Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T14:51:21.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Drag force in granular shear flows: regimes, scaling laws and implications for segregation

Published online by Cambridge University Press:  08 September 2022

Lu Jing Open the ORCID record for Lu Jing [Opens in a new window] ,
Julio M. Ottino Open the ORCID record for Julio M. Ottino [Opens in a new window] ,
Paul B. Umbanhowar Open the ORCID record for Paul B. Umbanhowar [Opens in a new window]  and
Richard M. Lueptow Open the ORCID record for Richard M. Lueptow [Opens in a new window]
Lu Jing
Affiliation:
Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, USA
Julio M. Ottino
Affiliation:
Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, USA Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, IL 60208, USA
Paul B. Umbanhowar
Affiliation:
Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA
Richard M. Lueptow*
Affiliation:
Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, USA Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, IL 60208, USA
*
†Email address for correspondence: r-lueptow@northwestern.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The drag force on a spherical intruder in dense granular shear flows is studied using discrete element method simulations. Three regimes of the intruder dynamics are observed depending on the magnitude of the drag force (or the corresponding intruder velocity) and the flow inertial number: a fluctuation-dominated regime for small drag forces; a viscous regime for intermediate drag forces; and an inertial (cavity formation) regime for large drag forces. The transition from the viscous regime (linear force-velocity relation) to the inertial regime (quadratic force-velocity relation) depends further on the inertial number. Despite these distinct intruder dynamics, we find a quantitative similarity between the intruder drag in granular shear flows and the Stokesian drag on a sphere in a viscous fluid for intruder Reynolds numbers spanning five orders of magnitude. Beyond this first-order description, a modified Stokes drag model is developed that accounts for the secondary dependence of the drag coefficient on the inertial number and the intruder size and density ratios. When the drag model is coupled with a segregation force model for intruders in dense granular flows, it is possible to predict the velocity of gravity-driven segregation of an intruder particle in shear flow simulations.

JFM classification

Granular media: Dry granular material Mixing: Granular mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 948 , 10 October 2022 , A24
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Present address: Institute for Ocean Engineering, Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, PR China.

References

REFERENCES

Albert, R., Pfeifer, M.A., Barabási, A.-L. & Schiffer, P. 1999 Slow drag in a granular medium. Phys. Rev. Lett. 82 (1), 205–208.CrossRefGoogle Scholar
Azéma, E. & Radjaï, F. 2014 Internal structure of inertial granular flows. Phys. Rev. Lett. 112 (7), 078001.CrossRefGoogle ScholarPubMed
Bancroft, R.S.J. & Johnson, C.G. 2021 Drag, diffusion and segregation in inertial granular flows. J. Fluid Mech. 924, A3.CrossRefGoogle Scholar
Candelier, R. & Dauchot, O. 2009 Creep motion of an intruder within a granular glass close to jamming. Phys. Rev. Lett. 103 (12), 128001.CrossRefGoogle ScholarPubMed
Chassagne, R., Maurin, R., Chauchat, J., Gray, J.M.N.T. & Frey, P. 2020 Discrete and continuum modelling of grain size segregation during bedload transport. J. Fluid Mech. 895, A30.CrossRefGoogle Scholar
Clark, A.H., Petersen, A.J. & Behringer, R.P. 2014 Collisional model for granular impact dynamics. Phys. Rev. E 89 (1), 012201.CrossRefGoogle ScholarPubMed
Clark, A.H., Thompson, J.D., Shattuck, M.D., Ouellette, N.T. & O'Hern, C.S. 2018 Critical scaling near the yielding transition in granular media. Phys. Rev. E 97 (6), 062901.CrossRefGoogle ScholarPubMed
da Cruz, F., Emam, S., Prochnow, M., Roux, J. & Chevoir, F. 2005 Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys. Rev. E 72, 021309.CrossRefGoogle ScholarPubMed
Das, P., Puri, S. & Schwartz, M. 2020 Intruder dynamics in a frictional granular fluid: a molecular dynamics study. Phys. Rev. E 102 (4), 042905.CrossRefGoogle Scholar
Ding, Y., Gravish, N. & Goldman, D.I. 2011 Drag induced lift in granular media. Phys. Rev. Lett. 106 (2), 028001.CrossRefGoogle ScholarPubMed
Duan, Y., Jing, L., Umbanhowar, P.B., Ottino, J.M. & Lueptow, R.M. 2022 Segregation forces in dense granular ows: closing the gap between single intruders and mixtures. J. Fluid Mech. 935, R1.CrossRefGoogle Scholar
Duan, Y., Umbanhowar, P.B., Ottino, J.M. & Lueptow, R.M. 2020 Segregation models for density-bidisperse granular flows. Phys. Rev. Fluids 5 (4), 044301.CrossRefGoogle Scholar
Duan, Y., Umbanhowar, P.B., Ottino, J.M. & Lueptow, R.M. 2021 Modelling segregation of bidisperse granular mixtures varying simultaneously in size and density for free surface flows. J. Fluid Mech. 918, A20.CrossRefGoogle Scholar
Félix, G. & Thomas, N. 2004 Evidence of two effects in the size segregation process in dry granular media. Phys. Rev. E 70 (5), 051307.CrossRefGoogle ScholarPubMed
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40 (1), 1–24.CrossRefGoogle Scholar
Fry, A.M., Umbanhowar, P.B., Ottino, J.M. & Lueptow, R.M. 2018 Effect of pressure on segregation in granular shear flows. Phys. Rev. E 97 (6), 062906.CrossRefGoogle ScholarPubMed
Geng, J. & Behringer, R.P. 2005 Slow drag in two-dimensional granular media. Phys. Rev. E 71 (1), 011302.CrossRefGoogle ScholarPubMed
Goossens, W.R.A. 2019 Review of the empirical correlations for the drag coefficient of rigid spheres. Powder Tech. 352, 350–359.CrossRefGoogle Scholar
Gravish, N., Umbanhowar, P.B. & Goldman, D.I. 2010 Force and flow transition in plowed granular media. Phys. Rev. Lett. 105 (12), 128301.CrossRefGoogle ScholarPubMed
Gray, J.M.N.T. 2018 Particle segregation in dense granular flows. Annu. Rev. Fluid Mech. 50 (1), 407–433.CrossRefGoogle Scholar
Guillard, F., Forterre, Y. & Pouliquen, O. 2016 Scaling laws for segregation forces in dense sheared granular flows. J. Fluid Mech. 807, R1.CrossRefGoogle Scholar
Jenkins, J.T. & Yoon, D.K. 2002 Segregation in binary mixtures under gravity. Phys. Rev. Lett. 88 (19), 194301.CrossRefGoogle ScholarPubMed
Jing, L., Kwok, C.Y. & Leung, Y.F. 2017 Micromechanical origin of particle size segregation. Phys. Rev. Lett. 118 (11), 118001.CrossRefGoogle ScholarPubMed
Jing, L., Kwok, C.Y., Leung, Y.F. & Sobral, Y.D. 2016 Characterization of base roughness for granular chute flows. Phys. Rev. E 94 (5), 052901.CrossRefGoogle ScholarPubMed
Jing, L., Ottino, J.M., Lueptow, R.M. & Umbanhowar, P.B. 2020 Rising and sinking intruders in dense granular flows. Phys. Rev. Res. 2 (2), 022069.CrossRefGoogle Scholar
Jing, L., Ottino, J.M., Lueptow, R.M. & Umbanhowar, P.B. 2021 A unified description of gravity- and kinematics-induced segregation forces in dense granular flows. J. Fluid Mech. 925, A29.CrossRefGoogle Scholar
Kamrin, K. 2019 Non-locality in granular flow: phenomenology and modeling approaches. Front. Phys. 7, 116.CrossRefGoogle Scholar
Kim, S. & Kamrin, K. 2020 Power-law scaling in granular rheology across flow geometries. Phys. Rev. Lett. 125 (8), 088002.CrossRefGoogle ScholarPubMed
Lerner, E., Düring, G. & Wyart, M. 2012 A unified framework for non-Brownian suspension flows and soft amorphous solids. Proc. Natl Acad. Sci. USA 109 (13), 4798–4803.CrossRefGoogle ScholarPubMed
Li, C., Zhang, T. & Goldman, D.I. 2013 A terradynamic of legged locomotion on granular media. Science 339 (6126), 1408–1412.CrossRefGoogle Scholar
Liu, S. & McCarthy, J.J. 2017 Transport analogy for segregation and granular rheology. Phys. Rev. E 96 (2), 020901(R).CrossRefGoogle ScholarPubMed
Liu, M. & Müller, C.R. 2021 Lift force acting on an intruder in dense, granular shear flows. Phys. Rev. E 104 (6), 064903.CrossRefGoogle Scholar
Nichol, K., Zanin, A., Bastien, R., Wandersman, E. & van Hecke, M. 2010 Flow-induced agitations create a granular fluid. Phys. Rev. Lett. 104 (7), 078302.CrossRefGoogle ScholarPubMed
Pacheco-Vázquez, F. & Ruiz-Suárez, J.C. 2010 Cooperative dynamics in the penetration of a group of intruders in a granular medium. Nat. Commun. 1 (8), 123.CrossRefGoogle Scholar
Papanastasiou, T.C. & Boudouvis, A.G. 1997 Flows of viscoplastic materials: models and computations. Comput. Struct. 64 (1), 677–694.CrossRefGoogle Scholar
Reddy, K.A., Forterre, Y. & Pouliquen, O. 2011 Evidence of mechanically activated processes in slow granular flows. Phys. Rev. Lett. 106 (10), 108301.CrossRefGoogle ScholarPubMed
Rognon, P. & Macaulay, M. 2021 Shear-induced diffusion in dense granular fluids. Soft Matt. 17 (21), 5271–5277.CrossRefGoogle ScholarPubMed
Rousseau, H., Chassagne, R., Chauchat, J., Maurin, R. & Frey, P. 2021 Bridging the gap between particle-scale forces and continuum modelling of size segregation: application to bedload transport. J. Fluid Mech. 916, A26.CrossRefGoogle Scholar
Saitoh, K. & Tighe, B.P. 2019 Nonlocal effects in inhomogeneous flows of soft athermal disks. Phys. Rev. Lett. 122 (18), 188001.CrossRefGoogle ScholarPubMed
Schlichting, H.T. 1979 Boundary-Layer Theory, 7th edn. McGraw-Hill.Google Scholar
Schlick, C.P., Fan, Y., Isner, A.B., Umbanhowar, P.B., Ottino, J.M. & Lueptow, R.M. 2015 Modeling segregation of bidisperse granular materials using physical control parameters in the quasi-2d bounded heap. AIChE J. 61 (5), 1524–1534.CrossRefGoogle Scholar
van Schrojenstein Lantman, M.P. 2019 A study on fundamental segregation mechanisms in dense granular flows. PhD thesis, University of Twente, Enschede.Google Scholar
van Schrojenstein Lantman, M.P., van der Vaart, K., Luding, S. & Thornton, A.R. 2021 Granular buoyancy in the context of segregation of single large grains in dense granular shear flows. Phys. Rev. Fluids 6 (6), 064307.CrossRefGoogle Scholar
Seguin, A., Bertho, Y., Martinez, F., Crassous, J. & Gondret, P. 2013 Experimental velocity fields and forces for a cylinder penetrating into a granular medium. Phys. Rev. E 87 (1), 012201.CrossRefGoogle ScholarPubMed
Seguin, A. & Gondret, P. 2017 Drag force in a cold or hot granular medium. Phys. Rev. E 96 (3), 032905.CrossRefGoogle ScholarPubMed
Staron, L. 2018 Rising dynamics and lift effect in dense segregating granular flows. Phys. Fluids 30 (12), 123303.CrossRefGoogle Scholar
Stone, M.B., Bernstein, D.P., Barry, R., Pelc, M.D., Tsui, Y.-K. & Schiffer, P. 2004 Stress propagation: getting to the bottom of a granular medium. Nature 427 (6974), 503–504.CrossRefGoogle Scholar
Thornton, A. 2021 A brief review of (multi-scale) modelling approaches to segregation. EPJ Web Conf. 249, 01004.CrossRefGoogle Scholar
Trewhela, T., Ancey, C. & Gray, J.M.N.T. 2021 An experimental scaling law for particle-size segregation in dense granular flows. J. Fluid Mech. 916, A55.CrossRefGoogle Scholar
Tripathi, A. & Khakhar, D.V. 2011 Numerical simulation of the sedimentation of a sphere in a sheared granular fluid: a granular Stokes experiment. Phys. Rev. Lett. 107 (10), 108001.CrossRefGoogle Scholar
Tripathi, A. & Khakhar, D.V. 2013 Density difference-driven segregation in a dense granular flow. J. Fluid Mech. 717, 643–669.CrossRefGoogle Scholar
Tripathi, A., Kumar, A., Nema, M. & Khakhar, D.V. 2021 Theory for size segregation in flowing granular mixtures based on computation of forces on a single large particle. Phys. Rev. E 103 (3), L031301.CrossRefGoogle ScholarPubMed
Umbanhowar, P.B., Lueptow, R.M. & Ottino, J.M. 2019 Modeling segregation in granular flows. Annu. Rev. Chem. Biomol. Engng 10 (1), 5.1–5.25.Google ScholarPubMed
van der Vaart, K., van Schrojenstein Lantman, M.P., Weinhart, T., Luding, S., Ancey, C. & Thornton, A.R. 2018 Segregation of large particles in dense granular flows suggests a granular Saffman effect. Phys. Rev. Fluids 3 (7), 074303.CrossRefGoogle Scholar
Weinhart, T., Hartkamp, R., Thornton, A.R. & Luding, S. 2013 Coarse-grained local and objective continuum description of three-dimensional granular flows down an inclined surface. Phys. Fluids 25 (7), 070605.CrossRefGoogle Scholar
White, F.M. 1974 Viscous Fluid Flow, 1st edn. McGraw-Hill.Google Scholar
Xiao, H., Umbanhowar, P.B., Ottino, J.M. & Lueptow, R.M. 2016 Modelling density segregation in flowing bidisperse granular materials. Proc. R. Soc. A 472 (2191), 20150856.CrossRefGoogle Scholar
Zheng, H., Wang, D., Barés, J. & Behringer, R.P. 2018 Sinking in a bed of grains activated by shearing. Phys. Rev. E 98 (1), 010901.CrossRefGoogle Scholar

Jing et al. Supplementary Movie 1

Vertical position of an intruder particle vs. time (left panel) and the corresponding animation (right panel) showing the intruder (red) and its contacting (blue) and non-contacting (grey) neighboring bed particles in an annular region around the intruder. The intruder has the same size and density as the bed particles ($R=1$ and $R_\rho=1$). The flow conditions are $P_0=1000\ \textrm{Pa}$ and $\dot\gamma_0=10\ \textrm{s}^{-1}$ ($I=0.08$), and the drag force (or, equivalently, the external driving force) is $F_d/P_0d_i^2=0.05$. Note the significant fluctuations and intermittency of the intruder position due to particle collisions at this small driving force.

Download Jing et al. Supplementary Movie 1(Video)
Video 1.8 MB

Jing et al. Supplementary Movie 2

Same as movie 1 except that $F_d/P_0d_i^2=5$. Note that cavity formation occurs in the wake of the red intruder particle and that the contacts (with blue bed particles) are concentrated at the leading edge of the intruder.

Download Jing et al. Supplementary Movie 2(Video)
Video 2.2 MB

Jing et al. Supplementary Movie 3

Same as movie 2 except that $R=4$ and that the flow conditions are $P_0=1000\ \textrm{Pa}$ and $\dot\gamma_0=20\ \textrm{s}^{-1}$ ($I=0.16$).

Download Jing et al. Supplementary Movie 3(Video)
Video 4.4 MB