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Mixing of the fluid phase in slowly sheared particle suspensions of cylinders

Published online by Cambridge University Press:  06 April 2017

Kjetil Thøgersen Open the ORCID record for Kjetil Thøgersen [Opens in a new window]  and
Marcin Dabrowski Open the ORCID record for Marcin Dabrowski [Opens in a new window]
Kjetil Thøgersen*
Affiliation:
Department of Physics, University of Oslo, Sem Sælands vei 24, NO-0316, Oslo, Norway
Marcin Dabrowski
Affiliation:
Department of Physics, University of Oslo, Sem Sælands vei 24, NO-0316, Oslo, Norway Computational Geology Laboratory, Polish Geological Institute - NRI, 53-122, Wrocław, Poland
*
†Email address for correspondence: kjetil.thogersen@fys.uio.no
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Abstract

We introduce a finite element model for neutrally buoyant particle suspensions of cylinders at zero Reynolds number and infinite Péclet number in the purely hydrodynamic limit, which allows us to access a high-accuracy fluid velocity field at any time during the simulation. We use the diffusive strip method to characterize the development of the concentration field in the fluid phase of sheared suspensions from initial thin filaments, and characterize the structures that form with their fractal dimension. We find that the growth of the fractal dimension of the filaments scales with the increase of mean square displacement in the fluid phase. Further, we measure the concentration distribution of tracers in the fluid phase, as well as the shear-induced self-diffusion coefficient in both the solid phase and the fluid phase. We demonstrate that the shear-induced self-diffusion coefficient is slightly larger in the fluid phase at infinite Péclet number. Finally, we investigate enhanced mass diffusivity in the fluid phase by systematic measurements of the shear-induced self-diffusion coefficient in the fluid phase for a wide range of fluid tracer Péclet numbers. We find that the functional dependence $D_{s}/D=1+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6FC}}Pe^{\unicode[STIX]{x1D701}}$ (where $D_{s}$ is the shear-induced self-diffusion coefficient, $D$ is the molecular diffusivity and $\unicode[STIX]{x1D719}$ is the particle volume fraction) fits the observations fairly well. We measure the constants $\unicode[STIX]{x1D6FD}=2.98\pm 0.39$, $\unicode[STIX]{x1D6FC}=1.61\pm 0.26$ and $\unicode[STIX]{x1D701}=0.900\pm 0.031$.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Mixing: Mixing Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 818 , 10 May 2017 , pp. 807 - 837
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© 2017 Cambridge University Press 

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References

Abade, G. C., Cichocki, B., Ekiel-Jeżewska, M. L., Nägele, G. & Wajnryb, E. 2011 Rotational and translational self-diffusion in concentrated suspensions of permeable particles. J. Chem. Phys. 134 (24), 244903.Google Scholar
Arp, P. & Mason, S. 1977 The kinetics of flowing dispersions. J. Colloid Interface Sci. 61 (1), 44–61.Google Scholar
Blanc, F., Peters, F. & Lemaire, E. 2011 Experimental signature of the pair trajectories of rough spheres in the shear-induced microstructure in noncolloidal suspensions. Phys. Rev. Lett. 107 (20), 208302.Google Scholar
Boschan, A., Aguirre, M. & Gauthier, G. 2015 Suspension flow: do particles act as mixers? Soft Matt. 11 (17), 3367–3372.Google Scholar
Bossis, G. & Brady, J. F. 1984 Dynamic simulation of sheared suspensions. I. General method. J. Chem. Phys. 80 (10), 5141–5154.Google Scholar
Boyer, F., Guazzelli, É. & Pouliquen, O. 2011 Unifying suspension and granular rheology. Phys. Rev. Lett. 107 (18), 188301.Google Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111–157.CrossRefGoogle Scholar
Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103–139.Google Scholar
Breedveld, V., van den Ende, D., Tripathi, A. & Acrivos, A. 1998 The measurement of the shear-induced particle and fluid tracer diffusivities in concentrated suspensions by a novel method. J. Fluid Mech. 375, 297–318.Google Scholar
Breedveld, V., Van Den Ende, D., Bosscher, M., Jongschaap, R. & Mellema, J. 2002 Measurement of the full shear-induced self-diffusion tensor of noncolloidal suspensions. J. Chem. Phys. 116 (23), 10529–10535.Google Scholar
Chang, C. & Powell, R. L. 1994 Self-diffusion of bimodal suspensions of hydrodynamically interacting spherical particles in shearing flow. J. Fluid Mech. 281, 51–80.Google Scholar
Corte, L., Chaikin, P., Gollub, J. & Pine, D. 2008 Random organization in periodically driven systems. Nat. Phys. 4 (5), 420–424.Google Scholar
Crouzeix, M. & Raviart, P.-A. 1973 Conforming and nonconforming finite element methods for solving the stationary stokes equations I. Rev. Fr. Autom., Inform., Rec. Opér. Math. 7 (3), 33–75.Google Scholar
Culbertson, C. T., Jacobson, S. C. & Ramsey, J. M. 2002 Diffusion coefficient measurements in microfluidic devices. Talanta 56 (2), 365–373.CrossRefGoogle ScholarPubMed
Cussler, E. L. 2009 Diffusion: Mass Transfer in Fluid Systems. Cambridge University Press.Google Scholar
Da Cunha, F. & Hinch, E. 1996 Shear-induced dispersion in a dilute suspension of rough spheres. J. Fluid Mech. 309, 211–223.Google Scholar
Dabrowski, M., Krotkiewski, M. & Schmid, D. 2008 Milamin: Matlab-based finite element method solver for large problems. Geochem. Geophys. Geosyst. 9 (4), Q04030.CrossRefGoogle Scholar
Deslouis, C., Ezzidi, A. & Tribollet, B. 1991 Mass transfer enhancement by suspensions in a shear flow. J. Appl. Electrochem. 21 (12), 1081–1086.Google Scholar
Drazer, G., Koplik, J., Khusid, B. & Acrivos, A. 2002 Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions. J. Fluid Mech. 460, 307–335.Google Scholar
Eckstein, E. C., Bailey, D. G. & Shapiro, A. H. 1977 Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79 (01), 191–208.Google Scholar
Freund, J. B. 2014 Numerical simulation of flowing blood cells. Annu. Rev. Fluid Mech. 46, 67–95.Google Scholar
Gallier, S., Lemaire, E., Lobry, L. & Peters, F. 2016 Effect of confinement in wall-bounded non-colloidal suspensions. J. Fluid Mech. 799, 100–127.CrossRefGoogle Scholar
Gallier, S., Lemaire, E., Peters, F. & Lobry, L. 2014 Rheology of sheared suspensions of rough frictional particles. J. Fluid Mech. 757, 514–549.Google Scholar
Gaspard, P. 2005 Chaos, Scattering and Statistical Mechanics, vol. 9. Cambridge University Press.Google Scholar
Glazner, A. F. 2014 Magmatic life at low Reynolds number. Geology 42 (11), 935–938.Google Scholar
Glowinski, R., Pan, T.-W., Hesla, T. I. & Joseph, D. D. 1999 A distributed Lagrange multiplier/fictitious domain method for particulate flows. Intl J. Multiphase Flow 25 (5), 755–794.Google Scholar
Goldshmit, E. & Nir, A. 1989 Effective conductivity of a dilute suspension at moderate particle Peclet numbers. Chem. Engng Commun. 82 (1), 163–175.Google Scholar
Haddadi, H. & Morris, J. F. 2014 Microstructure and rheology of finite inertia neutrally buoyant suspensions. J. Fluid Mech. 749, 431–459.Google Scholar
Hoogerbrugge, P. & Koelman, J. 1992 Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys. Lett. 19 (3), 155–160.Google Scholar
James, D. 1968 The thermal diffusivity of ice and water between - 40 and + 60C. J. Mater. Sci. 3 (5), 540–543.Google Scholar
Keller, K. 1971 Effect of fluid shear on mass transport in flowing blood. Fed. Proc. 30 (5), 1591.Google Scholar
Ladd, A. & Verberg, R. 2001 Lattice-Boltzmann simulations of particle–fluid suspensions. J. Stat. Phys. 104 (5–6), 1191–1251.Google Scholar
Leal, L. 1973 On the effective conductivity of a dilute suspension of spherical drops in the limit of low particle Peclet number. Chem. Engng Commun. 1 (1), 21–31.Google Scholar
Lee, C.-Y., Chang, C.-L., Wang, Y.-N. & Fu, L.-M. 2011 Microfluidic mixing: a review. Intl J. Mol. Sci. 12 (5), 3263–3287.Google Scholar
Leighton, D. & Acrivos, A. 1987a Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech. 177, 109–131.Google Scholar
Leighton, D. & Acrivos, A. 1987b The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.Google Scholar
Leshansky, A. M. & Brady, J. F. 2005 Dynamic structure factor study of diffusion in strongly sheared suspensions. J. Fluid Mech. 527, 141–169.Google Scholar
Leshansky, A. M., Morris, J. F. & Brady, J. F. 2008 Collective diffusion in sheared colloidal suspensions. J. Fluid Mech. 597, 305–341.Google Scholar
Marchioro, M. & Acrivos, A. 2001 Shear-induced particle diffusivities from numerical simulations. J. Fluid Mech. 443, 101–128.Google Scholar
Mauri, R. 2003 The constitutive relation of suspensions of noncolloidal particles in viscous fluids. Phys. Fluids 15 (7), 1888–1896.Google Scholar
Maury, B. 1999 Direct simulations of 2D fluid–particle flows in biperiodic domains. J. Comput. Phys. 156 (2), 325–351.Google Scholar
Meakin, P. 1998 Fractals, Scaling and Growth Far from Equilibrium, vol. 5. Cambridge University Press.Google Scholar
Meakin, P., Huang, H., Malthe-Sørenssen, A. & Thøgersen, K. 2013 Shale gas: opportunities and challenges. Environ. Geosci. 20 (4), 151–164.Google Scholar
Metzger, B. & Butler, J. E. 2010 Irreversibility and chaos: role of long-range hydrodynamic interactions in sheared suspensions. Phys. Rev. E 82 (5), 051406.Google Scholar
Metzger, B., Pham, P. & Butler, J. E. 2013a Irreversibility and chaos: role of lubrication interactions in sheared suspensions. Phys. Rev. E 87 (5), 052304.Google Scholar
Metzger, B., Rahli, O. & Yin, X. 2013b Heat transfer across sheared suspensions: role of the shear-induced diffusion. J. Fluid Mech. 724, 527–552.Google Scholar
Meunier, P. & Villermaux, E. 2010 The diffusive strip method for scalar mixing in two dimensions. J. Fluid Mech. 662, 134–172.Google Scholar
Nadim, A., Cox, R. & Brenner, H. 1986 Taylor dispersion in concentrated suspensions of rotating cylinders. J. Fluid Mech. 164, 185–215.Google Scholar
Nir, A. & Acrivos, A. 1976 The effective thermal conductivity of sheared suspensions. J. Fluid Mech. 78 (01), 33–48.CrossRefGoogle Scholar
Pham, P., Butler, J. E. & Metzger, B. 2016 Origin of critical strain amplitude in periodically sheared suspensions. Phys. Rev. Fluids 1 (2), 022201.Google Scholar
Pham, P., Metzger, B. & Butler, J. E. 2015 Particle dispersion in sheared suspensions: crucial role of solid–solid contacts. Phys. Fluids 27 (5), 051701.Google Scholar
Phillips, R. J., Armstrong, R. C., Brown, R. A., Graham, A. L. & Abbott, J. R. 1992 A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids A 4 (1), 30–40.Google Scholar
Pine, D., Gollub, J., Brady, J. & Leshansky, A. 2005 Chaos and threshold for irreversibility in sheared suspensions. Nature 438 (7070), 997–1000.Google Scholar
Rusconi, R. & Stone, H. A. 2008 Shear-induced diffusion of platelike particles in microchannels. Phys. Rev. Lett. 101 (25), 254502.Google Scholar
Schmid, D. W. & Podladchikov, Y. Y. 2003 Analytical solutions for deformable elliptical inclusions in general shear. Geophys. J. Intl 155 (1), 269–288.Google Scholar
Seto, R., Mari, R., Morris, J. F. & Denn, M. M. 2013 Discontinuous shear thickening of frictional hard-sphere suspensions. Phys. Rev. Lett. 111 (21), 218301.CrossRefGoogle ScholarPubMed
Shen, H. H., Hibler, W. D. & Leppäranta, M. 1987 The role of floe collisions in sea ice rheology. J. Geophys. Res. Oceans 92 (C7), 7085–7096.Google Scholar
Shewchuk, J. R. 1996 Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. In Applied Computational Geometry towards Geometric Engineering, pp. 203–222. Springer.Google Scholar
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115–146.Google Scholar
Sierou, A. & Brady, J. F. 2004 Shear-induced self-diffusion in non-colloidal suspensions. J. Fluid Mech. 506, 285–314.Google Scholar
Souzy, M., Pham, P. & Metzger, B. 2016 Taylor’s experiment in a periodically sheared particulate suspension. Phys. Rev. Fluids 1 (4), 042001.Google Scholar
Souzy, M., Yin, X., Villermaux, E., Abid, C. & Metzger, B. 2015 Super-diffusion in sheared suspensions. Phys. Fluids 27 (4), 041705.Google Scholar
Stroock, A. D., Dertinger, S. K., Ajdari, A., Mezić, I., Stone, H. A. & Whitesides, G. M. 2002 Chaotic mixer for microchannels. Science 295 (5555), 647–651.CrossRefGoogle ScholarPubMed
Thøgersen, K., Dabrowski, M. & Malthe-Sørenssen, A. 2016 Transient cluster formation in sheared non-Brownian suspensions. Phys. Rev. E 93 (2), 022611.CrossRefGoogle ScholarPubMed
Wang, L., Koch, D. L., Yin, X. & Cohen, C. 2009 Hydrodynamic diffusion and mass transfer across a sheared suspension of neutrally buoyant spheres. Phys. Fluids 21 (3), 033303.Google Scholar
Wang, N.-H. L. & Keller, K. 1985 Augmented transport of extracellular solutes in concentrated erythrocyte suspensions in Couette flow. J. Colloid Interface Sci. 103 (1), 210–225.Google Scholar
Whitesides, G. M. 2006 The origins and the future of microfluidics. Nature 442 (7101), 368–373.Google Scholar
Widom, B. 1966 Random sequential addition of hard spheres to a volume. J. Chem. Phys. 44 (10), 3888–3894.Google Scholar
Workamp, M., Saggiomo, V. & Dijksman, J. A. 2015 A simple low pressure drop suspension-based microfluidic mixer. J. Micromech. Microengng 25 (9), 094003.Google Scholar
Yamato, P., Tartese, R., Duretz, T. & May, D. A. 2012 Numerical modelling of magma transport in dykes. Tectonophysics 526, 97–109.Google Scholar
Yeo, K. & Maxey, M. R. 2010 Anomalous diffusion of wall-bounded non-colloidal suspensions in a steady shear flow. Europhys. Lett. 92 (2), 24008.Google Scholar
Zarraga, I. E. & Leighton, D. T. Jr. 2002 Measurement of an unexpectedly large shear-induced self-diffusivity in a dilute suspension of spheres. Phys. Fluids 14 (7), 2194–2201.Google Scholar
Zurita-Gotor, M., Bławzdziewicz, J. & Wajnryb, E. 2007 Swapping trajectories: a new wall-induced cross-streamline particle migration mechanism in a dilute suspension of spheres. J. Fluid Mech. 592, 447–469.Google Scholar