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Slender body theory for particles with non-circular cross-sections with application to particle dynamics in shear flows

Published online by Cambridge University Press:  02 September 2019

Neeraj S. Borker Open the ORCID record for Neeraj S. Borker [Opens in a new window]  and
Donald L. Koch Open the ORCID record for Donald L. Koch [Opens in a new window]
Neeraj S. Borker
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
Donald L. Koch*
Affiliation:
Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: dlk15@cornell.edu
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Abstract

This paper presents a theory to obtain the force per unit length acting on a slender filament with a non-circular cross-section moving in a fluid at low Reynolds number. Using a regular perturbation of the inner solution, we show that the force per unit length has $O(1/\ln (2A))+O(\unicode[STIX]{x1D6FC}/\ln ^{2}(2A))$ contributions driven by the relative motion of the particle and the local fluid velocity and an $O(\unicode[STIX]{x1D6FC}/(\ln (2A)A))$ contribution driven by the gradient in the imposed fluid velocity. Here, the aspect ratio ($A=l/a_{0}$) is defined as the ratio of the particle size ($l$) to the cross-sectional dimension ($a_{0}$) and $\unicode[STIX]{x1D6FC}$ is the amplitude of the non-circular perturbation. Using thought experiments, we show that two-lobed and three-lobed cross-sections affect the response to relative motion and velocity gradients, respectively. A two-dimensional Stokes flow calculation is used to extend the perturbation analysis to cross-sections that deviate significantly from a circle (i.e. $\unicode[STIX]{x1D6FC}\sim O(1)$). We demonstrate the ability of our method to accurately compute the resistance to translation and rotation of a slender triaxial ellipsoid. Furthermore, we illustrate novel dynamics of straight rods in a simple shear flow that translate and rotate quasi-periodically if they have two-lobed cross-section, and rotate chaotically and translate diffusively if they have a combination of two- and three-lobed cross-sections. Finally, we show the remarkable ability of our theory to accurately predict the motion of rings, retaining great accuracy for moderate aspect ratios (${\sim}10$) and cross-sections that deviate significantly from a circle, thereby making our theory a computationally inexpensive alternative to other Stokes flow solvers.

JFM classification

Low-Reynolds-number flows: Slender-body theory Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 877 , 25 October 2019 , pp. 1098 - 1133
Copyright
© 2019 Cambridge University Press 

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References

Acrivos, A. & Shaqfeh, E. S. G. 1988 The effective thermal conductivity and elongational viscosity of a nondilute suspension of aligned slender rods. Phys. Fluids 31, 1841–1844.10.1063/1.866681Google Scholar
Anczurowski, E. & Mason, S. G. 1968 Particle motions in sheared suspension. XXIV. Rotations of rigid spheroids and cylinders. Trans. Soc. Rheol. 12, 209–215.10.1122/1.549106Google Scholar
Batchelor, G. K. 1954 The skin friction on infinite cylinders moving parallel to their length. Q. J. Mech. Appl. Maths 7 (2), 179–192.10.1093/qjmam/7.2.179Google Scholar
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44 (3), 419–440.10.1017/S002211207000191XGoogle Scholar
Beckers, K. F., Koch, D. L. & Tester, J. W. 2015 Slender-body theory for transient heat conduction: theoretical basis, numerical implementation and case studies. Proc. R. Soc. Lond. A 471 (2184), 20150494.Google Scholar
Berg, H. C. & Anderson, R. A. 1973 Bacteria swim by rotating their flagellar filaments. Nature 245, 380–382.10.1038/245380a0Google Scholar
Borker, N. S., Stroock, A. D. & Koch, D. L. 2018 Controlling rotation and migration of rings in a simple shear flow through geometric modifications. J. Fluid Mech. 840, 379–407.10.1017/jfm.2018.20Google Scholar
Bray, D. 2000 Cell Movements. Garland.Google Scholar
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339–398.10.1146/annurev.fl.09.010177.002011Google Scholar
Brenner, H. 1964 The Stokes resistance of an arbitrary particle. 3. Shear fields. Chem. Engng Sci. 19, 631–651.10.1016/0009-2509(64)85052-1Google Scholar
Bretherton, F. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14 (2), 284–304.10.1017/S002211206200124XGoogle Scholar
Chen, H. S. & Acrivos, A. 1976 On the effective thermal conductivity of dilute suspensions containing highly conducting slender inclusions. Proc. R. Soc. Lond. A 349, 261–276.10.1098/rspa.1976.0072Google Scholar
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44 (4), 791–810.10.1017/S002211207000215XGoogle Scholar
Cox, R. G. 1971 The motion of long slender bodies in a viscous fluid. Part 2. Shear flow. J. Fluid Mech. 45 (4), 625–657.10.1017/S0022112071000259Google 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. Fluid 28, 013302.10.1063/1.4938239Google Scholar
Foulds, I. G. & Parameswaran, M. 2006 A planar self-sacrificial multilayer SU-8-based MEMS process utilizing a UV-blocking layer for the creation of freely moving parts. J. Micromech. Microengng 16 (10), 2109–2115.10.1088/0960-1317/16/10/026Google Scholar
Fredrickson, G. H. & Shaqfeh, E. S. G. 1989 Heat and mass transport in composites of aligned slender fibers. Phys. Fluids A 1, 3–20.10.1063/1.857546Google Scholar
Gao, W. & Wang, J. 2014 The environmental impact of micro/nanomachines: a review. ACS Nano 8 (4), 3170–3180.10.1021/nn500077aGoogle Scholar
Guasto, J. S., Rusconi, R. & Stocker, R. 2012 Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech. 44, 373–400.10.1146/annurev-fluid-120710-101156Google Scholar
Isla, A., Brostow, W., Bujard, B., Esteves, M., Rodriguez, J. R., Vargas, S. & Castano, V. M. 2003 Nanohybrid scratch resistant coatings for teeth and bone viscoelasticity manifested in tribology. Mat. Res. Innovat. 7, 110–114.10.1080/14328917.2003.11784770Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102 (715), 161–179.10.1098/rspa.1922.0078Google Scholar
Johnson, R. E. 1980 An improved slender-body theory for Stokes flow. J. Fluid Mech. 99 (2), 411–431.10.1017/S0022112080000687Google Scholar
Johnson, R. E. & Brokaw, C. J. 1979 Flagellar hydrodynamics. A comparison between resistive-force theory and slender-body theory. Biophys. J. 25, 113–127.10.1016/S0006-3495(79)85281-9Google Scholar
Jones, R. T.1946 Properties of low-aspect-ratio pointed wings at speeds below and above the speed of sound. NACA Tech. Rep. 835.Google Scholar
Keller, J. & Rubinow, S. 1976 Slender-body theory for slow viscous flow. J. Fluid Mech. 75 (4), 705–714.10.1017/S0022112076000475Google Scholar
Khayat, R. E. & Cox, R. G. 1989 Inertia effects on the motion of long slender bodies. J. Fluid Mech. 209, 435–462.10.1017/S0022112089003174Google Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics. Dover.Google Scholar
Kim, Y. J. & Rae, W. J. 1991 Separation of screw-sensed particles in a homogeneous shear field. Intl J. Multiphase Flow 17 (6), 717–744.10.1016/0301-9322(91)90053-6Google Scholar
Koens, L. & Lauga, E. 2016 Slender-ribbon theory. Phys. Fluids 28 (1), 013101.10.1063/1.4938566Google Scholar
Lamb, S. H. 1932 Hydrodynamics. The University Press.Google Scholar
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9, 305–317.10.1017/S0022112060001110Google Scholar
Lighthill, M. J. 1971 Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B 179, 125–138.Google Scholar
Lopez, M. & Graham, M. D. 2007 Shear-induced diffusion in dilute suspensions of spherical or non-spherical particles: effects of irreversibility and symmetry breaking. Phys. Fluids 19, 073602.10.1063/1.2750525Google Scholar
Mackaplow, M. B & Shaqfeh, E. S. G. 1996 A numerical study of the rheological properties of suspensions of rigid, non-Brownian fibres. J. Fluid Mech. 329, 155–186.10.1017/S0022112096008889Google Scholar
Mackaplow, M. B & Shaqfeh, E. S. G. 1998 A numerical study of the sedimentation of fibre suspensions. J. Fluid Mech. 376, 149–182.10.1017/S0022112098002663Google Scholar
Mackaplow, M. B, Shaqfeh, E. S. G. & Schiek, R. L. 1994 A numerical study of heat and mass transport in fibre suspensions. Proc. R. Soc. Lond. A 447, 77–110.10.1098/rspa.1994.0130Google Scholar
Munk, M. M.1924 The aerodynamic forces on airship hulls. NACA Tech. Rep. 184.Google Scholar
Newman, J. N. 1964 A slender-body theory for ship oscillations in waves. J. Fluid Mech. 18, 602–618.10.1017/S0022112064000441Google Scholar
Newman, J. N. 1970 Applications of slender-body theory in ship hydrodynamics. Annu. Rev. Fluid Mech. 2, 67–94.10.1146/annurev.fl.02.010170.000435Google Scholar
Paulsen, K. S., Di Carlo, D. & Chung, A. J. 2015 Optofluidic fabrication for 3D-shaped particles. Nat. Commun. 6, 6976.10.1038/ncomms7976Google Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods With the Software Library BEMLIB. Chapman and Hall/CRC.10.1201/9781420035254Google Scholar
Rahnama, M., Koch, D. L., Iso, Y. & Cohen, C. 1993 Hydrodynamic, translational diffusion in fiber suspensions subject to simple shear flow. Phys. Fluids A 5 (4), 849–862.10.1063/1.858890Google Scholar
Rahnama, M., Koch, D. L. & Shaqfeh, E. S. G. 1995 The effect of hydrodynamic interactions on the orientation distribution in a fiber suspension subject to simple shear flow. Phys. Fluids 7 (3), 487–506.10.1063/1.868647Google Scholar
Raney, J. R. & Lewis, J. A. 2015 Printing mesoscale architectures. MRS Bull. 40 (11), 943–950.10.1557/mrs.2015.235Google Scholar
Rocha, A. & Acrivos, A. 1973a On the effective thermal conductivity of dilute dispersions: general theory for inclusions of arbitrary shape. Q. J. Mech. Appl. Maths 26, 217–233.10.1093/qjmam/26.2.217Google Scholar
Rocha, A. & Acrivos, A. 1973b On the effective thermal conductivity of dilute dispersions: highly conducting inclusions of arbitrary shape. Q. J. Mech. Appl. Maths 26, 441–455.10.1093/qjmam/26.4.441Google Scholar
Sacanna, S., Korpics, M., Rodriguez, K., Colón-Meléndez, L., Kim, S.-H., Pine, D. J. & Yi, G.-R. 2013 Shaping colloids for self-assembly. Nat. Commun. 4, 1688.10.1038/ncomms2694Google Scholar
Shaqfeh, E. S. G. 1988 A nonlocal theory for the heat transport in composites containing highly conducting fibrous inclusions. Phys. Fluids 31, 2405–2425.10.1063/1.866594Google Scholar
Singh, V., Koch, D. L. & Stroock, A. D. 2013 Rigid ring-shaped particles that align in simple shear flow. J. Fluid Mech. 722, 121–158.10.1017/jfm.2013.53Google Scholar
Singh, V., Koch, D. L., Subramanian, G. & Stroock, A. D. 2014 Rotational motion of a thin axisymmetric disk in a low Reynolds number linear flow. Phys. Fluids 26 (3), 033303.10.1063/1.4868520Google Scholar
Soler, L. & Sánchez, S. 2014 Catalytic nanomotors for environmental monitoring and water remediation. Nanoscale 6 (13), 7175–7182.10.1039/C4NR01321BGoogle Scholar
Suarez, S. S. & Pacey, A. A. 2006 Sperm transport in the female reproductive tract. Hum. Reprod. Update 12 (1), 23–37.10.1093/humupd/dmi047Google Scholar
Tekce, H. S., Kumlutas, D. & Tavman, I. H. 2007 Effect of particle shape on thermal conductivity of copper reinforced polymer composites. J. Reinf. Plast. Comp. 26 (1), 113–121.10.1177/0731684407072522Google Scholar
Wang, J., Graham, M. D. & Klingenberg, D. J. 2014 Shear-induced diffusion in dilute curved fiber suspensions in simple shear flow. Phys. Fluids 26 (3), 033301.10.1063/1.4867171Google Scholar
Wang, J., Tozzi, E. J., Graham, M. D. & Klingenberg, D. J. 2012 Flipping, scooping, and spinning: drift of rigid curved nonchiral fibers in simple shear flow. Phys. Fluids 24, 123304.10.1063/1.4769980Google Scholar
Yarin, A. L., Gottlieb, O. & Roisman, I. V. 1997 Chaotic rotation of triaxial ellipsoids in simple shear flow. J. Fluid Mech. 340, 83–100.10.1017/S0022112097005260Google Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69 (02), 377–403.10.1017/S0022112075001486Google Scholar
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