# Numerical simulations of the motion of ellipsoids in planar Couette flow of Giesekus viscoelastic fluids

**Part of the following topical collections:**

## Abstract

The motion of neutrally buoyant ellipsoids in a planar Couette flow of Giesekus viscoelastic fluids between two narrowly set plates is numerically simulated with a fictitious domain method. The aspect ratio of the ellipsoid is 4 (i.e., prolate spheroids) and the Deborah number (*De*) ranges from 0 to 4.0. For a single ellipsoid initially placed in the mid-plane between the two plates, the ellipsoid major axis rotates around the vorticity axis in a kayaking mode at relatively low Deborah numbers, and is tilted in the flow-vorticity plane when the Deborah number exceeds a critical value, with the orientation being closer to the flow direction for a larger *De*. For a single ellipsoid initially not placed in the mid-plane, the ellipsoid undergoes lateral migration toward the nearby wall, and it is interesting that the ellipsoid turns its orientation to the vorticity axis at relatively small *De* and a direction close to the vorticity axis at large *De* (above 3.0), in contrast to the ellipsoid placed in the mid-plane without lateral migration, whose terminal orientation exhibits a kayaking motion at relatively small *De* and is close to the flow direction for *De* > 3. As a result, for the multiple-ellipsoid case, there exists a transient stage where the average orientation of the ellipsoids turns toward the vorticity axis for all nonzero Deborah numbers studied, and the orientation close to the vorticity axis can be often observed for the isolated ellipsoids. Both the particle interactions and the wall effect promote the ellipsoids to align with the flow direction. Particle aggregation and the dynamic aligning structures are observed at large Deborah numbers.

## Keywords

Lateral migration Planar Couette flow Giesekus viscoelastic fluid Ellipsoid## Notes

### Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant Nos. 11632016, 91752117).

## References

- Bartram E, Goldsmith HL, Mason SG (1975) Particle motions in non-Newtonian media, III. Further observations in viscoelastic fluids. Rheol Acta 14:776–782CrossRefGoogle Scholar
- Borzacchiello D et al (2016) Orientation kinematics of short fibres in a second-order viscoelastic fluid. Rheol Acta 55:397–409CrossRefGoogle Scholar
- Brunn P (1980) The motion of rigid particles in viscoelastic fluids. J Non-Newton Fluid Mech 7:271–288CrossRefGoogle Scholar
- Caserta S, D’Avino G, Greco F, Guido S, Maffettone PL (2010) Migration of a sphere in a viscoelastic fluid under planar shear flow: experiments and numerical predictions. Soft Matter 7:1100–1106CrossRefGoogle Scholar
- Cohen C, Chung B, Stasiak W (1987) Orientation and rheology of rod-like particles with weak Brownian diffusion in a 2nd-order fluid under simple shear-flow. Rheol Acta 26:217–232CrossRefGoogle Scholar
- Crowe CT et al (2011) Multiphase flows with droplets and particles. CRC Press, Boca Raton, FLCrossRefGoogle Scholar
- D’Avino G, Maffettone PL (2015) Particle dynamics in viscoelastic liquids. J Non-Newton Fluid Mech 215:80–104MathSciNetCrossRefGoogle Scholar
- D’Avino G, Tuccillo T, Hulsen MA, Greco F, Maffettone PL (2010a) Numerical simulations of particle migration in a viscoelastic fluid subjected to shear flow. Comput Fluids 39:709–721MathSciNetCrossRefGoogle Scholar
- D’Avino G, Maffettone PL, Greco F, Hulsen MA (2010b) Viscoelasticity-induced migration of a rigid sphere in confined shear flow. J Non-Newton Fluid Mech 165:466–474CrossRefGoogle Scholar
- D’Avino G, Hulsen MA, Greco F, Maffettone PL (2014) Bistability and metabistability scenario in the dynamics of an ellipsoidal particle in a sheared viscoelastic fluid. Phys Rev E 89:043006CrossRefGoogle Scholar
- D’Avino G, Greco F, Maffettone PL (2015) Rheology of a dilute viscoelastic suspension of spheroids in unconfined shear flow. Rheol Acta 54:915–928CrossRefGoogle Scholar
- D’Avino G, Greco F, Maffettone PL (2017) Particle migration due to viscoelasticity of the suspending liquid and its relevance in microfluidic devices. Annu Rev Fluid Mech 49:341–360MathSciNetCrossRefGoogle Scholar
- D’Avino G, Hulsen MA, Greco F, Maffettone PL (2019) Numerical simulations on the dynamics of a spheroid in a viscoelastic liquid in a wide-slit microchannel. J Non-Newton Fluid Mech 263:33–41MathSciNetCrossRefGoogle Scholar
- de Oliveira IS, den Otter WK, Briels WJ (2013) Alignment and segregation of bidisperse colloids in a shear-thinning viscoelastic fluid under shear flow. Europhys Lett 101:28002CrossRefGoogle Scholar
- Férec J, Bertevas E, Khoo BC, Ausias G, Thien NP (2017) Steady-shear rheological properties for suspensions of axisymmetric particles in second-order fluids. J Non-Newton Fluid Mech 239:62–72MathSciNetCrossRefGoogle Scholar
- Gauthier F, Goldsmith HL, Mason SG (1971) Particle motions in non-Newtonian media. I. Couette flow. Rheol Acta 10:344–364CrossRefGoogle Scholar
- Glowinski R, Pan TW, Hesla TI, Joseph DD (1999) A distributed lagrange multiplier/fictitious domain method for particulate flows. Int J Multiph Flow 25:755–794MathSciNetCrossRefGoogle Scholar
- Gunes DZ, Scirocco R, Mewis J, Vermant J (2008) Flow-induced orientation of nonspherical particles: effect of aspect ratio and medium rheology. J Non-Newton Fluid Mech 155:39–50CrossRefGoogle Scholar
- Harlen OG, Koch DL (1993) Simple shear-flow of a suspension of fibers in a dilute polymer-solution at high Deborah number. J Fluid Mech 252:187–207CrossRefGoogle Scholar
- Hobbie EK et al (2003) Orientation of carbon nanotubes in a sheared polymer melt. Phys Fluids 15:1196–1202CrossRefGoogle Scholar
- Huang H, Lu X (2017) An ellipsoidal particle in tube Poiseuille flow. J Fluid Mech 822:664–688MathSciNetCrossRefGoogle Scholar
- Huang PY, Feng J, Hu HH, Joseph DD (1997) Direct simulation of the motion of solid particles in Couette and Poiseuille flows of viscoelastic fluids. J Fluid Mech 343:73–94MathSciNetCrossRefGoogle Scholar
- Huang H, Yang X, Krafczyk M, Lu X (2012) Rotation of spheroidal particles in Couette flows. J Fluid Mech 692:369–394MathSciNetCrossRefGoogle Scholar
- Iso Y, Koch DL, Cohen C (1996a) Orientation in simple shear flow of semi-dilute fiber suspensions 1. Weakly elastic fluids. J Non-Newton Fluid Mech 62:115–134CrossRefGoogle Scholar
- Iso Y, Koch DL, Cohen C (1996b) Orientation in simple shear flow of semi-dilute fiber suspensions 2. Highly elastic fluids. J Non-Newton Fluid Mech 62:135–153CrossRefGoogle Scholar
- Jaensson NO, Hulsen MA, Anderson PD (2016) Direct numerical simulation of particle alignment in viscoelastic fluids. J Non-Newton Fluid Mech 235:125–142MathSciNetCrossRefGoogle Scholar
- Jeffery GB (1922) The motion of ellipsoidal particles immersed in a viscous fluid. Proc R Soc Ser A 102:161–179CrossRefGoogle Scholar
- Johnson SJ, Salem AJ, Fuller GG (1990) Dynamics of colloidal particles in sheared non-newtonian fluids. J Non-Newton Fluid Mech 34:89–121CrossRefGoogle Scholar
- Karnis A, Gldsmith HL, Masion SG (1966) The flow of suspensions through tubes. Part V: inertial effects. Can J Chem Eng 44:181–193CrossRefGoogle Scholar
- Leal LG (1975) Slow motion of slender rod-like particles in 2nd order fluid. J Fluid Mech 69:305–337CrossRefGoogle Scholar
- Leer BV (1979) Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J Comput Phys 32:101–136CrossRefGoogle Scholar
- Lin A, Han SP (2002) On the distance between two ellipsoids. SIAM J Optim 13:298–308MathSciNetCrossRefGoogle Scholar
- Lin J, Wang Y, Zhang P, Ku X (2018) Mixing and orientation behaviors of cylindrical particles in a mixing layer of an Oldroyd-B fluid. Chem Eng Sci 176:270–284CrossRefGoogle Scholar
- Lu X, Liu C, Hu G, Xuan X (2017) Particle manipulations in non-newtonian microfluidics: a review. J Colloid Interface Sci 500:182–201CrossRefGoogle Scholar
- Lyon MK, Mead DW, Elliott RE, Leal LG (2001) Structure formation in moderately concentrated viscoelastic suspensions in simple shear flow. J Rheol 45:881–890CrossRefGoogle Scholar
- Madani A et al (2010) Fractionation of non-Brownian rod-like particle suspensions in a viscoplastic fluid. Chem Eng Sci 65:1762–1772CrossRefGoogle Scholar
- Michele J, Patzold R, Donis R (1977) Alignment and aggregation effects in suspensions of spheres in non-Newtonian media. Rheol Acta 16:317–321CrossRefGoogle Scholar
- Pan TW, Chang CC, Glowinski R (2008) On the motion of a neutrally buoyant ellipsoid in a three-dimensional Poiseuille flow. Comput Methods Appl Mech Eng 197:2198–2209MathSciNetCrossRefGoogle Scholar
- Pasquino R, Snijkers F, Grizzuti N, Vermant J (2010) The effect of particle size and migration on the formation of flow-induced structures in viscoelastic suspensions. Rheol Acta 49:993–1001CrossRefGoogle Scholar
- Pasquino R, Panariello D, Grizzuti N (2013) Migration and alignment of spherical particles in sheared viscoelastic suspensions: a quantitative determination of the flow-induced self-assembly kinetics. J Colloid Interface Sci 394:49–54CrossRefGoogle Scholar
- Pasquino R, D’Avino G, Maffettone PL, Greco F, Grizzuti N (2014) Migration and chaining of noncolloidal spheres suspended in a sheared viscoelastic medium: experiments and numerical simulations. J Non-Newton Fluid Mech 203:1–8CrossRefGoogle Scholar
- Qi DW, Luo LS (2003) Rotational and orientational behaviour of three-dimensional spheroidal particles in Couette flows. J Fluid Mech 477:201–213CrossRefGoogle Scholar
- Rosén T, Do-Quang M, Aidun CK, Lundell F (2015) The dynamical states of a prolate spheroidal particle suspended in shear flow as a consequence of particle and fluid inertia. J Fluid Mech 771:115–158CrossRefGoogle Scholar
- Saffman PG (1956) On the motion of small spheroidal particles in a viscous liquid. J Fluid Mech 1:540–553MathSciNetCrossRefGoogle Scholar
- Scirocco R, Vermant J, Mewis J (2004) Effect of the viscoelasticity of the suspending fluid on structure formation in suspensions. J Non-Newton Fluid Mech 117:183–192CrossRefGoogle Scholar
- Trofa M, D’Avino G, Hulsen MA, Greco F, Maffettone PL (2016a) Numerical simulations of the dynamics of a slippery particle in Newtonian and viscoelastic fluids subjected to shear and poiseuille flows. J Non-Newton Fluid Mech 228:46–54MathSciNetCrossRefGoogle Scholar
- Trofa M, D’Avino G, Hulsen MA, Maffettone PL (2016b) The effect of wall slip on the dynamics of a spherical particle in Newtonian and viscoelastic fluids subjected to shear and poiseuille flows. J Non-Newton Fluid Mech 236:123–131MathSciNetCrossRefGoogle Scholar
- Wang P, Yu Z, Lin J (2018) Numerical simulations of particle migration in rectangular channel flow of Giesekus viscoelastic fluids. J Non-Newton Fluid Mech 262:142–168MathSciNetCrossRefGoogle Scholar
- Won D, Kim C (2004) Alignment and aggregation of spherical particles in viscoelastic fluid under shear flow. J Non-Newton Fluid Mech 117:141–146CrossRefGoogle Scholar
- Yu Z, Shao X (2007) A direct-forcing fictitious domain method for particulate flows. J Comput Phys 227:292–314CrossRefGoogle Scholar
- Yu Z, Wachs A (2007) A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids. J Non-Newton Fluid Mech 145:78–91CrossRefGoogle Scholar
- Yu Z, Phan-Thien N, Fan Y, Tanner R (2002) Viscoelastic mobility problem of a system of particles. J Non-Newton Fluid Mech 104:87–124CrossRefGoogle Scholar
- Yu Z, Wachs A, Peysson Y (2006) Numerical simulation of particle sedimentation in shear-thinning fluids with a fictitious domain method. J Non-Newton Fluid Mech 136:126–139CrossRefGoogle Scholar
- Yu Z, Phan TN, Roger IT (2007) Rotation of a spheroid in a Couette flow at moderate Reynolds numbers. Phys Rev E 76:026310CrossRefGoogle Scholar