2.1. Plane-shear geometry

We consider flows in a plane-shear geometry as illustrated in figure 1(a). It comprises two parallel walls shearing a two-dimensional assembly of grains, prescribing both shear rate and normal stress. The wall separation distance is $2H$ . They are made out of aligned grains having the same mechanical and physical properties as the flowing grains. The wall velocity in the shear direction $x$ , $V_{w}$ , is controlled such that the macroscopic shear rate,

is constant in time. The wall separation distance, $2H(t)$ , is free to vary in time in order to ensure that the external stress, ${\it\sigma}_{w}$ , is balanced by the total force of the grains contacting the wall, $F^{int}$ . $H(t)$ satisfies inertial dynamics, $M\ddot{H}=F^{int}-{\it\sigma}_{w}\,Ld$ , with $M$ being the total mass of the walls.

2.2. Grain properties

The grains are disks of density ${\it\rho}$ , mass $m$ and diameter $d$ . A polydispersity of $\pm 20\,\%$ is introduced in the grain diameter to avoid crystallisation. Grains interact with their neighbours through inelastic and frictional contacts. These interaction laws are detailed in appendix A. They only depend on the elastic coefficient of the grains, $E$ (Pa), Newton’s coefficient of restitution, $e$ , and the coefficient of friction between two grains, ${\it\mu}_{g}$ . Unless otherwise specified, the grain-to-grain friction and restitution coefficient are set to ${\it\mu}_{g}=0.5$ and $e=0.5$ , respectively. The value of the deformation number, ${\it\kappa}={\it\sigma}_{w}/E=10^{-3}$ , is fixed to be in the rigid-grain limit, characterised by small elastic deformations (see appendix A).

2.3. Numerical experiments

The numerical experiments comprise two steps: the preparation of a steady flow, and the steady flow during which measurements are performed. The preparation of the steady flow consists of applying both normal stress, ${\it\sigma}_{w}$ , and shear rate, $\dot{{\rm\Gamma}}_{M}$ , on a loose configuration of grains, initially set randomly with no contact. The system width, $H(t)$ , decreases and the solid fraction ${\it\phi}_{M}$ , the ratio of the grain surface area to the total system area, increases. They both eventually reach steady-state values with small time fluctuations.

In the following, the analysis will focus on steady flows in systems of varying width, $H/d=10$ , 20, and 40 (see figure 1 b), with different values of the macroscopic inertial number $I_{M}$ ,

ranging from $10^{-3}$ to $10^{-1}$ , which corresponds to flows in the dense regime (GDR MiDi 2004; da Cruz et al. Reference da Cruz, Emam, Prochnow, Roux and Chevoir2005).

2.4. Macroscopic quantities and profiles

In the following, the analysis will be based on quantities such as stresses, velocities, velocity gradient and solid fraction in the steady regime. These quantities will be averaged in time by considering a set of 100 snapshots, the system undergoing a shear deformation of one between two consecutive snapshots. We will distinguish the macroscopic quantities corresponding to a spatial average over the entire system, denoted by a subscript $M$ , from their profile along the transverse direction $y$ , obtained by averaging in the direction of shear, $x$ , and in time, using the procedure described in appendix B.

We denote as ${\it\phi}_{M}$ , ${\it\tau}_{M}$ , ${\it\sigma}_{M}$ and $\dot{{\rm\Gamma}}_{M}$ the macroscopic solid fraction, shear and normal stresses and shear rate; ${\it\phi}(y)$ , ${\it\tau}(y)$ , ${\it\sigma}(y)$ and $\dot{{\it\gamma}}(y)$ refers to their profiles. The relationship between macroscopic quantities and their profile is, taking the example of the solid fraction, ${\it\phi}_{M}=(1/2H)\int _{-H}^{H}{\it\phi}(y)\text{d}y$ .

3.1. Local friction and dilatancy laws

The local visco-plastic constitutive law describing dense granular flow can be expressed by a friction law combined with a dilatancy law (GDR MiDi 2004; da Cruz et al. Reference da Cruz, Emam, Prochnow, Roux and Chevoir2005; Jop et al. Reference Jop, Forterre and Pouliquen2006; Forterre & Pouliquen Reference Forterre and Pouliquen2008; Andreotti et al. Reference Andreotti, Forterre and Pouliquen2013). These two laws specify how the effective friction coefficient, ${\it\mu}$ , the ratio of the shear and normal stresses, and the solid fraction, ${\it\phi}$ , depend on the shear rate $\dot{{\it\gamma}}$ . In the dense regime, both laws are approximately linear functions of the local shear rate:

The coefficients ${\it\mu}_{s}$ , $a$ , ${\it\phi}_{0}$ and $b$ are positive numerical constants which depend on the nature of the grains and of their interactions, such as friction, shape and adhesion (da Cruz et al. Reference da Cruz, Emam, Prochnow, Roux and Chevoir2005; Rognon et al. Reference Rognon, Roux, Wolf, Naaim and Chevoir2006, Reference Rognon, Roux, Naaïm and Chevoir2007, Reference Rognon, Roux, Naaim and Chevoir2008).

3.2. Macroscopic friction and dilatancy laws

In steady sheared flows between two parallel plates, the momentum balance predicts that there should be no stress gradient in the transverse direction $y$ , ${\it\sigma}(y)={\it\sigma}_{w}={\it\sigma}_{M}$ and ${\it\tau}(y)={\it\tau}_{M}$ . According to (3.1) and (3.2), the local shear rate and solid fraction should thus be constant: $\dot{{\it\gamma}}(y)=\dot{{\rm\Gamma}}_{M}$ and ${\it\phi}(y)={\it\phi}_{M}$ . Integrating the local friction and dilatancy laws would thus lead to a macroscopic expression for the constitutive behaviour of the form

We measured such macroscopic laws in our systems (see figure 2). For a given system width $H$ , the macroscopic friction and dilatancy laws approximately exhibit a linear dependence on the macroscopic inertial number $I_{M}$ (2.2), consistent with (3.4) and (3.5). However, with the same inertial number $I_{M}$ different values of the macroscopic coefficient of friction and solid fraction are obtained in systems of differing width $H$ . Larger systems leads to higher coefficient of friction ${\it\mu}_{M}$ and lower solid fraction ${\it\phi}_{M}$ .

The local visco-plastic constitutive law as expressed in (3.4) and (3.5) does not predict this system size effect. It suggests that walls may affect the material rheology, and the next sections will focus on characterising and understanding such an effect.

Figure 2. Macroscopic constitutive behaviour measured in systems of differing sizes $H/d=$ $10$ (▪), 20 (●) and 40 (▴): (a) friction law and (b) dilatancy law.

4.1. Demonstrating wall effects

Let us first describe a particular flow obtained with $I_{M}=10^{-2}$ and $H/d=20$ . We observe that the local friction coefficient, ${\it\mu}(y)={\it\tau}(y)/{\it\sigma}(y)$ , is constant (see figure 3 a). This is consistent with the momentum balance applied to this geometry, which predicts no stress gradient in the $y$ -direction. The solid fraction ${\it\phi}(y)$ is also mostly constant, but slightly decreases in the wall vicinity. The important result is that, although stresses are constant throughout the layer, the velocity profile $v_{x}(y)$ is not linear (see figure 3 b) but exhibits an S-shape. The corresponding shear rate profile,

is thus not constant. It increases significantly when approaching the walls while staying approximately constant in the central region of the flow. Similar S-shape profiles in plane-shear flows were reported in da Cruz et al. (Reference da Cruz, Chevoir, Roux, Iordanoff, Dalmaz, Lubrecht, Dowson and Priest2004), GDR MiDi (2004), Miller et al. (Reference Miller, Rognon, Metzger and Einav2013) and Miller (Reference Miller2014).

The fact that the shear rate $\dot{{\it\gamma}}(y)$ varies near the wall while the stresses are constant shows that the local friction and dilatancy laws in (3.1) and (3.2) fail to describe the flow in these regions. An additional effect of the walls leads to an increase of the local shear rate in the wall vicinity.

Flows were computed varying two grain contact parameters: their coefficient of friction ${\it\mu}_{g}$ and the coefficient of restitution $e$ . Figure 4 shows that similar non-constant shear rate profiles are obtained when varying these parameters, even with frictionless grains ( ${\it\mu}_{g}=0$ ). This suggests that neither the friction nor restitution coefficient is responsible for the curvature of the velocity profile.

Figure 3. Time-averaged profiles along the $y$ -direction within a steady flow ( $H=20d$ , $I_{M}=0.01$ ). (a) Friction coefficient, ratio of shear to normal stresses ${\it\mu}(y)={\it\tau}(y)/{\it\sigma}(y)$ (bottom axis, dark grey curve, blue online) and solid fraction ${\it\phi}(y)$ (top axis, light grey curve, red online). (b) Velocity profile $v_{x}(y)$ normalised by the wall velocity $V_{w}$ (bottom axis, dark grey curve, blue online) and the linear velocity profile $V_{w}y/H$ is shown for comparison (dashed line); and shear rate $\dot{{\it\gamma}}(y)$ normalised by the nominal shear rate $\dot{{\rm\Gamma}}_{M}=V_{w}/H$ (top axis, light grey curve, red online).

Figure 4. Shear rate profiles along the $y$ -direction within steady flows ( $H=20d$ , $I_{w}=0.01$ ) for (a) differing contact restitution coefficient $e$ (0.1, dark grey, blue online; 0.5, light grey, red online; 0.9, black) and (b) differing contact friction coefficient ${\it\mu}_{g}$ (0, dark grey, blue online; 0.5, light grey, red online; 0.9, black).

4.2. Shear rate variation near walls

Figure 5(a) shows the variation in shear rate as a function of the distance to the wall ${\rm\Delta}y$ :

for $H=40d$ and $I_{M}=10^{-2}$ . The shear rate is maximum near the wall and approximately constant in a first layer of size ${\it\delta}\approx 2d$ . For ${\rm\Delta}y>{\it\delta}$ , the shear rate decreases when the distance to the wall ${\rm\Delta}y$ increases. In this zone, the shear rate decay can be represented by a power law:

For large distance to the wall, ${\rm\Delta}y>\ell$ , the shear rate seems again constant.

Figure 5(b) shows the shear rate profile $\dot{{\it\gamma}}({\rm\Delta}y)$ for a flow with the same inertial number as in figure 5(a), but in a narrower system, $H=20d$ . The first layer of constant shear rate for ${\rm\Delta}y<{\it\delta}\approx 2d$ and the second layer of decaying shear rate for ${\rm\Delta}y>{\it\delta}\approx 2d$ are still present. However, in this case the shear rate decays until the middle of the flow and there is thus no central zone with a constant shear rate.

Figure 5. Shear rate $\dot{{\it\gamma}}$ versus distance to the wall ${\rm\Delta}y$ for two simulations with similar inertial number, $I_{M}=10^{-2}$ , but different widths: (a) $H/d=40$ and (b) $H/d=20$ . The wall is at ${\rm\Delta}y=0$ while the centre of the sheared layer is at ${\rm\Delta}y=H$ . The shear rate, $\dot{{\it\gamma}}$ , is expressed in inverse unit of inertial time, $t_{i}=d\sqrt{{\it\rho}_{g}/{\it\sigma}_{M}}$ . Thick grey lines (red online) represent simulation results and thin black lines the empirical model defined in (4.4).

4.3. Empirical model capturing shear rate profiles

The profiles shown on figure 5 suggest how the shear rate can be modelled in each of the three zones, I, II and III. Accounting for the fact that the shear rate is continuous, these profiles can be captured by the following empirical model:

The numerical results shown in figures 5(a) and 5(b) are both captured by this empirical model using ${\it\delta}=2d$ , $\ell =20d$ and $\dot{{\it\gamma}}_{b}=0.75\,\dot{{\rm\Gamma}}_{M}$ , where $\dot{{\it\gamma}}_{b}$ is the bulk shear rate.

4.4. Range of wall perturbations

We denote as $\ell$ the range of the wall perturbations, the region comprising zones I and II ( $0<{\rm\Delta}y<\ell$ ) where the shear rate differs from the constant bulk value found in zone III.

For the flow conditions of figure 5(a), the wall perturbation range is remarkably large, $\ell \approx 20d$ . Within the same conditions, $I_{M}=10^{-2}$ , but in a narrower system, $H=20d$ , zone III no longer exists (see figure 5 b) which can be interpreted as an overlap of the perturbations from each wall.

Figure 6 shows the velocity and shear rate profiles for flows with various inertial numbers $I_{M}$ and various widths $H$ . For large systems, the wall perturbation range seems to increase as the inertial number decreases, and never exceeds the system width. For narrow systems, the wall perturbation spans the entire system width.

In the largest system, $H=40d$ , the size $\ell$ of the wall perturbation is estimated for flows of different internal number $I_{M}$ by adjusting the empirical model (4.4) to the measured shear rate profiles. Figure 7(a) shows the best fit of the empirical model to the measured shear rate profiles. All the shear rate profiles were captured by the model using similar values of ${\it\delta}$ ranging between 1.5 and 2. Conversely, the range of the wall perturbation, $\ell$ , was found to vary from $6.5d$ to $25d$ .

Figure 7(b) shows the values of wall perturbation range $\ell$ thus estimated as a function of the macroscopic inertial number $I_{M}$ . For macroscopic inertial number ranging from $6\times 10^{-3}$ to $10^{-1}$ , the size of the wall perturbation exhibits a power law:

For the lowest inertial numbers, $\ell$ could not be estimated since zone III did not exist even in the largest systems.

Figure 6. Time-averaged velocity and shear rate profiles for flows with various inertial numbers $I_{M}$ and widths $H$ as indicated by arrows. Same colour code as figure 3(b).

Figure 7. (a) Shear rate profiles versus distance to the wall, ${\rm\Delta}y$ , for three different inertial numbers, $I_{M}=10^{-2},3\times 10^{-2},10^{-1}$ . Grey curves (red online) represent simulation results and thin black lines the empirical model defined in (4.4). Symbols mark the width, $\ell$ , of the boundary layer for each system. (b) Width of the boundary layer, $\ell$ , versus inertial number, $I_{M}$ ; the line represents the function $\ell /d\approx 2I_{M}^{-1/2}$ .

5.1. Local constitutive law and viscosity length scale

The local constitutive laws, as expressed by the friction and dilatancy laws (3.1) and (3.2), fail to capture the wall perturbations as they predict a constant shear rate throughout the layer when the stresses are constant: $\dot{{\it\gamma}}=({\it\mu}-{\it\mu}_{s})/ad\sqrt{{\it\rho}/{\it\sigma}}$ . They nonetheless contain important information regarding the characteristic length scale involved into the diffusion of momentum. This length scale appears when one formulates these laws using the kinetic viscosity ${\it\nu}$ ( $\text{m}^{2}~\text{s}^{-1}$ ), which is also referred to as momentum diffusivity, defined as

Using the kinetic viscosity, the friction law becomes

which is equivalent to a Bingham constitutive law, with a yield stress ${\it\mu}_{s}{\it\sigma}$ increasing with the normal stress, and a kinetic viscosity ${\it\nu}\propto d\sqrt{P/{\it\rho}}=d^{2}\dot{{\it\gamma}}/I$ . This kinetic viscosity can be expressed as a function of a typical time scale, $\dot{{\it\gamma}}^{-1}$ , and a length scale $\ell _{{\it\nu}}$ as

Note that, considering the friction law (3.1), the typical length scale (5.4) can also be expressed as function of the quantity ${\it\mu}-{\it\mu}_{s}$ : $\ell _{{\it\nu}}\propto d/\sqrt{{\it\mu}-{\it\mu}_{s}}$ .

Equation (5.4) highlights an important feature of dense granular flows: at low inertial number, the momentum diffuses on a typical length scale much larger than the grain size $d$ . Remarkably, the scaling of this length with $I$ is identical to the scaling of ‘cooperativity’ length with $I$ , measured and predicted in different configurations within the KEP non-local model framework (Kamrin & Koval Reference Kamrin and Koval2012; Bouzid et al. Reference Bouzid, Trulsson, Claudin, Clément and Andreotti2013; Mansard et al. Reference Mansard, Colin, Chaudhuri and Bocquet2013).

Quantitative predictions of the viscosity in the zone affected by walls requires identifying the long-range mechanisms underpinning the momentum diffusion. Different analyses have been developed to address this question. Two non-local models, namely the NL process and KEP models, were derived from an analysis of the internal dynamics and the mode of stress propagation within the materials. By contrast, the EV model was developed from the analysis of the internal kinematics and the momentum transport within a non-affine displacement field. Let us now discuss these models and their ability to capture the observed long-range wall perturbations.

5.2. Non-local self-activated process (NL model)

This model, introduced in Pouliquen & Forterre (Reference Pouliquen and Forterre2001, Reference Pouliquen and Forterre2009), is based on the following ideas: (i) the shear between two layers of grains leads to topologic rearrangements, also called plastic events, that locally release the stresses; (ii) the associated stress relaxation induces some stress fluctuations that propagate through the network of contact with a typical length scale; (iii) the propagating stress fluctuations contribute to triggering plastic rearrangements far from their origin. In this model, the viscosity at a given point is derived from the sum of the stress fluctuations which have propagated to this point, originating from remote plastic events. As a consequence, the local viscosity depends on the rate of plastic events in the surroundings, and is lower when the shear rate in the surroundings is higher.

Consistent with Hartley & Behringer (Reference Hartley and Behringer2003), Majmudar & Behringer (Reference Majmudar and Behringer2005), da Cruz et al. (Reference da Cruz, Emam, Prochnow, Roux and Chevoir2005), Tordesillas, Zhang & Behringer (Reference Tordesillas, Zhang and Behringer2009), Tordesillas et al. (Reference Tordesillas, Lin, Zhang, Behringer and Shi2011), Azéma & Radjaï (Reference Azéma and Radjaï2014), figure 8 shows that contact force networks within the flows exhibit long length scales, which mechanically connect remote grains to the walls. Further, the contact network seems to be less connected for high values of the inertial number. This supports the relevance of relating non-local behaviour to the force network.

The NL model successfully predicts three observations that the local constitutive law alone is unable to capture: the shear rate profiles within flow in pipes; the dependence of the angle of arrest on the thickness of the granular layer in the inclined plane geometry; more generally, the existence of some flow in zones where the stresses are below the yield stress when there is some flow in the surroundings.

However, the role of the wall is not directly accounted for in the NL model formulation. An additional hypothesis needs to be made regarding the ability of the wall to reflect or absorb stress fluctuations. Considering that the wall absorbs the stress fluctuations, the model predicts that a layer near the wall is subjected to less fluctuation than a layer far from the wall, and its viscosity is higher. Accordingly, the predicted shear rate is lower near the wall, which is not consistent with our results. Assuming that the wall reflects stress fluctuations back into the flow, an opposite trend is predicted that is consistent with our results. Unfortunately, there is no clear rationale to determine whether or not and to what extent walls may reflect stress fluctuations. While the NL model may be able to capture our results, further understanding of the effect of the wall on stress propagation is needed to draw robust conclusions.

Figure 8. Snapshots of the contact network for various inertial numbers, $I_{w}$ , and system widths $H$ , as indicated by the arrows. The lines connect the centre of each pair of contacting grains. Their thickness is proportional to the normal contact force.

5.3. Non-local KEP model

This model was introduced in Goyon et al. (Reference Goyon, Colin, Ovarlez, Ajdari and Bocquet2008) and Bocquet, Colin & Ajdari (Reference Bocquet, Colin and Ajdari2009) to rationalise the flow of dense emulsions flowing in micro-channels, and was then adapted to dense granular flows in Kamrin & Koval (Reference Kamrin and Koval2012) and Bouzid et al. (Reference Bouzid, Trulsson, Claudin, Clément and Andreotti2013). It is based on the same conceptual ideas as the NL model, considering that (i) the local viscosity is inversely proportional to the local rate of plastic events; (ii) a plastic event at one location may trigger other plastic events at remote locations through the propagation of stress fluctuations through the contact network.

This model successfully predicts shear rate profiles of soft glassy materials such as emulsions and dry granular matter in geometries where the stresses are not spatially constant, such as pipes, inclined plane and plane shear with gravity, including in zones where the stresses are lower than the yield conditions predicted by their respective local constitutive laws.

In Mansard et al. (Reference Mansard, Colin, Chaudhuri and Bocquet2013), the non-local KEP model is able to capture wall effects in plane-shear flows of a model dense emulsion. This system has some similarities with the one discussed here: it comprises disks sheared between two rough walls in the absence of gravity, a configuration where stresses are spatially constant. However, it differs from our system in some important points: the disks are soft and interact by strongly dissipative contacts, and the shear was performed at fixed volume and the solid fraction investigated is large, ranging from 90 % to 110 %, which is made possible by large overlaps of soft disks. As a result of these differences, the model emulsion did not satisfy a visco-plastic constitutive law such as (3.1) and (3.2), but instead a Herschel–Bulkley constitutive law typical of soft glassy materials such as foams and emulsions. Nonetheless, in spite of these differences, the reported velocity profiles exhibit an S-shape qualitatively similar to that obtained in our systems. The KEP model, in the plane-shear configuration, can be expressed as

where ${\it\xi}$ is referred to as the ‘cooperativity’ length, and represents the spatial extent of the non-locality; $\dot{{\it\gamma}}(y)$ is the local shear rate, $\dot{{\it\gamma}}_{b}$ the bulk shear rate, which corresponds to the value of the shear rate in a flow with no heterogeneities, e.g. far from the wall in plane-shear flows. To predict the shear rate profile $\dot{{\it\gamma}}(y)$ , (5.5) can be solved by introducing a boundary condition, the value of the shear rate at the wall, $\dot{{\it\gamma}}_{w}$ . The predicted shear rate profile $\dot{{\it\gamma}}(y)$ is then

This prediction involves the bulk shear rate, $\dot{{\it\gamma}}_{b}$ , the shear rate at the wall, $\dot{{\it\gamma}}_{w}$ , and the cooperativity length, ${\it\xi}$ . In Mansard et al. (Reference Mansard, Colin, Chaudhuri and Bocquet2013), this equation was shown to predict S-shape velocity profiles matching the numerical results for a given value of the imposed shear stress and different values of system width $H$ , using a values of ${\it\xi}=2.5d$ and $\dot{{\it\gamma}}_{w}=1.7\dot{{\it\gamma}}_{b}$ .

Figure 9 (ai–aiii) shows a comparison of the KEP model prediction with our numerical results for three different flows. Equation (5.6) was plotted after measuring $\dot{{\it\gamma}}_{b}$ as the shear rate in the middle of the flow in the largest system, and $\dot{{\it\gamma}}_{w}$ as the maximum value of the shear rate near the walls. Numerical data are well represented using ${\it\xi}/d=5.4$ and $\dot{{\it\gamma}}_{w}/\dot{{\it\gamma}}_{b}=2.2$ for flow (ai), ${\it\xi}/d=5.4$ and $\dot{{\it\gamma}}_{w}/\dot{{\it\gamma}}_{b}=2.45$ for flow (aii), and ${\it\xi}/d=2.2$ and $\dot{{\it\gamma}}_{w}/\dot{{\it\gamma}}_{b}=2$ for flow (aiii).

A fundamental component of the non-local KEP model is identifying the inverse relation between the viscosity and the rate of plastic rearrangement. This was verified experimentally in Jop et al. (Reference Jop, Mansard, Chaudhuri, Bocquet and Colin2012) and numerically in Mansard et al. (Reference Mansard, Colin, Chaudhuri and Bocquet2013), by showing that zones of low viscosity exhibit a higher rate of particle rearrangement. Consistently, it was shown that zones of low viscosity exhibit higher level of velocity fluctuations, thought to be due to the higher rate of rearrangement (Mansard et al. Reference Mansard, Colin, Chaudhuri and Bocquet2013).

Figure 10, column (ii), shows the profile of velocity fluctuations ${\it\delta}v$ measured in our systems, defined as the standard deviation of the grain velocity fluctuations $\boldsymbol{v}_{i}^{\prime }=\boldsymbol{v}_{i}-\boldsymbol{v}(y_{i})$ . Results indicate that the velocity fluctuations are constant in the central region of the flow where the shear rate is constant. Near the walls, where the shear rate is higher and the apparent viscosity lower, the velocity fluctuations do not increase but decrease. This result differs from the measurements reported in Mansard et al. (Reference Mansard, Colin, Chaudhuri and Bocquet2013), which may be due to the differences in the systems considered.  While it does not affect the validity of the KEP model, which relies on the relationship between viscosity and rate of plastic rearrangement, this result suggests that there should exist a mode of grain rearrangement that allows for a higher rate of rearrangement together with lower velocity fluctuations when approaching the walls.

Figure 9. Comparison of shear rate profiles between numerical results (grey lines, red online) and and model prediction (black lines). (a) The KEP model prediction obtained from (5.6) is shown for three systems: (ai)  $H/d=40$ , $I_{M}=0.1$ ; (aii)  $H/d=40$ , $I_{M}=0.01$ ; and (aiii)  $H/d=20$ $I_{M}=0.01$ . (b) The EV model prediction obtained from (5.14) is shown for the same three systems, and with values of ${\it\beta}$ of (bi) 3.3, (bii) 2, and (biii) 1.85. Shear rates are expressed in units of inverse of inertial time.

Figure 10. Vorticity and vorticity length scale in three flows: (a)  $H/d=40,I_{M}=0.1$ ; (b)  $H/d=40,I_{M}=0.01$ ; (c)  $H/d=20,I_{M}=0.01$ . Profiles of shear rate $\dot{{\it\gamma}}$ , vorticity $w$ and $\unicode[STIX]{x1D60D}_{xy}$ are shown in column (i). Profiles of velocity fluctuations ${\it\delta}v$ are shown in column (ii). Profiles of the deduced vorticity length scale $\ell _{w}$ are shown in column (iii), measured according to (5.9) and normalised by the value at the centre of the flow, $\ell _{w}^{b}$ .

5.4. Vorticity and vorticity length scale

Let us now analyse the internal kinematics at the scale of the grains to seek a mechanism consistent with these observations. We focus on the local velocity gradient tensor defined as:

which can be measured locally around each grain following a method introduced by Marmottant, Raufaste & Graner (Reference Marmottant, Raufaste and Graner2008) and detailed in appendix B. By definition, the component $\unicode[STIX]{x1D60D}_{yx}$ is equal to the shear rate $\dot{{\it\gamma}}$ , and the anti-symmetric component of $\unicode[STIX]{x1D641}_{{\it\alpha}{\it\beta}}$ is the vorticity $w$ :

Figure 10, column (i), shows the profiles of $\unicode[STIX]{x1D60D}_{xy}(y)$ , $\dot{{\it\gamma}}(y)$ and $w(y)$ for three different flows ( $H/d=20$ and $40$ , $I_{M}=0.1$ and $0.01$ ). The component $\unicode[STIX]{x1D60D}_{xy}$ appears to always be much smaller than the shear rate $\dot{{\it\gamma}}$ . As a consequence, there is a direct link between the local shear rate and the vorticity: $w(y)\approx \dot{{\it\gamma}}(y)/2$ . Accordingly, the vorticity $w(y)$ is higher near the wall and constant in the centre of the flow. This property was consistently observed with all the flows investigated with different width $H/d$ and inertial number $I_{M}$ .

Figure 11 illustrates snapshots of the local vorticity fluctuation field $w_{i}^{\prime }=w_{i}-\dot{{\rm\Gamma}}_{M}/2$ . It shows layers of high vorticity near the wall (light grey, red online) and a lower vorticity in the centre of the flow (dark grey, blue online). Further, figure 11 shows that the local vorticity exhibits some spatially correlated patterns comprising large zones with higher or lower vorticity than their surroundings. Note that previous analyses focusing on velocity fluctuations instead of vorticity revealed similar patterns (Blair & Kudrolli Reference Blair and Kudrolli2001; Radjaï & Roux Reference Radjaï and Roux2002; Mueth Reference Mueth2003; Pouliquen Reference Pouliquen2004; Brito & Wyart Reference Brito and Wyart2007; Behringer et al. Reference Behringer, Daniels, Majmudar and Sperl2008; Lechenault et al. Reference Lechenault, Dauchot, Biroli and Bouchaud2008; Liu & Nagel Reference Liu and Nagel2010; Abedi & Rechenmacher Reference Abedi and Rechenmacher2011; Rechenmacher & Abedi Reference Rechenmacher and Abedi2011; Abedi, Rechenmacher & Orlando Reference Abedi, Rechenmacher and Orlando2012; Richefeu, Combe & Viggiani Reference Richefeu, Combe and Viggiani2012; Miller et al. Reference Miller, Rognon, Metzger and Einav2013; Miller Reference Miller2014). Measuring the typical size of these correlated structures is challenging given their seemingly complex shape and large size distribution. As a consequence, statistical indicators such as spatial correlations of the vorticity field were found not to be conclusive to measure such a length scale.

Figure 11. Vorticity field snapshots for different system widths, $H$ , and different inertial numbers, $I_{w}$ , as indicated by the arrows. The local vorticity around each grain is measured according to the method detailed in appendix B.

Figure 12. Vorticity length scale in the bulk of the flow, $\ell _{w}^{b}$ , as a function of the macroscopic inertial number, $I_{M}$ , measured in the largest systems, $H/d=40$ . The line represents the power law in (5.10).

Nonetheless, a typical vorticity length scale can be indirectly estimated from the combined measurements of the velocity fluctuation profile and the vorticity profile. The underlying idea is that a vortex of size $\ell _{w}$ rotating at a frequency $w$ induces a level of velocity fluctuations of the order of ${\it\delta}v\propto w\ell _{w}$ .  Thus, the typical length scale associated with the vorticity can be estimated at any location $y$ as

Figure 10, column (iii), shows the corresponding profiles of vorticity length scale $\ell _{w}(y)$ . In the bulk of the flow, where the shear rate is constant, the vorticity length is approximately constant: $\ell _{w}(y)\approx \ell _{w}^{b}$ . By contrast, close to the walls, the vorticity length scale decreases. Figure 12 shows the values of the bulk vorticity length $\ell _{w}^{b}$ measured in the largest system ( $H/d=40$ ) as a function of the macroscopic inertial number $I_{M}$ . It follows a power law:

which is remarkably similar to the wall perturbation range empirically determined in (4.5), to the length scale of granular momentum diffusivity deduced from the local friction law in (5.3), and to the cooperativity length scale measured and predicted in the KEP framework (Kamrin & Koval Reference Kamrin and Koval2012; Bouzid et al. Reference Bouzid, Trulsson, Claudin, Clément and Andreotti2013; Mansard et al. Reference Mansard, Colin, Chaudhuri and Bocquet2013).

To summarise, we extracted a vorticity length scale from the measurements of the velocity fluctuations. When approaching the walls, where the shear rate of the material increases, this length scale is found to decrease. This trend can be understood since, close to the walls, the lack of space mean that the correlated structures cannot freely develop: the presence of the wall tends to truncate the largest possible vortex in the flow. Let us now present a non-local model based on vorticity that could quantitatively capture the wall perturbations.

5.5. EV model

The EV model was first developed in the context of turbulent flows of Newtonian fluids to explain the increase in apparent viscosity observed between turbulent and laminar flow regimes. The underlying ideas are (i) there is an intrinsic viscosity ${\it\nu}_{0}$ due to the transfer of momentum by molecular collisions in both regimes and (ii) in turbulent regimes, vortices develop and lead to an additional mode of momentum transfer and an associated viscosity ${\it\nu}^{eddy}$ . The effective viscosity ${\it\nu}^{eff}$ is thus expressed as the sum of the two contributions:

Prandtl’s mixing length model provides an estimation of the typical momentum diffusivity associated with a vortex of size $\ell _{w}$ rotating at a frequency $w$ , as

In Staron et al. (Reference Staron, Lagrée, Josserand and Lhuillier2010) and Miller et al. (Reference Miller, Rognon, Metzger and Einav2013), the EV model was adapted to dense granular flows in the inclined-plane and plane-shear geometries. Both approaches consist of considering two mechanisms of momentum transfer in granular materials: pairwise grain collisions and large-scale vortex rotations. Accordingly, the apparent granular viscosity (5.3) can be expressed by

where ${\it\beta}$ denotes a numerical constant reflecting the actual contribution to momentum transfer of vortices. The first term represents the diffusion of momentum by grain collision and the second term that by rotation of vortices of size $\ell _{w}$ . Note that the corresponding effective length scale of momentum diffusion would then be $\ell _{{\it\nu}}=d\sqrt{1+{\it\beta}(\ell _{w}/d)^{2}}$ . Introducing this expression for viscosity into the friction law (3.1) leads to the following prediction of the shear rate profile:

with $A=({\it\mu}-{\it\mu}_{s}/a)(1/d\sqrt{{\it\rho}/{\it\sigma}})$ .

In Staron et al. (Reference Staron, Lagrée, Josserand and Lhuillier2010) and Miller et al. (Reference Miller, Rognon, Metzger and Einav2013), no vorticity length scale was measured or estimated. This length scale was assumed to be given by some function of the distance to the walls. Various functions of the distance to the wall were introduced, so that the model (5.13) would predict the correct viscosity variation near walls. However, there is no established rationale for the choice of this function, and different choices may equally well represent the results.

In our systems, a typical vorticity length scale $\ell _{w}$ was estimated from the analysis of the grain velocity fluctuations, and can thus directly be introduced in (5.14). Figure 9 (bi–biii) shows that the prediction of this model can successfully match the numerical results. The parameter $A$ was directly measured as the shear rate at the wall, $\dot{{\it\gamma}}(y=H)$ , using the fact that the vortex size $\ell _{w}(H)$ tends to zero, as shown on figure 10; ${\it\beta}$ is thus the only fitting parameter, with values of the order of 2.

The empirical model (4.4) for wall perturbations can be interpreted within the EV framework. Zone III corresponds to the region in the middle of the flow where the vorticity length scale can fully develop: it is not affected by the walls. Zone II corresponds to the region where the vorticity length scale is truncated by the proximity of the walls, yet it still contributes significantly to the effective viscosity. Zone I corresponds to the first layers of grains near the wall where the vorticity length scale is so small that the grain collisions become the dominant mode of momentum transfer. Given that the EV model, (5.14), correctly captures the shear rate profiles, i.e.  $\dot{{\it\gamma}}\propto {\rm\Delta}y^{-1/2}$ in zone II, the vorticity length must scale as $\ell _{w}(y)/d\propto ({\rm\Delta}y/d)^{1/2}$ when approaching the walls and not as $\ell _{w}(y)\propto {\rm\Delta}y$ as is usually assumed in EV models.