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Influence of small inertia on Jeffery orbits
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JFM Notebooks
Published online by Cambridge University Press: 19 January 2024
Abstract
We experimentally investigate the rotational dynamics of neutrally buoyant axisymmetric particles in a simple shear flow. A custom-built shearing cell and a multi-view shape-reconstruction method are used to obtain direct measurements of the orientation and period of rotation of particles having oblate and prolate shapes (such as spheroids and cylinders) of varying aspect ratios. By systematically changing the viscosity of the fluid, we examine the effect of inertia (which may be originated from either phase) on the dynamical behaviour of these suspended particles up to a particle Reynolds number of approximately one. While no significant effect on the period of rotation is found in this small-inertia regime, a systematic drift among several rotations toward limiting stable orbits is observed. Prolate particles are seen to drift towards the tumbling orbit in the plane of shear, whereas oblate particles are driven either to the tumbling or to the vorticity-aligned spinning orbits, depending on their initial orientation. These results are compared with recent small-inertia asymptotic theories, assessing their range of validity, as well as to numerical simulations in the small-inertia regime for both prolate and oblate particles.
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References
Abadi, M. 2015 TensorFlow: large-scale machine learning on heterogeneous systems. Software available from tensorflow.org.Google Scholar
Anczurowski, E. & Mason, S.G. 1968 Particle motions in sheared suspensions. XXIV. Rotation of rigid spheroids and cylinders. Trans. Soc. Rheol. 12 (2), 209–215.CrossRefGoogle Scholar
Bao, G., Hutchinson, J.W. & McMeeking, R.M. 1991 Particle reinforcement of ductile matrices against plastic flow and creep. Acta Metall. Mater. 39 (8), 1871–1882.CrossRefGoogle Scholar
Binder, R.C. 1939 The motion of cylindrical particles in viscous flow. J. Appl. Phys. 10 (10), 711–713.CrossRefGoogle Scholar
Bretherton, F.P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14 (2), 284–304.CrossRefGoogle Scholar
Burgers, J.M. 1938 Second report on viscosity and plasticity. Kon. Ned. Akad. Wet. Verhand. 16, 113–184.Google Scholar
Butler, J.E. & Snook, B. 2018 Microstructural dynamics and rheology of suspensions of rigid fibers. Annu. Rev. Fluid Mech. 50, 299–318.CrossRefGoogle Scholar
Byron, M., Einarsson, J., Gustavsson, K., Voth, G.A., Mehlig, B. & Variano, E. 2015 Shape-dependence of particle rotation in isotropic turbulence. Phys. Fluids 27 (3), 035101.CrossRefGoogle Scholar
Canny, J. 1986 A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 6, 679–698.CrossRefGoogle Scholar
Cox, R.G. 1971 The motion of long slender bodies in a viscous fluid. Part 2. Shear flow. J. Fluid Mech. 45 (4), 625–657.CrossRefGoogle Scholar
Dabade, V., Marath, N.K. & Subramanian, G. 2016 The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow. J. Fluid Mech. 791, 631–703.CrossRefGoogle Scholar
Du Roure, O., Lindner, A., Nazockdast, E.N. & Shelley, M.J. 2019 Dynamics of flexible fibers in viscous flows and fluids. Annu. Rev. Fluid Mech. 51, 539–572.CrossRefGoogle Scholar
Eberly, D. 1999 Least squares fitting of data by linear or quadratic structures. Geom. Tools, July.Google Scholar
Einarsson, J., Candelier, F., Lundell, F., Angilella, J.R. & Mehlig, B. 2015 a Effect of weak fluid inertia upon Jeffery orbits. Phys. Rev. E 91 (4), 041002.CrossRefGoogle ScholarPubMed
Einarsson, J., Candelier, F., Lundell, F., Angilella, J.R. & Mehlig, B. 2015 b Rotation of a spheroid in a simple shear at small Reynolds number. Phys. Fluids 27 (6), 063301.CrossRefGoogle Scholar
Einarsson, J., Mihiretie, B.M., Laas, A., Ankardal, S., Angilella, J.R., Hanstorp, D. & Mehlig, B. 2016 Tumbling of asymmetric microrods in a microchannel flow. Phys. Fluids 28 (1), 013302.CrossRefGoogle Scholar
Einstein, A. 1906 Eine neue Bestimmung der Moleküldimensionen. Ann. Physik 19, 289–306.CrossRefGoogle Scholar
Einstein, A. 1911 Berichtigung zu meiner arbeit: eine neue bestimmung der moleküldimensionen. Ann. Physik 34, 591–592.CrossRefGoogle Scholar
Forgacs, O.L. & Mason, S.G. 1959 Particle motions in sheared suspensions: X. Orbits of flexible threadlike particles. J. Colloid Sci. 14 (5), 473–491.CrossRefGoogle Scholar
Fries, J., Einarsson, J. & Mehlig, B. 2017 Angular dynamics of small crystals in viscous flow. Phys. Rev. Fluids 2, 014302.CrossRefGoogle Scholar
Glendinning, P. 1994 Stability, Instability and Chaos: an Introduction to the Theory of Nonlinear differential Equations, pp. 145–158. Cambridge University Press.CrossRefGoogle Scholar
Goldsmith, H.L. 1996 The microrheology of dispersions: application to blood cells. J. Jap. Soc. Biorheol. 10 (4), 15–36.Google Scholar
Goldsmith, H.L. & Mason, S.G. 1962 a The flow of suspensions through tubes. I. Single spheres, rods, and discs. J. Colloid Sci. 17 (5), 448–476.CrossRefGoogle Scholar
Goldsmith, H.L. & Mason, S.G. 1962 b Particle motions in sheared suspensions: XIII. The spin and rotation of disks. J. Fluid Mech. 12 (1), 88–96.CrossRefGoogle Scholar
Guasto, J.S., Rusconi, R. & Stocker, R. 2012 Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech. 44, 373–400.CrossRefGoogle Scholar
Gustavsson, K., Jucha, J., Naso, A., Lévêque, E., Pumir, A. & Mehlig, B. 2017 Statistical model for the orientation of nonspherical particles settling in turbulence. Phys. Rev. Lett. 119 (25), 254501.CrossRefGoogle ScholarPubMed
Harris, J.B. & Pittman, J.F.T. 1975 Equivalent ellipsoidal axis ratios of slender rod-like particles. J. Colloid Interface Sci. 50 (2), 280–282.CrossRefGoogle Scholar
Hoyt, J.W. 1972 Turbulent flow of drag-reducing suspensions. Tech. Rep.. Naval Undersea Center, San Diego, CA.Google Scholar
Huang, H., Yang, X., Krafczyk, M. & Lu, X.Y. 2012 Rotation of spheroidal particles in Couette flows. J. Fluid Mech. 692, 369–394.CrossRefGoogle Scholar
Jeffery, G.B. 1922 The rotation of two circular cylinders in a viscous fluid. Proc. R. Soc. Lond. A 101 (709), 169–174.Google Scholar
Karnis, A., Goldsmith, H.L. & Mason, S.G. 1963 Axial migration of particles in poiseuille flow. Nature 200 (4902), 159–160.CrossRefGoogle Scholar
Karnis, A., Goldsmith, H.L. & Mason, S.G. 1966 The flow of suspensions through tubes, v. inertial effects. Can. J. Chem. Engng 44 (4), 181–193.CrossRefGoogle Scholar
Leal, L.G. & Hinch, E.J. 1971 The effect of weak brownian rotations on particles in shear flow. J. Fluid Mech. 46 (4), 685–703.CrossRefGoogle Scholar
Lundell, F., Söderberg, L.D. & Alfredsson, P.H. 2011 Fluid mechanics of papermaking. Annu. Rev. Fluid Mech. 43, 195–217.CrossRefGoogle Scholar
Mao, W. & Alexeev, A. 2014 Motion of spheroid particles in shear flow with inertia. J. Fluid Mech. 749, 145–166.CrossRefGoogle Scholar
Marath, N.K. & Subramanian, G. 2017 The effect of inertia on the time period of rotation of an anisotropic particle in simple shear flow. J. Fluid Mech. 830, 165–210.CrossRefGoogle Scholar
Marath, N.K. & Subramanian, G. 2018 The inertial orientation dynamics of anisotropic particles in planar linear flows. J. Fluid Mech. 844, 357–402.CrossRefGoogle Scholar
Marchioli, C., Bhatia, H., Sardina, G., Brandt, L. & Soldati, A. 2019 Role of large-scale advection and small-scale turbulence on vertical migration of gyrotactic swimmers. Phys. Rev. Fluids 4, 124304.CrossRefGoogle Scholar
Metzger, B. & Butler, J.E. 2012 Clouds of particles in a periodic shear flow. Phys. Fluids 24 (2), 021703.CrossRefGoogle Scholar
Moses, K.B., Advani, S.G. & Reinhardt, A. 2001 Investigation of fiber motion near solid boundaries in simple shear flow. Rheol. Acta 40 (3), 296–306.CrossRefGoogle Scholar
Ni, R., Kramel, S., Ouellette, N.T. & Voth, G.A. 2015 Measurements of the coupling between the tumbling of rods and the velocity gradient tensor in turbulence. J. Fluid Mech. 766, 202–225.CrossRefGoogle Scholar
Paschkewitz, J.S., Dubief, Y., Dimitropoulos, C.D., Shaqfeh, E.S.G. & Moin, P. 2004 Numerical simulation of turbulent drag reduction using rigid fibres. J. Fluid Mech. 518, 281–317.CrossRefGoogle Scholar
Qi, D. & Luo, L.S. 2003 Rotational and orientational behaviour of three-dimensional spheroidal particles in Couette flows. J. Fluid Mech. 477, 201–213.CrossRefGoogle Scholar
Quilez, I. 2016 Cylinder - bounding box. Available at: https://iquilezles.org/articles/diskbbox.Google Scholar
Rosén, T., Einarsson, J., Nordmark, A., Aidun, C.K., Lundell, F. & Mehlig, B. 2015 Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers. Phys. Rev. E 92 (6), 063022.CrossRefGoogle ScholarPubMed
Ross, P.S., et al. 2021 Pervasive distribution of polyester fibres in the arctic ocean is driven by Atlantic inputs. Nat. Commun. 12 (1), 1–9.CrossRefGoogle ScholarPubMed
Saffman, P.G. 1956 On the motion of small spheroidal particles in a viscous liquid. J. Fluid Mech. 1 (5), 540–553.CrossRefGoogle Scholar
Sheikh, M.Z., Gustavsson, K., Lopez, D., Lévêque, E., Mehlig, B., Pumir, A. & Naso, A. 2020 Importance of fluid inertia for the orientation of spheroids settling in turbulent flow. J. Fluid Mech. 886, A9.CrossRefGoogle Scholar
Stover, C.A. & Cohen, C. 1990 The motion of rodlike particles in the pressure-driven flow between two flat plates. Rheol. Acta 29 (3), 192–203.CrossRefGoogle Scholar
Subramanian, G. & Koch, D.L. 2005 Inertial effects on fibre motion in simple shear flow. J. Fluid Mech. 535, 383–414.CrossRefGoogle Scholar
Subramanian, G. & Koch, D.L. 2006 Inertial effects on the orientation of nearly spherical particles in simple shear flow. J. Fluid Mech. 557, 257–296.CrossRefGoogle Scholar
Taylor, G.I. 1923 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 103 (720), 58–61.Google Scholar
Trevelyan, B.J. & Mason, S.G. 1951 Particle motions in sheared suspensions. I. Rotations. J. Colloid Sci. 6 (4), 354–367.CrossRefGoogle Scholar
Voth, G.A. & Soldati, A. 2017 Anisotropic particles in turbulence. Annu. Rev. Fluid Mech. 49 (1), 249–276.CrossRefGoogle Scholar
Wang, Z., Xu, C.-X. & Zhao, L. 2021 Turbulence modulations and drag reduction by inertialess spheroids in turbulent channel flow. Phys. Fluids 33 (12), 123313.CrossRefGoogle Scholar
Zettner, C.M. & Yoda, M. 2001 Moderate-aspect-ratio elliptical cylinders in simple shear with inertia. J. Fluid Mech. 442, 241–266.CrossRefGoogle Scholar
Zhao, L. & Andersson, H.I. 2016 Why spheroids orient preferentially in near-wall turbulence. J. Fluid Mech. 807, 221–234.CrossRefGoogle Scholar
Di Giusto et al. supplementary movie 1
Jeffery orbits for a fibre with equivalent aspect ratio r_eq=7.4 at small particle Reynolds number Re_p=0.08, run 5 of panel (a) of Figure 6;
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Di Giusto et al. supplementary movie 2
Jeffery orbits for a fibre with equivalent aspect ratio r_eq=7.4 at small particle Reynolds number Re_p=0.08, run 6 of panel (a) of Figure 6;
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Di Giusto et al. supplementary movie 3
Jeffery orbits for a fibre with equivalent aspect ratio r_eq=7.4 at small particle Reynolds number Re_p=0.08, run 8 of panel (a) of Figure 6;
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Di Giusto et al. supplementary movie 4
Jeffery orbits for a fibre with equivalent aspect ratio r_eq=7.4 at large particle Reynolds number Re_p=1.0, run 13 of panel (b) of Figure 6;
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Di Giusto et al. supplementary movie 5
Jeffery orbits for a fibre with equivalent aspect ratio r_eq=7.4 at large particle Reynolds number Re_p=1.0, run 14 of panel (b) of Figure 6;
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Di Giusto et al. supplementary movie 6
Jeffery orbits for a fibre with equivalent aspect ratio r_eq=7.4 at large particle Reynolds number Re_p=1.0, run 3 of panel (b) of Figure 6;
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Di Giusto et al. supplementary movie 7
Jeffery orbits for a spheroid with aspect ratio r=0.6 at small particle Reynolds number Re_p=0.02, run 1 of panel (c) of Figure 6;
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Di Giusto et al. supplementary movie 8
Jeffery orbits for a spheroid with aspect ratio r=0.6 at small particle Reynolds number Re_p=0.02, run 3 of panel (c) of Figure 6;
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Di Giusto et al. supplementary movie 9
Jeffery orbits for a spheroid with aspect ratio r=0.6 at small particle Reynolds number Re_p=0.02, run 4 of panel (c) of Figure 6;
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Di Giusto et al. supplementary movie 10
Jeffery orbits for a spheroid with aspect ratio r=0.6 at moderate particle Reynolds number Re_p=0.43, run 10 of panel (d) of Figure 6;
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Di Giusto et al. supplementary movie 11
Jeffery orbits for a spheroid with aspect ratio r=0.6 at moderate particle Reynolds number Re_p=0.43, run 8 of panel (d) of Figure 6;
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Di Giusto et al. supplementary movie 12
Jeffery orbits for a spheroid with aspect ratio r=0.6 at moderate particle Reynolds number Re_p=0.43, run 5 of panel (d) of Figure 6;
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Di Giusto et al. supplementary movie 13
Jeffery orbits for a disk with equivalent aspect ratio r=0.18 at small particle Reynolds number Re_p=0.05, run 1 of panel (e) of Figure 6;
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Di Giusto et al. supplementary movie 14
Jeffery orbits for a disk with equivalent aspect ratio r=0.18 at small particle Reynolds number Re_p=0.05, run 2 of panel (e) of Figure 6;
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Di Giusto et al. supplementary movie 15
Jeffery orbits for a disk with equivalent aspect ratio r=0.18 at small particle Reynolds number Re_p=0.05, run 5 of panel (e) of Figure 6;
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Di Giusto et al. supplementary movie 16
Jeffery orbits for a disk with equivalent aspect ratio r=0.18 at large particle Reynolds number Re_p=1.32, run 1 of panel (e) of Figure 6;
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Di Giusto et al. supplementary movie 17
Jeffery orbits for a disk with equivalent aspect ratio r=0.18 at large particle Reynolds number Re_p=1.32, run 2 of panel (e) of Figure 6;
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Di Giusto et al. supplementary movie 18
Jeffery orbits for a disk with equivalent aspect ratio r=0.18 at large particle Reynolds number Re_p=1.32, run 9 of panel (e) of Figure 6;
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Di Giusto et al. supplementary movie 19
Five different Jeffery Orbits for a spheroid of aspect ratio r=10, see Figure 1.
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Di Giusto et al. supplementary material 20
Di Giusto et al. supplementary material
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