Journal Club for July 2024: Size-Segregation in Dense, Bidisperse Granular Materials

Harkirat Singh ( California Institute of Technology) and David Henann (Brown University)

 

Challenges

Granular materials (large collections of discrete particles) appear in many forms across industries (e.g., pharmaceutical, agriculture, and construction) and in nature. Predictive models for granular materials are crucial to design handling processes and to understand geophysical phenomena, such as avalanches and landslides, but bulk granular materials display a variety of behaviors that challenge modeling efforts. One class of difficulties relates to particle segregation in granular mixtures, whereby particles with different properties (e.g., size, density, shape, or particle friction) will demix based on their properties during flow or other excitation, such as shaking. The complex, spatially-heterogeneous distributions in particle size that can arise during the segregation process [e.g., 1–7] pose a substantial challenge to modeling efforts, and significant effort over the last few decades has gone into developing continuum models for the evolution of particle-size distribution in a number of flow geometries (e.g., see the reviews of Gray [8] and Umbanhowar et al. [9]).

The focus of this journal club entry is segregation based on size in flowing bidisperse mixtures, consisting of particles with two different sizes but otherwise identical properties. We begin by highlighting two specific challenges for dense, bidisperse granular media. First, as per current understanding, there are two important driving forces for size-segregation in dense granular flows: (1) pressure gradients, typically arising due to gravity, and (2) shear- strain-rate gradients. In the presence of pressure gradients, the mechanisms of kinetic sieving and squeeze expulsion [1, 8] result in a net flux of small particles to high-pressure regions and large particles to low-pressure regions, such as near a free surface. Pressure-gradient-driven segregation is widely recognized as a dominant driving force and is the focus of most size-segregation models in the literature [e.g., 6, 10–20]. Apart from pressure gradients, gradients in shear-strain-rate can also drive size-segregation in dense flows, in which large particles segregate towards more rapidly shearing regions [3, 21, 22], and comparatively fewer continuum modeling works have been dedicated to capturing this driving force [23–25].

The second challenge is coupling segregation modeling with rheology. The majority of prior modeling works do not involve rheological constitutive equations capable of predicting flow fields across different geometries. Instead, a certain flow field is assumed or measured from experiments or discrete element method (DEM) simulations and then prescribed as input to the segregation model. This is because modeling the rheological behavior of dense granular materials across flow geometries is a substantial challenge itself. For example, flowing, dense granular materials form clear, experimentally robust features (e.g., shear bands), which can have a variety of possible widths and decay nontrivially into the surrounding nearly-rigid material, and nonlocal constitutive modeling is required to capture this rheological behavior.

The interplay between flow and both mechanisms of size-segregation is illustrated in Fig. 1 in a split-bottom cell [26]. The geometry consists of an annular cell with rough walls and bottom and an open top, filled with grains to a height H. The bottom is split at a radius Rs, and the inner portion is then rotated at an angular velocity Ω, holding the outer portion fixed. During steady flow, a shear band emanates from the split along the bottom of the cell. The shear band gradually moves inward with increasing height, accompanied by a broadening of the shear-band width (due to nonlocal, cooperative effects) before terminating at the top surface. This flow field drives segregation, in which large grains make their way towards the shear band, where the strain rate is highest, and towards the free surface, where the pressure is lowest, as illustrated by DEM simulation results in Fig. 1.

Figure 1: Illustration of the coupling between flow and size-segregation in a split-bottom cell. In a bidisperse mixture, large grains tend to move towards areas of low pressure (such as a free surface) and where deformation rates are high (such as in shear bands).

Continuum modeling

Next, we highlight the essential ingredients for continuum modeling of size-segregation and flow in dense, bidisperse granular materials. The key principle that governs the segregation dynamics is mass conservation of the constituents. In a bidisperse system, the state of the mixture is typically described by concentration fields, denoted as cl and cs for the large and small grains, respectively, with cl + cs = 1. The concentration fields represent the fraction of grain volume occupied by either the large or small species. Then, utilizing the concentration of large grains cl as the field variable to describe the segregation dynamics, the mass conservation equation becomes Dcl/Dt + ∂wil/∂xi = 0, where D(•)/Dt is the material time derivative, and wil is the relative volume flux of large grains. The constitutive equation for the relative volume flux should take into account contributions due to diffusion, strain- rate-gradient driven segregation, and pressure-gradient driven segregation in dense granular flows. In prior works, Gray and co-authors [e.g., 10–12, 14, 18] as well as Lueptow, Ottino, Umbanhowar, and co-authors [e.g., 6, 15, 16, 27–29] have made important contributions to understanding the pressure-gradient-driven mechanism of size-segregation in dense flows, including how this flux should scale with the concentration, strain rate, pressure, and grain sizes. Hill and co-authors [23, 24] highlighted the importance of strain-rate-gradient-driven segregation and advanced modeling efforts for this mechanism.

The rheological behavior of dense granular mixtures is described by a constitutive equation for the Cauchy stress tensor. The inertial, or μ(I), rheology [30–33], where μ is the stress ratio and I is the inertial number, is a common viscoplastic modeling approach for dense granular flow and uses dimensional arguments to relate the stress state to the state of strain rate at a point through a local constitutive equation. The μ(I) rheology works well in homogeneous shearing and certain other dense inertial flows, such as flow down an incline. However, the μ(I) rheology cannot capture a broad set of inhomogeneous flows spanning the quasi-static and dense inertial flow regimes, and a number of nonlocal continuum modeling approaches have been developed to capture the features of dense, inhomogeneous flows [34]. The focus of our work has been the nonlocal granular fluidity (NGF) model [35, 36], which has been successfully applied to predicting dense flows of monodisperse grains in a wide variety of flow geometries (see prior journal club entries here and here).

Only a few works [17, 19, 25, 37, 38] have sought to couple segregation modeling with rheological constitutive equations for a dense, bidisperse granular medium. To this end, in a series of recent papers [25, 37], we have integrated nonlocal rheology, granular diffusion, strain-rate-gradient-driven, and pressure-gradient-driven segregation mechanisms within a single framework. In the next section, we highlight the capability of our recent modeling work [25, 37] by testing it in three different flow configurations, namely, (1) vertical chute flow, in which only the strain-rate-gradient-driven mechanism is present, (2) inclined plane flow, in which the strain-rate-gradient-driven and pressure-gradient-driven mechanisms oppose one another, and (3) planar shear flow with gravity, in which the strain-rate-gradient-driven and pressure-gradient-driven mechanisms cooperate. We compare continuum model predictions against data generated using DEM simulations.

 

Illustrative examples

 

Vertical chute flow: In vertical chute flow, an initially well-mixed bidisperse mixture flows down a vertical channel under the action of gravity as shown in Fig. 2. A pressure Pw is applied by the outer walls, so that the pressure field is uniform, and the only drivers of segregation are strain-rate gradients, which are large near the walls and minimal in the center of the channel, as indicated by the qualitative sketch of the velocity field in Fig. 2. As a result, the large grains segregate towards the outer walls leaving behind small-grain rich regions just inside, as shown in the DEM ‘segregated-state’. The middle region remains well-mixed because the strain-rates are too small to drive substantial segregation. The large-grain concentration field cl from the continuum model prediction is compared against DEM simulation results at very short time, short time, moderate time, and long time. Furthermore, the model is able to capture the rapidly flowing region near the outer wall and creeping region farther away from the outer wall as shown in the rightmost column of Fig. 2. In short, the coupled continuum model showcases its capability to predict steady-state flow and segregation dynamics simultaneously in the vertical chute flow configuration.

 

Figure 2: Flow and size-segregation in vertical chute flow. DEM snapshots of initial and segregated states are shown in the left column. Large grains are dark gray, and small grains are light gray. Quantitative comparisons of the evolution of the large-grain concentration field cl and the steady-state flow field are shown in the middle and right columns, respectively. For details refer to [25].

Inclined plane flow: While vertical chute flow isolates the strain-rate-gradient-driven segregation mechanism, inclined plane flow is a widely studied flow configuration in the literature, in which both driving mechanisms of segregation are present. This configuration is illustrated in Fig. 3 and consists of a semi-infinite layer of thickness H of a dense, bidisperse granular mixture flowing down an inclined plane with surface inclination angle θ under the action of gravity. Pressure gradients, induced by gravity, drive large grains towards the top, while strain-rate gradients drive them towards the bottom surface, since the strain rate is greatest there. Therefore, the two mechanisms oppose each other. Interestingly, pressure gradients turn out to be the dominant mechanism in this flow configuration, and large grains segregate towards the free surface as shown in ’segregated state’ in the leftmost column of Fig. 3. Quantitative comparisons of the model-predicted evolution of the concentration field and the steady-state flow field against DEM simulation results are shown in the middle and rightmost columns of Fig. 3. Testing the model in the inclined plane flow configuration illustrates that the continuum framework can account for both the shear-strain-rate-gradient and the pressure-gradient mechanisms of segregation.

Figure 3: Flow and size-segregation in inclined plane flow. Plots are arranged in the same manner as in Fig. 2. For details refer to [37].

 

Planar shear flow with gravity: Planar shear flow with gravity is a kinematically driven flow configuration, wherein pressure gradients and strain-rate gradients cooperatively contribute to the segregation dynamics. As illustrated in Fig. 4, a rough wall sits atop a bidisperse granular bed, imparting a pressure of Pw, and is dragged tangential to the top surface at a velocity vw, driving shear flow. The resulting flow field decays rapidly underneath the top wall as sketched in the ‘initial state’ of the DEM setup, giving rise to high strain-rates near the top wall. As a consequence, large grains segregate towards the top wall leaving beneath a narrow band of small grains, as can be seen in the DEM ‘segregated state’. Further beneath the surface, the concentration field hardly evolves and remains well-mixed since the strain rates are very small. To contrast with inclined plane flow, while the pressure-gradient-driven and strain-rate-gradient-driven mechanisms of segregation oppose one another in inclined plane flow, they are cooperative mechanisms in planar shear flow with gravity, i.e., both drive large grains toward the top wall. Quantitative comparisons of the evolution of the large-grain concentration field and the steady-state flow field between the continuum model and DEM simulation results are shown in the second and third columns of Fig. 4. Collectively, the coupled continuum model is capable of accounting for the contributions of the pressure-gradient-driven and strain-rate-gradient-driven segregation mechanisms and, therefore, captures the transient evolution of the segregation process and the flow fields across several flow geometries.

 

Figure 4: Flow and size-segregation in planar shear flow with gravity. Plots are arranged in the same manner as in Fig. 4.  For details refer to [37].

 

Summary and outlook

 

Despite their ubiquity, dense granular flows pose challenges to modeling efforts due to their peculiar behaviors, including the coexistence of fluid-like and solid-like behavior, the size-dependence of flows, nonlocal effects, and segregation, among many other interesting features. Our work integrates models for nonlocal rheology and segregation phenomena in bidisperse dense granular mixtures and has been tested across several flow configurations.

There remain many important directions for future work, several of which are highlighted below.

• First, we have only discussed cases in which the velocity and concentration fields vary along one dimension. It is yet to be tested if the coupled model is predictive in multi-dimensional flow configurations, for instance, split-bottom flow.
• We have focused on bidisperse granular mixtures, but many granular systems are polydisperse in nature. There are a number of experimental [39–41] and DEM-based [13, 41–43] studies that investigate segregation in polydisperse mixtures. On the theoretical front, Gray and Ancey [12] have developed a theoretical framework for multidisperse granular systems with a finite number of particle sizes, and Marks et al. [13] and Schlick et al. [44] have considered systems with a continuous distribution of particle sizes. There is a need to develop a theoretical framework that integrates models for rheology and size-segregation in polydisperse granular mixtures.
• Granular mixtures can also undergo segregation due to mismatches in other grain properties, such as mismatches in density, shape, or particle friction. Density-driven segregation [45–48] can be rationalized in terms of buoyancy and drag forces wherein less dense particles rise to the free surface. Several works [46, 47] have sought to integrate density-based segregation with size-based segregation within a single frame-work. Apart from density and size mismatches, shear-driven granular mixtures can undergo segregation when the particles differ only in their surface friction coefficients [49], wherein the smoother particles segregate towards the high shear rate zones leaving the rougher particles behind in the low shear rate regions. Lastly, mismatches in grain shape can also lead to interesting segregation patterns [50, 51]. These segregation driving factors need to be integrated with appropriate rheological models for dense granular media.

• In dilute granular flows (i.e., the collisional gas regime), in which particle interactions are dominated by collisions rather than enduring contacts, a granular-kinetic-theory-based approach [52–56] is commonly employed to model segregation phenomenology. Dilute flows can lead to a reversal of the nature of segregation when compared to dense flows [22]. Developing a predictive continuum model that can bridge the dilute and the dense regimes remains a challenge. 

 

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