Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T23:10:47.442Z Has data issue: false hasContentIssue false
  • English
  • Français

Tensorial rheological model for concentrated non-colloidal suspensions: normal stress differences

Published online by Cambridge University Press:  13 July 2020

Olivier Ozenda Open the ORCID record for Olivier Ozenda [Opens in a new window] ,
Pierre Saramito  and
Guillaume Chambon Open the ORCID record for Guillaume Chambon [Opens in a new window]
Olivier Ozenda
Affiliation:
Lab. J. Kuntzmann, CNRS and Université Grenoble Alpes, CS 40700, 38058Grenoble CEDEX 9, France Université Grenoble Alpes, INRAE, UR ETGR, 2 rue de la Papeterie, 38402St-Martin-d’Hères, France
Pierre Saramito*
Affiliation:
Lab. J. Kuntzmann, CNRS and Université Grenoble Alpes, CS 40700, 38058Grenoble CEDEX 9, France
Guillaume Chambon
Affiliation:
Université Grenoble Alpes, INRAE, UR ETGR, 2 rue de la Papeterie, 38402St-Martin-d’Hères, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Only few rheological models in the literature simultaneously capture the two main non-Newtonian trends of non-colloidal suspensions, namely finite normal stress differences and transient effects. We address this issue by extending a previously proposed minimal model accounting for microstructure anisotropy through a conformation tensor, which was shown to correctly predict transient effects (Ozenda, Saramito & Chambon, J. Rheol., vol. 62 (4), 2018, pp. 889–903). A systematic sensitivity study was performed to provide insights into the physical interpretation of the various model terms. This new model is compared to a large experimental dataset involving varying volume fractions, from dilute to concentrated cases. Both apparent viscosity and normal stress differences in steady state are quantitatively reproduced in the whole range of volume fraction, and qualitative agreement for transient evolution of apparent viscosity during shear reversal is obtained. Furthermore, the model is validated against particle pressure measurements that were not used for parameter identification. Even if the proposed constitutive equation for the Cauchy stress tensor is more difficult to interpret than in the minimal model, this study opens the way for the use of conformation tensor rheological models in applications where the effect of normal stress differences is prominent, like elongational flows or particle migration processes.

JFM classification

Complex Fluids: Suspensions Non-Newtonian Flows: Rheology Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 898 , 10 September 2020 , A25
Copyright
© The Author(s), 2020. 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.)

References

Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order c 2. J. Fluid Mech. 56 (03), 401–427.CrossRefGoogle Scholar
Baumgarten, A. S. & Kamrin, K. 2019 A general fluid–sediment mixture model and constitutive theory validated in many flow regimes. J. Fluid Mech. 861, 721–764.CrossRefGoogle Scholar
Blanc, F., Lemaire, E., Meunier, A. & Peters, F. 2013 Microstructure in sheared non-Brownian concentrated suspensions. J. Rheol. 57 (1), 273–292.CrossRefGoogle Scholar
Blanc, F., Peters, F. & Lemaire, E. 2011 Local transient rheological behavior of concentrated suspensions. J. Rheol. 55 (4), 835–854.CrossRefGoogle Scholar
Boyer, F., Pouliquen, O. & Guazzelli, E. 2011 Dense suspensions in rotating-rod flows: normal stresses and particle migration. J. Fluid Mech. 686, 5–25.CrossRefGoogle Scholar
Brady, J. F. & Bossis, G. 1985 The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech. 155, 105–129.CrossRefGoogle Scholar
Chacko, R. N., Mari, R., Fielding, S. M. & Cates, M. E. 2018 Shear reversal in dense suspensions: the challenge to fabric evolution models from simulation data. J. Fluid Mech. 847, 700–734.CrossRefGoogle Scholar
Couturier, E., Boyer, F., Pouliquen, O. & Guazzelli, E. 2011 Suspensions in a tilted trough: second normal stress difference. J. Fluid Mech. 686, 26–39.CrossRefGoogle Scholar
Dai, S. & Tanner, R. I. 2017 Elongational flows of some non-colloidal suspensions. Rheol. Acta 56 (1), 63–71.CrossRefGoogle Scholar
Dai, S.-C., Bertevas, E., Qi, F. & Tanner, R. I. 2013 Viscometric functions for noncolloidal sphere suspensions with Newtonian matrices. J. Rheol. 57 (2), 493–510.CrossRefGoogle Scholar
Dbouk, T. 2016 A suspension balance direct-forcing immersed boundary model for wet granular flows over obstacles. J. Non-Newtonian Fluid Mech. 230, 68–79.CrossRefGoogle Scholar
Dbouk, T., Lobry, L. & Lemaire, E. 2013 Normal stresses in concentrated non-Brownian suspensions. J. Fluid Mech. 715, 239–272.CrossRefGoogle Scholar
Deboeuf, A., Gauthier, G., Martin, J., Yurkovetsky, Y. & Morris, J. F. 2009 Particle pressure in a sheared suspension: a bridge from osmosis to granular dilatancy. Phys. Rev. Let. 102 (10), 108301.CrossRefGoogle Scholar
Denn, M. M. & Morris, J. F. 2014 Rheology of non-Brownian suspensions. Annu. Rev. Chem. Biomol. Engng 5, 203–228.CrossRefGoogle ScholarPubMed
Einstein, A. 1906 Eine neue bestimmung der moleküldimensionen. Ann. Phys. Ser. 4 19, 289–306.CrossRefGoogle Scholar
Gadala-Maria, F.1979 The rheology of concentrated suspensions. PhD thesis, Stanford University.Google Scholar
Gadala-Maria, F. & Acrivos, A. 1980 Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol. 24 (6), 799–814.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.CrossRefGoogle Scholar
Goddard, J. D. 1982 Memory materials without characteristic time and their relation to the rheology of certain particle suspensions. Adv. Colloid Interface Sci. 17 (1), 241–262.CrossRefGoogle Scholar
Goddard, J. D. 2006 A dissipative anisotropic fluid model for non-colloidal particle dispersions. J. Fluid Mech. 568, 1–17.CrossRefGoogle Scholar
Gordon, R. J. & Schowalter, W. R. 1972 Anisotropic fluid theory: a different approach to the dumbbell theory of dilute polymer solutions. J. Rheol. 16, 79–97.Google Scholar
Guazzelli, E. & Pouliquen, O. 2018 Rheology of dense granular suspensions. J. Fluid Mech. 852, P1.CrossRefGoogle Scholar
Haddadi, H., Shojaei-Zadeh, S., Connington, K. & Morris, J. F. 2014 Suspension flow past a cylinder: particle interactions with recirculating wakes. J. Fluid Mech. 760, R2.CrossRefGoogle Scholar
Hand, G. L. 1962 A theory of anisotropic fluids. J. Fluid Mech. 13 (1), 33–46.CrossRefGoogle Scholar
Hulsen, M. A. 1990 A sufficient condition for a positive definite configuration tensor in differential models. J. Non-Newtonian Fluid Mech. 38 (1), 93–100.CrossRefGoogle Scholar
Jackson, R. 2000 The Dynamics of Fluidized Particles. Cambridge University Press.Google Scholar
Kolli, V. G., Pollauf, E. J. & Gadala-Maria, F. 2002 Transient normal stress response in a concentrated suspension of spherical particles. J. Rheol. 46 (1), 321–334.CrossRefGoogle Scholar
Lehoucq, R., Weiss, J., Dubrulle, B., Amon, A., le Bouil, A., Crassous, J., Amitrano, D. & Graner, F. 2015 Analysis of image versus position, scale and direction reveals pattern texture anisotropy. Front. Phys. 2, 84.CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.CrossRefGoogle Scholar
Mari, R., Seto, R., Morris, J. F. & Denn, M. M. 2014 Shear thickening, frictionless and frictional rheologies in non-Brownian suspensions. J. Rheol. 58 (6), 1693–1724.CrossRefGoogle Scholar
Maron, S. H. & Pierce, P. E. 1956 Application of Ree–Eyring generalized flow theory to suspensions of spherical particles. J. Colloid Sci. 11 (1), 80–95.Google Scholar
Miller, R. M. & Morris, J. F. 2006 Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions. J. Non-Newtonian Fluid Mech. 135 (2–3), 149–165.CrossRefGoogle Scholar
More, J. J., Garbow, B. S. & Hillstrom, K. E.1980 User guide for MINIPACK-1.[in fortran]. Tech. Rep. Argonne National Lab., IL, USA. doi:10.2172/6997568.CrossRefGoogle Scholar
Morris, J. F. & Boulay, F. 1999 Curvilinear flows of noncolloidal suspensions: the role of normal stresses. J. Rheol. 43 (5), 1213–1237.CrossRefGoogle Scholar
Newville, M., Stensitzki, T., Allen, D. B., Rawlik, M., Ingargiola, A. & Nelson, A. 2016 Lmfit: non-linear least-square minimization and curve-fitting for Python. Astrophys. Source Code Lib. Available at https://ui.adsabs.harvard.edu/abs/2016ascl.soft06014N.Google Scholar
Nott, P. R. & Brady, J. F. 1994 Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199.CrossRefGoogle Scholar
Nott, P. R., Guazzelli, E. & Pouliquen, O. 2011 The suspension balance model revisited. Phys. Fluids 23 (4), 043304.CrossRefGoogle Scholar
Ozenda, O., Saramito, P. & Chambon, G. 2018 A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in Newtonian fluids. J. Rheol. 62 (4), 889–903.CrossRefGoogle Scholar
Pailha, M. & Pouliquen, O. 2009 A two-phase flow description of the initiation of underwater granular avalanches. J. Fluid Mech. 633, 115–135.CrossRefGoogle Scholar
Peters, F., Ghigliotti, G., Gallier, S., Blanc, F., Lemaire, E. & Lobry, L. 2016 Rheology of non-Brownian suspensions of rough frictional particles under shear reversal: a numerical study. J. Rheol. 60 (4), 715–732.CrossRefGoogle Scholar
Phan-Thien, N. 1995 Constitutive equation for concentrated suspensions in Newtonian liquids. J. Rheol. 39 (4), 679–695.CrossRefGoogle Scholar
Radhakrishnan, K. & Hindmarsh, A. C.1993 Description and use of lsode, the livermore solver for ordinary differential equations. Tech. Rep. L.L.N. Lab., Livermore, CA (USA).CrossRefGoogle Scholar
Royer, J. R., Blair, D. L. & Hudson, S. D. 2016 Rheological signature of frictional interactions in shear thickening suspensions. Phys. Rev. Let. 116 (18), 188301.CrossRefGoogle ScholarPubMed
Saramito, P. 2016 Complex Fluids: Modelling and Algorithms. Springer.CrossRefGoogle Scholar
Sierou, A. & Brady, J. F. 2002 Rheology and microstructure in concentrated noncolloidal suspensions. J. Rheol. 46, 1031–1056.CrossRefGoogle Scholar
Singh, A., Mari, R., Denn, M. M. & Morris, J. F. 2018 A constitutive model for simple shear of dense frictional suspensions. J. Rheol. 62 (2), 457–468.CrossRefGoogle Scholar
Singh, A. & Nott, P. R. 2003 Experimental measurements of the normal stresses in sheared Stokesian suspensions. J. Fluid Mech. 490, 293–320.CrossRefGoogle Scholar
Stickel, J. J., Phillips, R. J. & Powell, R. L. 2006 A constitutive model for microstructure and total stress in particulate suspensions. J. Rheol. 50 (4), 379–413.CrossRefGoogle Scholar
Stickel, J. J., Phillips, R. J. & Powell, R. L. 2007 Application of a constitutive model for particulate suspensions: time-dependent viscometric flows. J. Rheol. 51 (6), 1271–1302.CrossRefGoogle Scholar
Yapici, K., Powell, R. L. & Phillips, R. J. 2009 Particle migration and suspension structure in steady and oscillatory plane Poiseuille flow. Phys. Fluids 21 (5), 053302.CrossRefGoogle Scholar
Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J. et al. 2020 SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nat. Meth. 17, 261–272.CrossRefGoogle ScholarPubMed
Zarraga, I. E., Hill, D. A. & Leighton, D. T. Jr 2000 The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J. Rheol. 44 (2), 185–220.CrossRefGoogle Scholar