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Investigation into the rheology of a solid sphere suspension in second-order fluid using a cell model

Liam J. Escott* and Helen J. Wilson

  • Department of Mathematics, UCL, Gower Street, London WC1E 6BT, United Kingdom

  • *liam.escott.12@ucl.ac.uk
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Phys. Rev. Fluids 5, 083301 – Published 3 August, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.083301

Abstract

We consider a suspension of solid spheres in a viscoelastic suspending fluid. We use the second-order fluid model to capture the first effects of viscoelasticity in the suspension and a mean-field cell model to express the influence of the solid particles. We create a semianalytical constitutive equation for the whole suspension, a second-order fluid with modified material parameters. We find that, in simple shear, the ratio of first normal stress difference to shear stress is independent of volume fraction, but that the second normal stress difference can change sign as the volume fraction of solids increases. Our model is valid for all flows for which the second-order fluid expansion is applicable.

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References (48)

  1. L. Bergström, Shear thinning and shear thickening of concentrated ceramic suspensions, Colloids Surf. A 133, 151 (1998).
  2. A. J. Liu and S. R. Nagel, Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales (CRC Press, Boca Raton, FL, 2001).
  3. N. Fernandez, R. Mani, D. Rinaldi, D. Kadau, M. Mosquet, H. Lombois-Burger, J. Cayer-Barrioz, H. J. Herrmann, N. D. Spencer, and L. Isa, Microscopic Mechanism for Shear Thickening of Non-Brownian Suspensions, Phys. Rev. Lett. 111, 108301 (2013).
  4. M. Wyart and M. E. Cates, Discontinuous Shear Thickening Without Inertia in Dense Non-Brownian Suspensions, Phys. Rev. Lett. 112, 98302 (2014).
  5. R. Seto, G. G. Giusteri, and A. Martiniello, Microstructure and thickening of dense suspensions under extensional and shear flows, J. Fluid Mech. 825, R3 (2017).
  6. X. Xu, S. A. Rice, and A. R. Dinner, Relation between ordering and shear thinning in colloidal suspensions, Proc. Natl. Acad. Sci. USA 110, 1 (2013).
  7. A. Vázquez-Quesada, A. Mahmud, S. Dai, M. Ellero, and R. I. Tanner, Investigating the causes of shear-thinning in non-colloidal suspensions: Experiments and simulations, J. Non-Newtonian Fluid Mech. 248, 1 (2017).
  8. D. J. Highgate and R. W. Whorlow, Rheological properties of suspensions of spheres in non-Newtonian media, Rheol. Acta 9, 569 (1970).
  9. T. Kataoka, T. Kitano, M. Sasahara, and K. Nishijima, Viscosity of particle filled polymer melts, Rheol. Acta 17, 149 (1978).
  10. H. A. Barnes, A review of the rheology of filled viscoelastic systems, Rheol. Rev. 2003, 1 (2003).
  11. M. Liard, N. S. Martys, W. L. George, D. Lootens, and P. Hebraud, Scaling laws for the flow of generalized Newtonian colloidal suspensions, J. Rheol. 58, 1993 (2014).
  12. S. Mall-Gleissle, W. Gleissle, G. H. McKinley, and H. Buggisch, The normal stress behavior of suspensions with viscoelastic matrix fluids, Rheol. Acta 41, 61 (2002).
  13. R. Tanner, S. Dai, F. Qi, and K. Housiadas, Viscometric functions of semidilute noncolloidal suspensions of spheres in a viscoelastic matrix, J. Non-Newtonian Fluid Mech. 201, 130 (2013).
  14. R. Tanner, F. Qi, and S. Dai, Scaling the normal stresses in concentrated non-colloidal suspensions of spheres, Rheol. Acta 52, 291 (2013).
  15. S. Dai, F. Qi, and R. I. Tanner, Viscometric functions of concentrated non-colloidal suspensions of spheres in a viscoelastic matrix, J. Rheol. 58, 183 (2014).
  16. D. L. Koch, E. F. Lee, and I. Mustafa, Stress in a dilute suspension of spheres in a dilute polymer solution subject to simple shear flow at finite Deborah numbers, Phys. Rev. Fluids 1, 013301 (2016).
  17. S. Krishnan, E. S. G. Shaqfeh, and G. Iaccarino, Fully resolved viscoelastic particulate simulations using unstructured grids, J. Comput. Phys. 338, 313 (2017).
  18. A. Einstein, Eine neue bestimmung der moleküldimensionen, Annalen der Physik 324, 289 (1906).
  19. G. K. Batchelor and J. T. Green, The determination of the bulk stress in a suspension of spherical particles to order c2, J. Fluid Mech. 56, 401 (1972).
  20. M. Cohen and D. Turnbull, Molecular transport in liquids and glasses, J. Chem. Phys. 31, 1164 (1959).
  21. Z. Cheng, J. Zhu, P. M. Chaikin, S. E. Phan, and W. B. Russel, Nature of the divergence in low shear viscosity of colloidal hard-sphere dispersions, Phys. Rev. E 65, 041405 (2002).
  22. I. M. Krieger and T. J. Dougherty, A mechanism for non-Newtonian flow in suspensions of rigid spheres, Trans. Soc. Rheol. 3, 137 (1959).
  23. P. N. Kaloni and V. Statsna, Steady-shear rheological behavior of the suspension of spherical particles in a second-order fluid, Polym. Eng. Sci. 23, 465 (1983).
  24. K. Sun and K. Jayamaran, Bulk rheology of dilute suspensions in viscoelastic liquids, Rheol. Acta 23, 84 (1984).
  25. R. T. Mifflin, Dissipation in a dilute system of spheres in a second-order fluid, J. Non-Newtonian Fluid Mech. 17, 267 (1985).
  26. N. A. Patankar and H. H. Hu, Rheology of a suspension of particles in viscoelastic fluids, J. Non-Newtonian Fluid Mech. 96, 427 (2001).
  27. F. Greco, G. D'Avino, and P. L. Maffettone, Rheology of a dilute suspension of rigid spheres in a second-order fluid, J. Non-Newtonian Fluid Mech. 147, 1 (2007).
  28. K. D. Housiadas and R. I. Tanner, On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid, J. Non-Newtonian Fluid Mech. 162, 88 (2009).
  29. D. L. Koch and G. Subramanian, Corrigendum to “The stress in a dilute suspension of spheres suspended in a second-order fluid subject to a linear velocity field,” J. Non-Newtonian Fluid Mech. 153, 202 (2008).
  30. D. L. Koch and G. Subramanian, The stress in a dilute suspension of spheres suspended in a second-order fluid subject to a linear velocity field, J. Non-Newtonian Fluid Mech. 138, 87 (2006).
  31. J. M. Rallison, The stress in a dilute suspension of liquid spheres in a second-order fluid, J. Fluid Mech. 693, 500 (2012).
  32. R. Simha, A treatment of the viscosity of concentrated suspensions, J. Appl. Phys. 23, 1020 (1952).
  33. J. Happel, Viscosity of suspensions of uniform spheres, J. Appl. Phys. 28, 1288 (1957).
  34. H. Ohshima, Sedimentation potential in a concentrated suspension of spherical colloidal particles, J. Colloid Interface Sci. 208, 295 (1998).
  35. J. M. Ding and H. J. Keh, The electrophoretic mobility and electric conductivity of a concentrated suspension of colloidal spheres with arbitrary double-layer thickness, J. Colloid Interface Sci. 236, 180 (2001).
  36. R. Pal, Evaluation of theoretical viscosity models for concentrated emulsions at low capillary numbers, Chem. Eng. J. 81, 15 (2001).
  37. H. Usui, Prediction of dispersion characteristics and rheology in dense slurries, J. Chem. Eng. Jpn. 35, 815 (2002).
  38. M. Yang and E. S. G. Shaqfeh, Mechanism of shear thickening in suspensions of rigid spheres in Boger fluids. Part I: Dilute suspensions, J. Rheol. 62, 1363 (2018).
  39. J. Einarsson, M. Yang, and E. S. G. Shaqfeh, The Einstein viscosity with fluid elasticity, Phys. Rev. Fluids 3, 013301 (2017).
  40. M. S. Chong, A. E. Perry, B. J. Cantwell, M. S. Chong, and A. E. Perry, A general classification of three-dimensional flow fields, Phys. Fluids A 2, 765 (1990).
  41. J. D. Sherwood, Cell models for suspension viscosity, Chem. Eng. Sci. 61, 6727 (2006).
  42. R. S. Rivlin and J. L. Ericksen, in Stress-Deformation Relations for Isotropic Materials, edited by G. I. Barenblatt, D. D. Joseph (Springer, New York, 1997).
  43. M. Yang and E. S. G. Shaqfeh, Mechanism of shear thickening in suspensions of rigid spheres in Boger fluids. Part II: Suspensions at finite concentration, J. Rheol. 62, 1379 (2018).
  44. R. J. Phillips and L. Talini, Chaining of weakly interacting particles suspended in viscoelastic fluids, J. Non-Newtonian Fluid Mech. 147, 175 (2007).
  45. J. H. Peery, Fluid mechanics of rigid and deformable particles in shear flows at low Reynolds numbers, Ph.D. thesis, Princeton University (1966).
  46. B. P. Ho and L. G. Leal, Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid, J. Fluid Mech. 76, 783 (1976).
  47. M. Yang, S. Krishnan, and E. S. G. Shaqfeh, Numerical simulations of the rheology of suspensions of rigid spheres at low volume fraction in a viscoelastic fluid under shear, J. Non-Newtonian Fluid Mech. 234, 51 (2016).
  48. D. Park, Direct calculation of inter-particle distance in suspension by image processing, Powder Technol. 330, 252 (2018).

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Phys. Rev. Fluids 5, 083301– Published 3 August, 2020
  • Received 28 June 2019
  • Accepted 26 June 2020
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