Shear-induced particle migration in the flow of particle suspensions through a sudden plane expansion

  • Ivan Rosa de Siqueira
  • Márcio Silveira da Carvalho
Technical Paper
First Online:
Received:
Accepted:
  • 29 Downloads

Abstract

This work investigates the creeping flow of particle suspensions through an abrupt 1:4 plane expansion. The suspension is composed by non-Brownian, rigid spherical particles suspended in a viscous Newtonian liquid. The flow is described by mass and momentum conservation equations together with a convection–diffusion transport equation that accounts for shear-induced particle migration according to the diffusive flux model. The resulting set of fully coupled, nonlinear differential equations is solved with a stabilized finite element method. A detailed parametric study is presented in terms of suspension bulk concentration, particle Péclet number and ratio of diffusivity coefficients, and the numerical results show significant effects of these parameters on the particle concentration field in the flow.

Keywords

Suspensions Particle migration Sudden expansion Finite element method 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors would like to acknowledge the financial support from the Brazilian Research Council (CNPq) and the Industrial Partnership for Research in Interfacial Materials & Engineering (IPRIME) of the University of Minnesota.

References

  1. 1.
    Tadros TF (2011) Rheology of dispersions: principles and applications. John Wiley & Sons, New JerseyGoogle Scholar
  2. 2.
    Tadros TF (2012) Dispersion of powders in liquids and stabilization of suspensions. John Wiley & Sons, New JerseyCrossRefGoogle Scholar
  3. 3.
    Barnes HA (1980) A survey of industrial problems and academic progress. Dispersion Rheology. R.I.C. Industrial Division Report, 1981Google Scholar
  4. 4.
    Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids. Fluid mechanics, vol 1. John Wiley & Sons, New JerseyGoogle Scholar
  5. 5.
    Macosko CW (1994) Rheology: principles, measurements and applications. Wiley, New JerseyGoogle Scholar
  6. 6.
    Stickel JJ, Powell RL (2005) Fluid mechanics and rheology of dense suspensions. Ann Rev Fluid Mech 37:129–149MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bahiraei M (2016) Particle migration in nanofluids: a critical review. Int J Therm Sci 109:90–113CrossRefGoogle Scholar
  8. 8.
    Leighton D, Acrivos A (1987) The shear-induced migration of particles in concentrated suspensions. J Fluid Mech 181:415–439CrossRefGoogle Scholar
  9. 9.
    Abbott JR, Tetlow N, Graham AL, Altobelli SA, Fukushima E, Mondy LA, Stephens TS (1991) Experimental observations of particle migration in concentrated suspensions: Couette flow. J Rheol 35:773–797CrossRefGoogle Scholar
  10. 10.
    Altobelli SA, Givler RC (1991) Velocity and concentration measurements of suspensions by nuclear magnetic resonance imaging. J Rheol 35:721–734CrossRefGoogle Scholar
  11. 11.
    Graham AL, Altobelli SA (1991) NMR imaging of shear-induced diffusion and structure in concentrated suspensions. J Rheol 35:191–201CrossRefGoogle Scholar
  12. 12.
    Sinton SA, Chow A (1991) NMR flow imaging of fluids and solid suspensions in poiseuille flow. J Rheol 35:735–772CrossRefGoogle Scholar
  13. 13.
    Phillips RJ, Armstrong RC, Brown RC, Graham AL, Abbott JR (1992) A constitutive equation for concentrated suspensions that accounts for shear-induced particle migrations. Phys Fluids 4:30–40CrossRefMATHGoogle Scholar
  14. 14.
    Koh CJ, Hookham P, Leal LG (1994) An experimental investigation of concentrated suspension flows in a rectangular channel. J Fluid Mech 266:1–32CrossRefGoogle Scholar
  15. 15.
    Phan-Thien N, Graham AL, Altobelli SA, Abbott JR, Mondy LA (1995) Hydrodynamic particle migration in a concentrated suspension undergoing flow between rotating eccentric cylinders. Ind Eng Chem Res 34:3187–3194CrossRefGoogle Scholar
  16. 16.
    Hampton RE, Mammoli AA, Graham AL, Tetlow N, Altobelli SA (1997) Migration of particles undergoing pressure-driven flow in a circular conduit. J Rheol 41:621–640CrossRefGoogle Scholar
  17. 17.
    Subia SR, Ingber MS, Mondy LA, Altobelli SA, Graham AL (1998) Modelling of concentrated suspensions using a continuum constitutive equation. J Fluid Mech 373:193–219CrossRefMATHGoogle Scholar
  18. 18.
    Butler JE, Bonnecaze RT (1999) Imaging of particle shear migration with electrical impedance tomography. Phys Fluids 11:1982–1994CrossRefMATHGoogle Scholar
  19. 19.
    Han M, Kim C, Kim M, Lee S (1999) Particle migration in tube flow of suspensions. J Rheol 43:1157–1174CrossRefGoogle Scholar
  20. 20.
    Norman JT, Nayak HV, Bonnecaze RT (2005) Migration of buoyant particles in low-Reynolds-number pressure-driven flows. J Fluid Mech 523:1–35CrossRefMATHGoogle Scholar
  21. 21.
    Timberlake DM, Morris JF (2005) Particle migration and free-surface topography in inclined plane flow of a suspension. J Fluid Mech 538:309–341CrossRefMATHGoogle Scholar
  22. 22.
    van Dinther AMC, Schroen CGPH, Vergeldt FJ, van der Sman RGM (2012) Suspension flow in microfluidic devices a review of experimental techniques focusing on concentration and velocity gradients. Adv Colloid Interface Sci 173:23–34CrossRefGoogle Scholar
  23. 23.
    Graham AL, Mammoli AA, Busch MB (1998) Effects of demixing on suspension rheometry. Rheol Acta 37:139–150CrossRefGoogle Scholar
  24. 24.
    Tetlow N, Graham AL, Ingber MS, Rubia SR, Mondy LA, Altobelli SA (1998) Particle migration in a Couette apparatus: experiment and modeling. J Rheol 42:307–327CrossRefGoogle Scholar
  25. 25.
    Chapman B (1990) Shear-induced migration phenomena in concentrated suspensions. PhD thesis, University of Notre-Dame, Notre DameGoogle Scholar
  26. 26.
    Chow AW, Sinton SW, Iwamiya JH, Stephens TS (1994) Shear-induced migration in Couette and parallel-plate viscometers: NMR imaging and stress measurements. Phys Fluids 6:2561–2576CrossRefGoogle Scholar
  27. 27.
    Krishnan GP, Beimfohr S, Leighton D (1996) Shear-induced radial segregation in bidisperse suspensions. J Fluid Mech 321:371–393CrossRefGoogle Scholar
  28. 28.
    Kim JM, Lee SG, Kim C (2008) Numerical simulations of particle migration in suspension flows: frame-invariant formulation of curvature-induced migration. J Nonnewton Fluid Mech 150:162–176CrossRefMATHGoogle Scholar
  29. 29.
    Miller RM, Morris JF (2006) Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions. J Nonnewton Fluid Mech 135:149–165CrossRefMATHGoogle Scholar
  30. 30.
    Rebouças RB, Siqueira IR, de Souza Mendes PR, Carvalho MS (2016) On the pressure-driven flow of suspensions: particle migration in shear sensitive liquids. J Nonnewton Fluid Mech 234:178–187MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ahmed GMY, Singh A (2011) Numerical simulation of particle migration in asymmetric bifurcation channel. J Nonnewton Fluid Mech 166:42–51CrossRefMATHGoogle Scholar
  32. 32.
    Yadav S, Reddy MM, Singh A (2015) Shear-induced particle migration in three-dimensional bifurcation channel. Int J Multiph Flow 76:1–12CrossRefGoogle Scholar
  33. 33.
    Campana DM, Silva LDV, Carvalho MS (2017) Slot coating flows of non-colloidal particle suspensions. AIChE J 63:1122–1131CrossRefGoogle Scholar
  34. 34.
    Siqueira IR, Carvalho MS (2017) Particle migration in planar die-swell flows. J Fluid Mech 825:49–68MathSciNetCrossRefGoogle Scholar
  35. 35.
    Siqueira IR, Rebouças RB, Carvalho MS (2017) Particle migration and alignment in slot coating flows of elongated particle suspensions. AIChE J 63:3187–3198CrossRefGoogle Scholar
  36. 36.
    Siqueira IR, Rebouças RB, Carvalho MS (2017) Migration and alignment in the flow of elongated particle suspensions through a converging–diverging channel. J Nonnewton Fluid Mech 243:56–63MathSciNetCrossRefGoogle Scholar
  37. 37.
    Rao RR, Mondy LA, Baer TA, Altobelli SA, Stephens TS (2002) NMR measurements and simulations of particle migration in non-Newtonian fluids. Chem Eng Commun 189(1):1–22CrossRefGoogle Scholar
  38. 38.
    Lavrenteva OM, Nir A (2016) Shear-induced particles migration in a Bingham fluid. J Nonnewton Fluid Mech 238:80–91MathSciNetCrossRefGoogle Scholar
  39. 39.
    Manica R, de Bortoli AL (2004) Simulation of sudden expansion flows for power-law fluids. J Nonnewton Fluid Mech 121:35–40CrossRefMATHGoogle Scholar
  40. 40.
    Ternik P, Marn J, Zunič Z (2006) Non-Newtonian fluid flow through a planar symmetric expansion: shear-thickening fluids. J Nonnewton Fluid Mech 135:136–148CrossRefMATHGoogle Scholar
  41. 41.
    Ternik P (2009) Planar sudden symmetric expansion flows and bifurcation phenomena of purely viscous shear-thinning fluids. J Nonnewton Fluid Mech 157:15–25CrossRefMATHGoogle Scholar
  42. 42.
    Dhinakaran S, Oliveira MSN, Pinho FT, Alves MA (2013) Steady flow of power-law fluids in a 1:3 planar sudden expansion. J Nonnewton Fluid Mech 198:48–58CrossRefGoogle Scholar
  43. 43.
    Missirlis K, Assimacopoulos D, Mitsoulis E (1998) A finite volume approach in the simulation of viscoelastic expansion flows. J Nonnewton Fluid Mech 78:91–118CrossRefMATHGoogle Scholar
  44. 44.
    Oliveira PJ (2003) Asymmetric flows of viscoelastic fluids in symmetric planar expansion geometries. J Nonnewton Fluid Mech 114:33–63CrossRefMATHGoogle Scholar
  45. 45.
    Rocha GN, Poole RJ, Oliveira PJ (2007) Bifurcation phenomena in viscoelastic flows through a symmetric 1:4 expansion. J Nonnewton Fluid Mech 141:1–17CrossRefMATHGoogle Scholar
  46. 46.
    Poole RJ, Alves MA, Oliveira PJ, Pinho FT (2007) Plane sudden expansion flows of viscoelastic liquids. J Nonnewton Fluid Mech 145:79–91CrossRefMATHGoogle Scholar
  47. 47.
    Poole RJ, Pinho FT, Alves MA, Oliveira PJ (2009) The effect of expansion ratio for creeping expansion flows of UCM fluids. J Nonnewton Fluid Mech 163:35–44CrossRefMATHGoogle Scholar
  48. 48.
    Sousa PC, Coelho PM, Oliveira MSN, Alves MA (2011) Laminar flow in three-dimensional squaren-square expansions. J Nonnewton Fluid Mech 166:1033–1048CrossRefMATHGoogle Scholar
  49. 49.
    Vradis GC, Ötügen MV (1997) The axisymmetric sudden expansion flow of a non-Newtonian viscoplastic fluid. J Fluids Eng 110:193–200CrossRefGoogle Scholar
  50. 50.
    Jay P, Magnin A, Piau JM (2002) Viscoplastic fluid flow through a sudden axisymmetric expansion. AIChE J 47(10):2155–2166CrossRefGoogle Scholar
  51. 51.
    Alexandrou AN, McGilvreay TM, Burgos G (2001) Steady Herschel-Bulkley fluid flow in three-dimensional expansions. J Nonnewton Fluid Mech 100:77–96CrossRefMATHGoogle Scholar
  52. 52.
    Nassar B, de Souza Mendes PR, Naccache MF (2011) Flow of elasto-viscoplastic liquids through an axisymmetric expansion–contraction. J Nonnewton Fluid Mech 166:386–394CrossRefMATHGoogle Scholar
  53. 53.
    Hermany L, dos Santos DD, Frey S, Naccache MF, de Souza Mendes PR (2013) Flow of yield-stress liquids through an axisymmetric abrupt expansion–contraction. J Nonnewton Fluid Mech 201:1–9CrossRefGoogle Scholar
  54. 54.
    Link FB, Frey S, Thompson RL, Naccache MF, de Souza Mendes PR (2015) Plane flow of thixotropic elasto-viscoplastic materials through a 1:4 sudden expansion. J Nonnewton Fluid Mech 220:162–174MathSciNetCrossRefGoogle Scholar
  55. 55.
    Leonard EF, West AC, Shapley NC, Larsen MU (2004) Dialysis without membrances: How and why? Blood Purif 22:92–100CrossRefGoogle Scholar
  56. 56.
    Faivre M, Abkarian M, Bickraj K, Stone HA (2006) Geometrical focusing of cells in a microfluidic device: an approach to separate blood plasma. Birheology 43:147–159Google Scholar
  57. 57.
    Jaggi RD, Sandoz R, Effenhauser CS (2007) Microfluidic depletion of red blood cells from whole blood in high-aspect-ratio microchannels. Microfluid Nanofluid 3:47–53CrossRefGoogle Scholar
  58. 58.
    Arola DF, Powell RL, McCarthy MJ, Li T-Q, Ödberg L (1998) NMR imaging of pulp suspension flowing through an abrupt pipe expansion. AIChE J 44:2597–2606CrossRefGoogle Scholar
  59. 59.
    Jossic L, Magnin A (2005) Structuring of gelled suspensions flowing through a sudden three-dimensional expansion. J Nonnewton Fluid Mech 127:201–212CrossRefGoogle Scholar
  60. 60.
    Moraczewski T, Tanga H, Shapley NC (2005) Flow of a concentrated suspension through an abrupt axisymmetric expansion measured by nuclear magnetic resonance imaging. J Rheol 49:1409–1428CrossRefGoogle Scholar
  61. 61.
    Moraczewski T, Shapley NC (2006) The effect of inlet conditions on concentrated suspension flows in abrupt expansions. Phys Fluids 18:123303CrossRefMATHGoogle Scholar
  62. 62.
    Moraczewski T, Shapley NC (2007) Pressure drop enhancement in a concentrated suspension flowing through an abrupt axisymmetric contraction–expansion. Phys Fluids 19:103304CrossRefMATHGoogle Scholar
  63. 63.
    Xi C, Shapley NC (2008) Flows of concentrated suspensions through an asymmetric bifurcation. J Rheol 52:625–647CrossRefGoogle Scholar
  64. 64.
    Nott PR, Brady JF (1994) Pressure-driven flow of suspensions: simulation and theory. J Fluid Mech 275:157–199CrossRefMATHGoogle Scholar
  65. 65.
    Mendoza CI, Santamaría-Holek I (2009) The rheology of hard sphere suspensions at arbitrary volume fractions: an improved differential viscosity model. J Chem Phys 130:044904CrossRefGoogle Scholar
  66. 66.
    Einstein A (1911) Berichtigung zu meiner arbeit: Eine nene bestimmung der molekuldimension. Annalen der Physik 34:591–592CrossRefMATHGoogle Scholar
  67. 67.
    Krieger IM (1972) Rheology of monodisperse latices. Adv Colloid Interface Sci 3:111–136CrossRefGoogle Scholar
  68. 68.
    Szady MJ, Salamon TR, Liu AW, Armstrong RC, Brown RA (1995) A new mixed finite element method for viscoelastic flows governed by differential constitutive equations. J Nonnewton Fluid Mech 59:215–243CrossRefGoogle Scholar
  69. 69.
    Guénette R, Fortin M (1995) A new mixed finite element method for computing viscoelastic flows. J Nonnewton Fluid Mech 60:27–52CrossRefGoogle Scholar
  70. 70.
    Pasquali M, Scriven LE (2002) Free surface flows of polymer solutions with models based on the conformation tensor. J Nonnewton Fluid Mech 108:363–409CrossRefMATHGoogle Scholar
  71. 71.
    Duff IS, Erisman AM, Reid JK (1989) Direct method for sparse matrices. OUP Oxford, OxfordMATHGoogle Scholar
  72. 72.
    Bolstad JH, Keller HB (1986) A multigrid continuation method for elliptic problems. J Sci Stat Comput 7:1081–1104MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • Ivan Rosa de Siqueira
    • 1
  • Márcio Silveira da Carvalho
    • 2
  1. 1.Department of Chemical and Biomolecular EngineeringRice UniversityHoustonUSA
  2. 2.Department of Mechanical EngineeringPontifícia Universidade Católica do Rio de JaneiroRio de JaneiroBrazil

Personalised recommendations