Rheology of Thickly-Coated Granular-Fluid Systems

Rheology of Thickly-Coated Granular-Fluid Systems

Teng Man    Qingfeng Feng    K. M. Hill kmhill@umn.edu Department of Civil, Environmental, and Geo- Engineering, University of Minnesota, Minneapolis, Minnesota, USA. State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, China
July 20, 2019
Abstract

We investigate the link between particle-scale dynamics and bulk behaviors of thickly-coated particle-fluid flows using computational simulations. We find that, similar to dense fully-saturated slurries, the form the rheology takes in these systems can carry signatures of interparticle collisions and/or interparticle viscous dynamics that vary solid fraction. However, we find significant qualitative and quantitative differences in the transitions of the system between what might be called viscous, collisional, and visco-collisional behaviors. We show how these transitions arise from changes in the “fabric,” e.g. strong force network, and the “granular temperature,” or fluctuation energy, and suggest extension of these frameworks to better elucidate other particle-fluid behaviors.

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47.57.Gc, 81.05.Rm, 83.10.Pp, 83.80.Hj

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preprint: APS/123-QED

Thickly-coated particle-fluid flows are ubiquitous in natural and man-made structures, from muddy geophysical flows to concrete and other construction materials. Yet our physics-based understanding of these flows is lacking. The complexity of modeling these flows lies in part in mesoscale ephemeral structures that influence internal resistance to flow. In the last decade, significant progress has been made in modeling closely related systems – dry particle flows Pouliquen and Forterre (2002); MiDi (2004); Pouliquen et al. (2006); Jop et al. (2006), suspensions Boyer et al. (2011), and slurries Cassar et al. (2005); Trulsson et al. (2012) – by considering the rheology in the context of a non-dimensionalizing shear rate using a local, collisional or viscous, timescale:

(1a) (1b)

Here, is the normal stress; and are the particle size and material density; is the dynamic fluid viscosity. What we label here as and , are typically referred to as inertial numbers and are also often expressed as stress ratios. takes a form proportional to the square root of Bagnold’s classic dispersive stressBagnold (1954), essentially an interparticle collisional stress () divided by ; is a viscous stress () divided by . Simulations and experiments Pouliquen and Forterre (2002); MiDi (2004); Pouliquen et al. (2006); Jop et al. (2006); Boyer et al. (2011); Cassar et al. (2005) have demonstrated that behaviors of a wide range of wet and dry particle flows can be efficiently expressed in terms of the dependence of , and solid fraction on the relevant inertial number and/or (e.g., Table 1).

Trulsson et al. Trulsson et al. (2012) built on this previous work by hypothesizing that the shear stress () in dense slurries may be expressed in terms of a linear superposition of contact and viscous stresses scaled with a single function of solid fraction :

(2)

(a constant) and are empirically determined. We note that the significance of this suggestion lies in part on the implication that the relative contribution to internal resistance in particle-fluid systems by interparticle contacts (e.g., via ) and fluid-mediated interactions (e.g., via ) is independent of . Trulsson et al. Trulsson et al. (2012) showed that using Eqn. (2) they could express in terms of a superposition of inertial numbers:

(3a)
(3b)

They found the dependence of and on in slurries to be analogous to those previously obtained for particle-fluid flows dominated by or alone (Table 1).

In this Letter, we computationally model behaviors of thickly-coated particle-fluid systems. We find that their rheology can be expressed using similar constructs as for fully saturated flows. Specifically, we find somewhat analogous contributions to the rheology by contact and viscous particle-scale interactions previously seen in slurries Trulsson et al. (2012). However, in our coated systems, we find that there are multiple transitions, or dynamic pathways, between collisional, viscous, and visco-collisional behaviors that are mediated by particle concentration and variations in strong force networks.

To study thickly-coated particle-fluid systems, we use 3-d discrete element method (DEM) simulations Cundall and Strack (1979). Our particles interact via model contact forces whenever particle surfaces touch and via model viscous liquid forces whenever particle surfaces are sufficiently close. The interparticle contact forces depend on particle properties according to Hertz-Mindlin contact theories, Coulomb friction laws, and a damping component as described in Refs. Tsuji et al. (1992); Yohannes and Hill (2010); Hill and Yohannes (2011). To model the effect of the viscosity of the coating, we treat the fluid as one that moves with the particles, rather than in a separate phase, i.e., in the form of a lubrication model representing the viscous drag force on interparticle movement Pitois et al. (2000):

(4a)
(4b)

and are normal and tangential components, respectively, to the plane of contact between particles and ; ; are contacting particle diameters; is the smallest distance between neighboring particle surfaces; and are the normal and tangential velocities of these neighboring surfaces. and the bracketed term represent non-infinite fluid volume effects. We use regularizing length scales of and (see Supplemental Material I). Table 2 provides the range of particle and fluid properties we used. In each simulation we sheared 6400 particles.

dry
beads Pouliquen and Forterre (2002)
suspen-
sions Boyer et al. (2011)
slurriesTrulsson et al. (2012)
Table 1: Fits and fit parameters for data in Fig. 1
(mm) () (m/s)
2650
Table 2: Input Parameters
Figure 1: (a-b) plots of (a) vs.  and (b) vs. for representative data sets. (a-b) Symbols indicate approximate values according to: 0.74, 0.28, 0.1, 0.034, to , to , to , . Solid lines show the best fit line of relationships for (a) and (b) suggested in Ref. Jop et al. (2005) for dry spherical particles. (c-d) plots of (c) vs.  and (d) vs.  for representative data sets. (c-d) Solid lines show the best fit line of relationships for (c) and (d) suggested in Ref. Boyer et al. (2011). (e-f) plots of (e) and (f) for all simulated data. (e-f) Solid lines are the best fit relationships for and suggested in Ref. Trulsson et al. (2012) for slurries. We obtain the best collapse when , significantly smaller than that for 2-d slurries in Ref. Trulsson et al. (2012) indicating for our systems a higher sensitivity to viscous rather than collisional effects. (a-f) The equations and parameters for fits are provided in Table 1. (e) inset: sketch of simulations. The simulated cell size in the y- and z- directions are fixed at 30 mm and 58.3 mm, respectively; the steady state value for varies.

To measure the rheology of these systems, we shear them in a rectangular box (Fig. 1) with periodic boundary conditions in the - and - directions and roughened walls (using “glued” particles) in the -direction. We move one of the rough walls in the -direction (only) with a constant velocity . We apply a constant normal stress () to the other roughened wall and allow it to move only in the -direction. We run this simulation until the system reaches a statistically steady state and calculate the steady state average value of . We performed ten sets of simulations (see Supplemental Material II) designed to vary approximately from to and , when non-zero, from to .

We find that our thickly-coated granular flows behave remarkably similarly to previously measured particle- and particle-fluid flows (Fig. 1): (1) for sufficiently low values of (), and are both essentially independent of and similar to those of dry granular flows Jop et al. (2005); (2) for sufficiently low values of (), and are both independent of and similar to those in suspensions and slurries Boyer et al. (2011); (3) for moderate-to-high values of and , and depend both on and and appear to collapse best with . However, this is not as clean as a collapse as it first appears, apparent in a parametric plot of vs.  (Fig. 2(a))

Rather than the monotonic relationship between and , is non-monotonic for these system in the range . corresponds to the lower limit of for the highest value of at which our systems remain uniform. Within this range, is bounded at the low end with a predictive relationship for collisionally-dominated flows . The upper bound for appears to be dependent on both collisional and viscous effects (Fig. 2(a) caption). We can transition systems from the lower limit curve to the higher one ( with an increasing at constant (e.g., red arrow in Fig. (2(a) for ) by increasing . With an initial increase in , increases at a nearly constant to the curve. If we increase further still, increases further while decreases along the curve.

Figure 2: (a) main plot: parametric plot of vs.  for all simulations performed herein. The lines are: (dashed) (line 1, Table 1); (dot-dahsed) line 3, Table 1; (solid) ; (red arrow) constant- pathway for increasing . (b) Ratio of computational to theoretical lubrication stresses vs. . solid line: ; dashed line: , respectively. (c) Ratio of computational contract stress to two different theoretical stresses. inset: vs. ; main plot: vs. . solid line: . Insets to (a): instantaneous “force diagrams” from a slice of the shear cell (); each line connects centers of contacting particles for which the interparticle force , the mean force. [, ] = (i) [0.68, 0.033] (ii) [0.69, 0.0034] (iii) [0.0075, 0.012] (iv) [0.25, 0.0037].

For a representation of force structures under these transitions, for two pairs of experiments with similar (’s) and substantially different ’s we plot a representative “strong force network” – line segments for particle pairs for which the interparticle contact force is greater than average (Fig. 2(a)). For each pair on the same limit curve or , increasing simultaneously decreases and both connectivity and density of high force pairs. At the same time, enduring extended frictional contact networks are replaced by more isolated collisional contacts. In contrast, for each pair with similar ’s, increasing limit curve increases both connectivity and density of high force pairs. Based on these observations, we hypothesize that as a system transitions from the low limit curve to the high limit curve increasing at constant , modest increases in viscous damping of interparticle movement decreases separation events of contacting particles, similar to an effective “stickiness”, increasing relative strong contact force connectivity. This results in an increase in relative to without much change in . Once the upper limit curve is reached, further increases in reduces particle contact events transitioning the system back to one where interparticle contacts are more isolated and collisional.

From our data, we calculate a contact and viscous shear stresses ( and ) from the -component of all contact forces and viscous forces between each wall particle and adjacent free particles. We find that can be expressed as a single-valued function of , (Fig. 2(b)) though cannot (Fig. 2(c), inset). Rather, is much closer to a single-valued function of , ) (Fig. 2(c) and caption). We note that this -dependent behavior suggest that, in these systems, while non-contact viscous interactions are influenced by alone, contact interactions are influenced by both and . Considering the significantly different forms of and , we suggest that in place of Eqn. (2-3) for thickly-coated systems we write:

(5a)
(5b)

Equation 5(b) provides an effective, if empirical, expression for the data in Fig. 2(a). Further, in certain limits and with Figs. 2(b-c) it provides intuition to the structure of . For , , so for these data signifying a regime where viscous and contact effects influence the dynamics. For all data on the (upper bound) curve in Fig. 2(a) correspond to cases where (a Stokes number). For these data signifying a regime where viscous effects dominate systems on the upper curve for . While the functional forms for this upper bound curve differ, at , and , so there is a smooth transition. Now considering the lower bound curve: for , and so , dependent primarily on contact effects, as Fig. 2(a) and caption imply. For larger ’s on this curve, decrease and the functional form , the same form as the upper bound curve in this region of .

This general picture is supported by consideration of the relationship between coordination number (average number of contacts per particle), , and other bulk parameters (Figs. 3(a-c)). As indicated in Fig. 3(a) (dark red arrows), for the same moderate increase in for which the system remained primarily on the low limit curve, maintains a near-constant small value. We also check the fraction of particles potentially available for a connected strong force network (Fig. 3(b)); is the part of the system solid fraction comprised of particles contacting more than one particle (NR stands for non-rattler particles, as in Ref. Bi et al. (2011))). For highest and , indicating that on average only of the particles have more than one contact, supporting the picture of a system dominated by occasional collisions. Then first decreases as increases (from 0.2 to 0.1), signifying the more highly connected force network. With further increase, rises as drops again, signifying the return to a less well-connected contact network. We find when we plot vs. , we get a convincing collapse, suggesting as others have found for somewhat different particulate systems Bi et al. (2011), and are much more deterministic of system behavior than .

Figure 3: (a) Coordination number plotted vs. . red arrows: pathways of increasing for constant . (b) , the fraction of particles with contacts plotted vs. ; symbols indicate value of . (c) vs. : ; Dashed curve: . (d) Normalized granular temperature, plotted vs. log(), (=0.61).

To summarize, we find that thickly-coated particle-fluid flows behave in many ways similarly to fully saturated particle-fluid flows, particularly in their response to changing either viscous or contact effects constant. However, simultaneous variation of both effects reveals concentration-dependent transitions in bulk behaviors reflected in interparticle forces and force networks not previously reported for similar particle-fluid systems. Ongoing work includes investigations of the transitions between particle-scale interactions of these systems and dynamics in analogous fully-saturated systems.

We finish with two additional observations. First, somewhat unexpectedly to us, the granular temperature plotted vs.  does not help us distinguish between collisional and visco-collisional flows. Rather, we for nearly all of our systems, has the same functional dependence on , not dissimilar from that predicted by Bagnold for moderate-to-dense granular-fluid flows Bagnold (1954) (see supplement). The exceptions we found (not surprisingly) were for and for which we found increases with increasing (Fig. 3(c)). Finally, for future refreence we note the wealth of related studies at the “jamming” (liquid-to-solid) transition from which we can seek additional insight for transitions in our flowing systems. In their 2-d experimental jamming studies, Bi et al. Bi et al. (2011) found a similar functional form and slope change in vs.  as we did in our flowing system (Fig. 3(c)) as well as a slope change with a change of fabric structures. Links like this gives insight to a more broadly unified physical framework for dense particle-fluid deformation and flows.

We gratefully acknowledge the funding for this research provided by the NSF under the grant EAR-1451957 on ”Entrainment and Deposition of Surface Material by Particle-Laden Flows”, the UMN Center of Transportation Studies and the CEGE Sommereld Fellowship and computing resources provided by SAFL at UMN. The authors also thank Prof. Jia-Liang Le for helpful discussions.

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