is the typical Reynolds number at the onset of the deceleration phase. Since $\sigma _c \sim \eta _0 \dot {\gamma }_c$ and both $\varOmega _c$ and $V_{f}$ scale like $\dot {\gamma }_c$, the Reynolds number is essentially the square of the ratio between the cylinder size $R$ and the characteristic viscous length $\sqrt {\eta _0/\rho \dot {\gamma }_c}$ of the suspension. In the experiments $Re \approx 5$ implies $R_{f,rel}\gtrsim 12R$, which means that the release stress, $\approx \sigma _c$, is not expected to be reached before the front arrives at the outer wall ($R_{out} \simeq 6.8R$), even for the thickest suspension layer. This is consistent with the observation that, in all the experiments, the re-acceleration of the cylinder ($t=t_{rel}$) is posterior to the arrival of the front at the outer wall ($t=t_{out}$).

In this case the interaction of the front with the wall has to be considered. High stresses should develop as the thickened region reaches the wall, and one could expect that the cylinder would re-accelerate only after a reflected shear front propagating inward (i.e. a stop front with even higher stresses and a low velocity for $r>R_f$) has reached the cylinder or decreased the stress below $\sigma _c$. Such an inward-propagating wave of low velocity is indeed observed systematically in experiments (see figure 8a). However, modelling the influence of a stop front on the cylinder deceleration under the same assumptions as in § 3.3.3 does not match, quantitatively, the observations for $t>t_{out}$. Indeed, as discussed in Appendix A, if the dissipation law of the front is assumed to be unchanged by the reflection on the wall, the cylinder deceleration is expected to be slightly larger for a reflected front than for an ever outward-propagating front (sufficiently large vessel), whereas the observed deceleration is actually milder. This suggests that departure from axial symmetry may have a more significant effect on the stress field in the suspension after the interaction of the front with the wall than before, or that the front interaction with the wall leads to either an elastic response of the suspension (Larsen et al. Reference Larsen, Kim, Zukoski and Weitz2010) or to out-of-plane motion, facilitating the release of stress.

Figure 8.

Late part of the deceleration phase: release. (a) Velocity profiles in the suspension (azimuthal velocity component, same direction as in figure 6) between the front arrival time, $t_{out}$ ($R_{f} = R_{out}$), and the release time, $t_{rel}$ (onset of re-acceleration, $\dot {\varOmega } = 0$). (b) Release velocity. The mean observed value is $\varOmega _{rel} \simeq 0.24 \varOmega _c$. (c) Release time relative to the front arrival time (time origin at the onset of the deceleration phase). Symbols: experiments. The solid line indicates the upper bound $t_{rel} \leq 2t_{out}$ of (3.15), see text. The mean observed value is $t_{rel} \simeq 2.0t_{out}$.

Nevertheless, the observation of an inward-propagating stop front suggests that the typical re-acceleration time (relative to the onset of deceleration) should follow

(3.15)\begin{equation} t_{rel} \lesssim 2\frac{R_{out}-R}{V_{f}}.\end{equation}

Equation (3.15) is found to be consistent with the measurements presented in figure 8(b). The observed re-acceleration time is found to be $t_{rel} \approx 2t_{out}$ for all layer thicknesses without significant dependence on the latter. Similarly, the cylinder release velocity is found to be close to $\varOmega _{rel} \approx \varOmega _c/4$ without significant dependence on $h$. While this is compatible with the idea that the front reflection at the wall leads to the stress release, which allows the cylinder re-acceleration, further investigations would be required to be conclusive.