Hostname: page-component-7f64f4797f-7vssv Total loading time: 0 Render date: 2025-11-04T11:42:29.473Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigid ring-shaped particles that align in simple shear flow

Published online by Cambridge University Press:  28 March 2013

Vikram Singh ,
Donald L. Koch  and
Abraham D. Stroock
Vikram Singh
Affiliation:
Department of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
Donald L. Koch*
Affiliation:
Department of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
Abraham D. Stroock
Affiliation:
Department of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA Kavli Institute of Cornell, Cornell University, Ithaca, NY 14853, USA
*
†Email addresses for correspondence: dlk15@cornell.edu, vs225@cornell.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Most rigid, torque-free, low-Reynolds-number, axisymmetric particles undergo a time-periodic tumbling motion in a simple shear flow, with their axes of symmetry following a set of closed Jeffery orbits. We have identified a class of rigid, ring-like particles whose axes of symmetry instead reach a permanent alignment near the velocity gradient direction with the plane of the particle aligning near the flow–vorticity plane. An asymptotic analysis for small particle aspect ratio (ratio of length parallel to the axis of symmetry to diameter perpendicular to the axis) shows that an appropriate asymmetry of the ring cross-section with a thinner outer edge and thicker inner edge leads to a tendency to rotate in a direction opposite to the vorticity; this tendency can balance the usual rotation rate associated with the finite thickness of the particle. Boundary integral computations for finite particle aspect ratios are used to determine the conditions of aspect ratio and degree of asymmetry that lead to the aligning behaviour and the final orientation of the axis of symmetry of the aligned particles. The aligning particle follows an equation of motion similar to the Leslie–Erickson equation for the director of a small-molecule nematic liquid crystal. However, whereas the alignment of the director arises from intermolecular interactions, the ring-like particle aligns solely due to its intrinsic rotational motion in a low-Reynolds-number flow.

JFM classification

Low-Reynolds-number flows: Boundary integral methods Low-Reynolds-number flows: Low-Reynolds-number flows Complex Fluids: Suspensions

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 722 , 10 May 2013 , pp. 121 - 158
Copyright
©2013 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.)

Article purchase

Temporarily unavailable

References

Badaire, S., Cottin-Bizonne, C., Woody, J. W., Yang, A. & Stroock, A. D. 2007 Shape selectivity in the assembly of lithographically designed colloidal particles. J. Am. Chem. Soc 129, 40–41.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.CrossRefGoogle Scholar
Bhattacharyya, D. S. 1966 Orientation of mineral lineation along the flow direction in rocks. Tectonophysics 3, 559–564.Google Scholar
Brenner, H. 1964 The Stokes resistance of an arbitrary particle. 3. shear fields. Chem. Engng Sci. 19, 631–651.CrossRefGoogle Scholar
Bretherton, F. P. 1962a The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284–304.CrossRefGoogle Scholar
Bretherton, F. P. 1962b Slow viscous motion round a cylinder in a simple shear. J. Fluid Mech. 12, 591–613.CrossRefGoogle Scholar
Chen, S. B. & Koch, D. L. 1996 Isotropic-nematic phase transitions in aqueous solutions of weakly charged, rodlike polyelectrolytes. J. Chem. Phys. 104 (1), 359–374.CrossRefGoogle Scholar
Chen, S. B. & Koch, D. L. 1997 Rheology of highly aligned nematic liquid crystals. J. Non-Newtonian Fluid Mech. 69, 273.CrossRefGoogle Scholar
Chien, S. 1987 Red-cell deformability and its relevance to blood-flow. Annu. Rev. Physiology 49, 177–192.CrossRefGoogle ScholarPubMed
Chwang, A. T. 1975 Hydrodynamics of low-Reynolds-number flow. Part 3. Motion of a spheroidal particle in quadratic flows. J. Fluid Mech. 72 (1), 17–34.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
Deschamps, J., Kantsler, V. & Steinberg, V. 2009 Phase diagram of single vesicle dynamical states in shear flow. Phys. Rev. Lett. 102 (11), 11444–11447.CrossRefGoogle ScholarPubMed
Edwards, D. A., Hanes, J., Caponetti, G., Hrkach, J., Ben-Jebria, A., Eskew, M. L., Mintzes, J., Deaver, D., Lotan, N. & Langer, R. 1997 Large porous particles for pulmonary drug delivery. Science 276 (5320), 1868–1872.CrossRefGoogle ScholarPubMed
Einstein, A. 1906 Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. Leipzig 19, 289.CrossRefGoogle Scholar
Euliss, L. E., DuPont, J. A., Gratton, S. & DeSimone, J. 2006 Imparting size, shape, and composition control of materials for nanomedicine. Chem. Soc. Rev. 35, 1095–1104.CrossRefGoogle ScholarPubMed
Forgacs, O. L. & Mason, S. G. 1959 Particle motions in sheared suspensions ix. Spin and deformation of threadlike particles. J. Colloid Sci. 14, 457–472.CrossRefGoogle Scholar
Gay, N. C. 1967 The motion of rigid particles embedded in a viscous fluid during pure shear deformation of the fluid. Tectonophysics 5 (2), 81–88.CrossRefGoogle Scholar
Ghosh, S. K. & Ramberg, H. 1976 Reorientation of inclusions by combination of combination of pure and simple shear. Tectonophysics 34, 1–70.CrossRefGoogle Scholar
Goldsmith, H. L. 1993 Poiseuille lecture: from papermaking fibers to human red blood cells. Biorheology 30, 165–190.Google Scholar
Goldsmith, H. L. & Mason, S. G. 1962 The flow of suspensions through tubes i. Single spheres, rods, and discs. J. Colloid Sci. 17, 448–476.CrossRefGoogle Scholar
Goren, S. L. & O’Neill, M. E. 1980 Asymmetric creeping motion of an open torus. J. Fluid Mech. 101, 97–110.CrossRefGoogle Scholar
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65, 365–400.CrossRefGoogle Scholar
Ho, B. P. & Leal, L. G. 1976 Migration of rigid spheres in a 2-dimensional unidirectional shear-flow of a 2nd-order fluid. J. Fluid Mech. 76, 783–799.Google Scholar
Iso, Y., Koch, D. L. & Cohen, C. 1996 Orientation in simple shear flow of semi-dilute fiber suspensions 2. Highly elastic fluids. J. Non-Newtonian Fluid Mech. 62, 135–153.CrossRefGoogle Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102 (715), 161–179.Google Scholar
Jogun, S. M. & Zukoski, C. F. 1999 Rheology and microstructure of dense suspensions of plate-shaped colloidal particles. J. Rheol. 43 (4), 847–871.CrossRefGoogle Scholar
Johnson, R. E. & Wu, T. Y. 1979 Hydromechanics of low-Reynolds-number flow. Part 5. Motion of a slender torus. J. Fluid Mech. 95, 263–277.CrossRefGoogle Scholar
Kantsler, V. & Steinberg, V. 2005 Orientation and dynamics of a vesicle in tank-treading motion in shear flow. Phys. Rev. Lett. 95 (25).CrossRefGoogle ScholarPubMed
Kim, S. & Karilla, S. J. 1991 Microhydrodynamics. Dover.Google Scholar
K Veerapaneni, S., Young, Y.-N., Vlahovska, P. M. & Blawzdzeiwicz, J. 2011 Dynamics of a compound vesicle in shear flow. Phys. Rev. Lett. 106 (15), 158103.CrossRefGoogle Scholar
Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.Google Scholar
Larson, R. G. 1991 Arrested tumbling in shearing flows of liquid crystal polymers. Macromolecules 23, 3383–3392.Google Scholar
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.Google Scholar
Leal, L. G. 1980 Particle motion in viscous fluid. Annu. Rev. Fluid Mech. 12, 435–476.CrossRefGoogle Scholar
Leal, L. G. 2007 Advanced Transport Phenomena. Cambridge University Press.CrossRefGoogle Scholar
Lee, S. H. & Leal, L. G. 1982 The motion of a sphere in the presence of a deformable interface. ii. a numerical study of the translation of a sphere normal to an interface. J. Colloid Interface Sci. 87 (1), 81–106.CrossRefGoogle Scholar
Lewis, J. A. 2006 Direct ink writing of 3d functional materials. Adv. Funct. Mater. 16, 2193–2204.CrossRefGoogle Scholar
Mody, N. A. & King, M. R. 2005 Three-dimensional simulations of a platelet-shaped spheroid near a wall in shear flow. Phys. Fluids 17, 113302.CrossRefGoogle Scholar
Nir, A. & Acrivos, A. 1973 Creeping motion of 2 arbitrary-sized touching spheres in a linear shear field. J. Fluid Mech. 59, 209–223.CrossRefGoogle Scholar
Olla, P. 1999 Simplified model for red cell dynamics in small blood vessels. Phys. Rev. Lett. 82 (2), 453–456.CrossRefGoogle Scholar
Pell, W. H. & Payne, L. E. 1960 On Stokes flow about a torus. Mathematika 7, 78–92.CrossRefGoogle Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods With the Software Library BEMLIB. Chapman and Hall/CRC.CrossRefGoogle Scholar
Rafai, S., Jibuti, L. & Peyla, P. 2010 Effective viscosity of microswimmer suspensions. Phys. Rev. Lett. 104, 098102.CrossRefGoogle ScholarPubMed
Singh, V., Koch, D. L. & Stroock, A. D. 2011 Ideal rate of collision of cylindrical particles. Langmuir 27, 1813–1823.CrossRefGoogle Scholar
Singh, V., Koch, D. L., Subramanian, G. & Stroock, A. D. 2013 Rotational motion of a thin axisymmetric disk in a low Reynolds number linear flow. Phys. Fluids (in preparation).CrossRefGoogle Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Cambridge Philos. Trans. 9, 8–106.Google Scholar
Stover, C. A. & Cohen, C. 1990 The motion of rodlike particles in the pressure-driven flow between two flat plates. Rheol. Acta 29, 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
Tchen, C. M. 1954 Motion of small particles in skew shape suspended in a viscous liquid. Am. Inst. Phys. 25, 463–473.Google Scholar
Thaokar, R. M., Schiessel, H. & Kulic, I. M. 2007 Hydrodynamics of a rotating torus. Eur. Phys. J. 60, 325–336.CrossRefGoogle Scholar
Torza, S., Cox, R. G. & Mason, S. G. 1972 Particle motions in sheared suspensions xxvii. transient and steady deformation and burst of liquid drops. J. Colloid Interface Sci. 38 (2), 396–411.CrossRefGoogle Scholar
Vasseur, P. & Cox, R. G. 1976 The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech. 78, 385–413.CrossRefGoogle Scholar
Willis, D. G. 1977 Kinematic model of preferred orientation. Geol. Soc. Am. Bull. 88 (6), 883–894.2.0.CO;2>CrossRefGoogle Scholar
Yamakawa, H. & Yamaei, J. 1973 Application of Kirkwood theory of transport in polymer solutions to rigid assemblies of beads. J. Chem. Phys. 58, 2049–2055.CrossRefGoogle Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape- numerical-method of solution. J. Fluid Mech. 69, 377–403.CrossRefGoogle Scholar