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Structure of the streaming flow generated by a sphere in a fluid undergoing rectilinear oscillation

Published online by Cambridge University Press:  03 November 2023

Peijing Li Open the ORCID record for Peijing Li [Opens in a new window] ,
Jesse F. Collis Open the ORCID record for Jesse F. Collis [Opens in a new window] ,
Douglas R. Brumley Open the ORCID record for Douglas R. Brumley [Opens in a new window] ,
Lennart Schneiders Open the ORCID record for Lennart Schneiders [Opens in a new window]  and
John E. Sader Open the ORCID record for John E. Sader [Opens in a new window]
Peijing Li
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
Jesse F. Collis
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
Douglas R. Brumley
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
Lennart Schneiders
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
John E. Sader*
Affiliation:
Graduate Aerospace Laboratories and Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: jsader@caltech.edu
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Abstract

A solid body in a viscous fluid undergoing oscillatory motion naturally produces a steady secondary flow due to convective inertia. This phenomenon is embodied in the streaming flow generated by a sphere in an unbounded fluid executing rectilinear oscillations. We review the considerable literature on this canonical problem and summarise exact and asymptotic formulas in the small-amplitude limit. These analytical formulas are used to explore the characteristic flow structure of this problem and clarify previously unreported features. A single, toroidal-shaped vortex exists in each hemisphere regardless of the oscillation frequency, which can drive a counter-flow away from the sphere. The vortex centre moves monotonically away from the sphere with decreasing oscillation frequency, and engulfs the entire flow domain for $\beta \equiv \omega R^2/\nu < 16.317$, where $\omega$ is the angular oscillation frequency, $R$ the sphere radius, and $\nu$ the fluid kinematic viscosity. This seemingly abrupt change in flow structure at the critical frequency $\beta _{critical} =16.317$, and its quantification, appear to have not been reported previously. We perform a direct numerical simulation of the Navier–Stokes equations, to (1) confirm existence of this critical frequency at finite amplitude, and (2) examine its variation with amplitude. This reveals a universal relationship between the critical frequency and oscillation amplitude, clarifying previous reports on the structure of this streaming flow. The critical frequency is shown to be identical for the streaming flow and the cycle-averaged particle paths, establishing that the critical frequency is accessible directly using standard measurements.

JFM classification

Micro-/Nano-fluid dynamics: Microscale transport Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 974 , 10 November 2023 , A37
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Alassar, R.S. 2008 Acoustic streaming on spheres. Intl J. Non-Linear Mech. 43 (9), 892–897.CrossRefGoogle Scholar
Bertelsen, A., Svardal, A. & Tjøtta, S. 1973 Nonlinear streaming effects associated with oscillating cylinders. J. Fluid Mech. 59 (3), 493–511.CrossRefGoogle Scholar
Bhosale, Y., Vishwanathan, G., Upadhyay, G., Parthasarathy, T., Juarez, G. & Gazzola, M. 2022 Multicurvature viscous streaming: flow topology and particle manipulation. Proc. Natl Acad. Sci. 119 (36), e2120538119.CrossRefGoogle ScholarPubMed
Blackburn, H.M. 2002 Mass and momentum transport from a sphere in steady and oscillatory flows. Phys. Fluids 14 (11), 3997–4011.CrossRefGoogle Scholar
Carlsson, F., Sen, M. & Löfdahl, L. 2004 Steady streaming due to vibrating walls. Phys. Fluids 16 (5), 1822–1825.CrossRefGoogle Scholar
Chan, F.K., Bhosale, Y., Parthasarathy, T. & Gazzola, M. 2022 Three-dimensional geometry and topology effects in viscous streaming. J. Fluid Mech. 933, A53.CrossRefGoogle Scholar
Chang, E.J. & Maxey, M.R. 1994 Unsteady flow about a sphere at low to moderate Reynolds number. Part 1. Oscillatory motion. J. Fluid Mech. 277, 347–379.CrossRefGoogle Scholar
Collis, J.F., Chakraborty, D. & Sader, J.E. 2017 Autonomous propulsion of nanorods trapped in an acoustic field. J. Fluid Mech. 825, 29–48.CrossRefGoogle Scholar
Collis, J.F., Chakraborty, D. & Sader, J.E. 2022 Autonomous propulsion of nanorods trapped in an acoustic field – corrigendum. J. Fluid Mech. 935, E1.CrossRefGoogle Scholar
Davidson, B.J. & Riley, N. 1971 Cavitation microstreaming. J. Sound Vib. 15 (2), 217–233.CrossRefGoogle Scholar
Dhara, N. 1982 The unsteady flow around an oscillating sphere in a viscous fluid. J. Phys. Soc. Japan 51 (12), 4095–4103.CrossRefGoogle Scholar
Dombrowski, T. & Klotsa, D. 2020 Kinematics of a simple reciprocal model swimmer at intermediate Reynolds numbers. Phys. Rev. Fluids 5 (6), 063103.CrossRefGoogle Scholar
Fabre, D., Jalal, J., Leontini, J.S. & Manasseh, R. 2017 Acoustic streaming and the induced forces between two spheres. J. Fluid Mech. 810, 378–391.CrossRefGoogle Scholar
Gemmell, B.J., Jiang, H. & Buskey, E.J. 2015 A tale of the ciliate tail: investigation into the adaptive significance of this sub-cellular structure. Proc. R. Soc. B 282 (1812), 20150770.CrossRefGoogle ScholarPubMed
Kaneko, A. & Honji, H. 1979 Double structures of steady streaming in the oscillatory viscous flow over a wavy wall. J. Fluid Mech. 93 (4), 727–736.CrossRefGoogle Scholar
Kotas, C.W., Yoda, M. & Rogers, P.H. 2006 Visualizations of steady streaming at moderate Reynolds numbers. Phys. Fluids 18 (9), 091102.CrossRefGoogle Scholar
Kotas, C.W., Yoda, M. & Rogers, P.H. 2007 Visualization of steady streaming near oscillating spheroids. Exp. Fluids 42 (1), 111–121.CrossRefGoogle Scholar
Kotas, C.W., Yoda, M. & Rogers, P.H. 2008 Steady streaming flows near spheroids oscillated at multiple frequencies. Exp. Fluids 45 (2), 295–307.CrossRefGoogle Scholar
Lane, C.A. 1955 Acoustical streaming in the vicinity of a sphere. J. Acoust. Soc. Am. 27 (6), 1082–1086.CrossRefGoogle Scholar
Li, J., Mayorga-Martinez, C.C., Ohl, C.-D. & Pumera, M. 2022 Ultrasonically propelled micro- and nanorobots. Adv. Funct. Mater. 32 (5), 2102265.CrossRefGoogle Scholar
Lippera, K., Dauchot, O., Michelin, S. & Benzaquen, M. 2019 No net motion for oscillating near-spheres at low Reynolds numbers. J. Fluid Mech. 866, R1.CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245 (903), 535–581.Google Scholar
Longuet-Higgins, M.S. 1998 Viscous streaming from an oscillating spherical bubble. Proc. R. Soc. Lond. A 454 (1970), 725–742.CrossRefGoogle Scholar
Lutz, B.R., Chen, J. & Schwartz, D.T. 2005 Microscopic steady streaming eddies created around short cylinders in a channel: flow visualization and Stokes layer scaling. Phys. Fluids 17 (2), 023601.CrossRefGoogle Scholar
Mei, R. 1994 Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds number. J. Fluid Mech. 270, 133–174.CrossRefGoogle Scholar
Mohaghar, M., Adhikari, D. & Webster, D.R. 2019 Characteristics of swimming shelled Antarctic pteropods (Limacina helicina antarctica) at intermediate Reynolds number regime. Phys. Rev. Fluids 4, 111101.CrossRefGoogle Scholar
Mohanty, S., Khalil, I.S.M. & Misra, S. 2020 Contactless acoustic micro/nano manipulation: a paradigm for next generation applications in life sciences. Proc. R. Soc. A 476 (2243), 20200621.CrossRefGoogle ScholarPubMed
Nadal, F. & Lauga, E. 2014 Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies. Phys. Fluids 26 (8), 082001.CrossRefGoogle Scholar
Nadal, F. & Michelin, S. 2020 Acoustic propulsion of a small, bottom-heavy sphere. J. Fluid Mech. 898, A10.CrossRefGoogle Scholar
Otto, F., Riegler, E.K. & Voth, G.A. 2008 Measurements of the steady streaming flow around oscillating spheres using three dimensional particle tracking velocimetry. Phys. Fluids 20 (9), 093304.CrossRefGoogle Scholar
Parthasarathy, T., Chan, F.K. & Gazzola, M. 2019 Streaming-enhanced flow-mediated transport. J. Fluid Mech. 878, 647–662.CrossRefGoogle Scholar
Raney, W.P., Corelli, J.C. & Westervelt, P.J. 1954 Acoustical streaming in the vicinity of a cylinder. J. Acoust. Soc. Am. 26 (6), 1006–1014.CrossRefGoogle Scholar
Rayleigh, Lord 1884 On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems. Phil. Trans. R. Soc. Lond. 175, 1–21.Google Scholar
Ren, C., Cheng, L., Tong, F., Xiong, C. & Chen, T. 2019 Oscillatory flow regimes around four cylinders in a diamond arrangement. J. Fluid Mech. 877, 955–1006.CrossRefGoogle Scholar
Ren, L., Wang, W. & Mallouk, T.E. 2018 Two forces are better than one: combining chemical and acoustic propulsion for enhanced micromotor functionality. Acc. Chem. Res. 51 (9), 1948–1956.CrossRefGoogle ScholarPubMed
Riley, N. 1966 On a sphere oscillating in a viscous fluid. Q. J. Mech. Appl. Maths 19 (4), 461–472.CrossRefGoogle Scholar
Riley, N. 2001 Steady streaming. Annu. Rev. Fluid Mech. 33 (1), 43–65.CrossRefGoogle Scholar
Sarpkaya, T. & Storm, M. 1985 In-line force on a cylinder translating in oscillatory flow. Appl. Ocean Res. 7 (4), 188–196.CrossRefGoogle Scholar
Stokes, G.G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–106.Google Scholar
Stuart, J.T. 1966 Double boundary layers in oscillatory viscous flow. J. Fluid Mech. 24 (4), 673–687.CrossRefGoogle Scholar
Valdez-Garduño, M., Leal-Estrada, M., Oliveros-Mata, E.S., Sandoval-Bojorquez, D.I., Soto, F., Wang, J. & Garcia-Gradilla, V. 2020 Density asymmetry driven propulsion of ultrasound-powered Janus micromotors. Adv. Funct. Mater. 30 (50), 2004043.CrossRefGoogle Scholar
Wang, C.-Y. 1965 The flow field induced by an oscillating sphere. J. Sound Vib. 2 (3), 257–269.CrossRefGoogle Scholar
Westervelt, P.J. 1953 a Acoustic streaming near a small obstacle. J. Acoust. Soc. Am. 25, 1123.CrossRefGoogle Scholar
Westervelt, P.J. 1953 b Hydrodynamic flow and Oseen's approximation. J. Acoust. Soc. Am. 25 (5), 951–953.CrossRefGoogle Scholar
Wu, J. & Du, G. 1997 Streaming generated by a bubble in an ultrasound field. J. Acoust. Soc. Am. 101 (4), 1899–1907.CrossRefGoogle Scholar
Wybrow, M.F., Yan, B. & Riley, N. 1996 Oscillatory flow over a circular cylinder close to a plane boundary. Fluid Dyn. Res. 18 (5), 269–288.CrossRefGoogle Scholar
Zhou, C., Zhao, L., Wei, M. & Wang, W. 2017 Twists and turns of orbiting and spinning metallic microparticles powered by megahertz ultrasound. ACS Nano 11 (12), 12668–12676.CrossRefGoogle ScholarPubMed
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