Hostname: page-component-89b8bd64d-72crv Total loading time: 0 Render date: 2026-05-11T23:42:39.530Z Has data issue: false hasContentIssue false
  • English
  • Français

Sphere oscillating in a rarefied gas

Published online by Cambridge University Press:  30 March 2016

Ying Wan Yap  and
John E. Sader
Ying Wan Yap
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
John E. Sader*
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
*
†Email address for correspondence: jsader@unimelb.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Flow generated by an oscillating sphere in a quiescent fluid is a classical problem in fluid mechanics whose solution is used ubiquitously. Miniaturisation of mechanical devices to small scales and their operation at high frequencies in fluid, which is common in modern nanomechanical systems, can preclude the use of the unsteady Stokes equation for continuum flow. Here, we explore the combined effects of gas rarefaction and unsteady motion of a sphere, within the framework of the unsteady linearised Boltzmann–BGK (Bhatnagar–Gross–Krook) equation. This equation is solved using the method of characteristics, and the resulting solution is valid for any oscillation frequency and arbitrary degrees of gas rarefaction. The resulting force provides the non-continuum counterpart to the (continuum) unsteady Stokes drag on a sphere. In contrast to the Stokes solution, where the flow is isothermal, non-continuum effects lead to a temperature jump at the sphere surface and non-isothermal flow. Unsteady effects and heat transport are found to mix strongly, leading to marked differences relative to the steady case. The solution to this canonical flow problem is expected to be of significant practical value in many applications, including the optical trapping of nanoparticles and the design and application of nanoelectromechanical systems. It also provides a benchmark for computational and approximate methods of solution for the Boltzmann equation.

JFM classification

Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics Micro-/Nano-fluid dynamics: Non-continuum effects

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 794 , 10 May 2016 , pp. 109 - 153
Check for updates
Copyright
© 2016 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

Aoki, K. & Sone, Y. 1987 Temperature field induced around a sphere in a uniform flow of a rarefied gas. Phys. Fluids 30 (7), 2286–2288.CrossRefGoogle Scholar
Baker, R. M. L. & Charwat, A. F. 1958 Transitional correction to the drag of a sphere in free molecule flow. Phys. Fluids 1, 73–82.CrossRefGoogle Scholar
Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. Phys. Rev. 94 (3), 511–525.CrossRefGoogle Scholar
Cercignani, C. 2000 Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge University Press.Google Scholar
Cercignani, C. & Pagani, C. D. 1966 Variational approach to boundary-value problems in kinetic theory. Phys. Fluids 9 (6), 1167–1173.CrossRefGoogle Scholar
Cercignani, C. & Pagani, C. D. 1968 Flow of a rarefied gas past an axisymmetric body: I. General remarks. Phys. Fluids 11 (8), 1395–1399.CrossRefGoogle Scholar
Cercignani, C., Pagani, C. D. & Bassanini, P. 1968 Flow of a rarefied gas past an axisymmetric body: II. Case of a sphere. Phys. Fluids 11 (7), 1399–1403.CrossRefGoogle Scholar
Chun, J. & Koch, D. L. 2005 A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number. Phys. Fluids 17 (10), 107107.CrossRefGoogle Scholar
Cunningham, E. 1910 On the velocity of steady fall of spherical particles through fluid medium. Proc. R. Soc. Lond. A 83 (563), 357.Google Scholar
Einstein, A. 1905 Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322 (8), 549–560.CrossRefGoogle Scholar
Ekinci, K. L. & Roukes, M. L. 2005 Nanoelectromechanical systems. Rev. Sci. Instrum. 76 (6), 061101.CrossRefGoogle Scholar
Epstein, P. S. 1923 On the resistance experienced by spheres in their motion through gases. Phys. Rev. 23 (6), 710.CrossRefGoogle Scholar
Gu, X. J. & Emerson, D. R. 2011 Modeling oscillatory flows in the transition regime using a high-order moment method. Microfluid. Nanofluid. 10 (2), 389–401.CrossRefGoogle Scholar
Hansen, J. S. & Ottesen, J. T. 2006 Molecular dynamics simulations of oscillatory flows in microfluidic channels. Microfluid. Nanofluid. 2 (4), 301–307.CrossRefGoogle Scholar
Ivchenko, I. N., Loyalka, S. K. & Tompson, R. V. Jr. 2007 Analytical Methods for Problems of Molecular Transport. Springer.CrossRefGoogle Scholar
Jensen, K., Kim, K. & Zettl, A. 2008 An atomic-resolution nanomechanical mass sensor. Nature Nanotechnol. 3 (9), 533–537.CrossRefGoogle ScholarPubMed
Ladiges, D. R. & Sader, J. E. 2015 Frequency-domain Monte Carlo method for linear oscillatory gas flows. J. Comput. Phys. 284, 351–366.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon.Google Scholar
Law, W. S. & Loyalka, S. K. 1986 Motion of a sphere in rarefied gas: II. Role of temperature variation in the Knudsen layer. Phys. Fluids 29 (11), 3886–3888.CrossRefGoogle Scholar
Lea, K. C. & Loyalka, S. K. 1982 Motion of a sphere in rarefied gas. Phys. Fluids 25 (9), 1550–1557.CrossRefGoogle Scholar
Li, M., Tang, H. T. & Roukes, M. L. 2007 Ultra-sensitive nems-based cantilevers for sensing, scanned probe and very high-frequency applications. Nature Nanotechnol. 2, 114–120.CrossRefGoogle ScholarPubMed
Liu, V.-C., Pang, S.-C. & Jew, H. 1965 Sphere drag in flows of almost-free molecules. Phys. Fluids 8 (5), 788.CrossRefGoogle Scholar
Loyalka, S. K. & Tompson, R. V. 2009 The velocity slip problem: accurate solutions of the BGK model integral equation. Eur. J. Mech. (B/Fluids) 28 (2), 211–213.CrossRefGoogle Scholar
Manela, A. & Hadjiconstantinou, N. G. 2008 Gas motion induced by unsteady boundary heating in a small-scale slab. Phys. Fluids 20 (11), 117104.CrossRefGoogle Scholar
Manela, A. & Hadjiconstantinou, N. G. 2010 Gas-flow animation by unsteady heating in a microchannel. Phys. Fluids 22 (6), 062001.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
Mei, R. & Adrian, R. J. 1992 Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number. J. Fluid Mech. 237, 323–341.CrossRefGoogle Scholar
Millikan, R. A. 1923 The general law of a fall of a small spherical body through a gas, and its bearing upon the nature of molecular reflection from surfaces. Phys. Rev. 22, 1.CrossRefGoogle Scholar
Nassios, J. & Sader, J. E. 2012 Asymptotic analysis of the Boltzmann–BGK equation for oscillatory flows. J. Fluid Mech. 708, 197–249.CrossRefGoogle Scholar
Nassios, J. & Sader, J. E. 2013 High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation. J. Fluid Mech. 729, 1–46.CrossRefGoogle Scholar
Neukirch, L. P., Gieseler, J., Quidant, R., Novotny, L. & Vamivakas, A. N. 2013 Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond. Opt. Lett. 38 (16), 2976–2979.CrossRefGoogle ScholarPubMed
Nourazar, S. S. & Ganjaei, A. A. 2010 A new scheme developed for the numerical simulation of the Boltzmann equation using the direct simulation Monte-Carlo scheme for the flow about a sphere. J. Mech. Sci. Technol. 24 (10), 1989–1996.CrossRefGoogle Scholar
O’Connell, A. D., Hofheinz, M., Ansmann, M., Bialczak, R. C., Lenander, M., Lucero, E., Neeley, M., Sank, D., Wang, H., Weides, M., Wenner, J., Martinis, J. M. & Cleland, A. N. 2010 Quantum ground state and single-phonon control of a mechanical resonator. Nature 464 (7289), 697–703.CrossRefGoogle ScholarPubMed
Pelton, M., Chakraborty, D., Malachosky, E., Guyot-Sionnest, P. & Sader, J. E. 2013 Viscoelastic flows in simple liquids generated by vibrating nanostructures. Phys. Rev. Lett. 111, 244502.CrossRefGoogle ScholarPubMed
Pelton, M., Sader, J. E., Burgin, J., Liu, M., Guyot-sionnest, P. & Gosztola, D. 2009 Damping of acoustic vibrations in gold nanoparticles. Nature Nanotechnol. 4, 492–495.CrossRefGoogle ScholarPubMed
Phillips, W. F. 1975 Drag on a small sphere moving through a gas. Phys. Fluids 18 (9), 1089.CrossRefGoogle Scholar
Riley, N. 1966 On a sphere oscillating in a viscous fluid. Q. J. Mech. Appl. Maths XIX (Pt. 4), 461–472.CrossRefGoogle Scholar
Sharipov, F. & Kalempa, D. 2007 Gas flow near a plate oscillating longitudinally with an arbitrary frequency. Phys. Fluids 19 (1), 017110.CrossRefGoogle Scholar
Sharipov, F. & Kalempa, D. 2008 Oscillatory Couette flow at arbitrary oscillation frequency over the whole range of the Knudsen number. Microfluid. Nanofluid. 4, 363–374.CrossRefGoogle Scholar
Sharipov, F. & Seleznev, V. 1998 Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data 27 (3), 657–706.CrossRefGoogle Scholar
Sone, Y. 2002 Kinetic Theory and Fluid Dynamics. Birkhauser.CrossRefGoogle Scholar
Sone, Y. & Aoki, K. 1976 Forces on a spherical particle in a slightly rarefied gas. In Progress in Astronautics and Aeronautics, (ed. Potter, J. L.), vol. 51, Part I, p. 417. AIAA.Google Scholar
Stokes, G. G. 1880 Mathematical and Physical Papers, vol. 1. Cambridge University Press.Google Scholar
Sutherland, W. 1905 A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. Phil. Mag. 9 (54), 781–785.CrossRefGoogle Scholar
Taheri, P., Rana, A. S., Torrilhon, M. & Struchtrup, H. 2009 Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics. Contin. Mech. Thermodyn. 21 (6), 423–443.CrossRefGoogle Scholar
Takata, S. & Hattori, M. 2012 Asymptotic theory for the time-dependent behavior of a slightly rarefied gas over a smooth solid boundary. J. Stat. Phys. 147 (6), 1182–1215.CrossRefGoogle Scholar
Takata, S., Sone, Y. & Aoki, K. 1993 Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 5 (3), 716.CrossRefGoogle Scholar
Torrilhon, M. 2010 Slow gas microflow past a sphere: analytical solution based on moment equations. Phys. Fluids 22 (7), 072001.CrossRefGoogle Scholar
Tzou, D. Y., Beraun, J. E. & Chen, J. K. 2002 Ultrafast deformation in femtosecond laser heating. Trans. ASME J. Heat Transfer 124 (2), 284–292.CrossRefGoogle Scholar
Welander, P. 1954 On the temperature jump in a rarefied gas. Ark. Fys. 7, 507–533.Google Scholar
Willis, D. R. 1966 Sphere drag at high Knudsen number and low Mach number. Phys. Fluids 9 (12), 2522.CrossRefGoogle Scholar
Yap, Y. W. & Sader, J. E. 2012 High accuracy numerical solutions of the Boltzmann Bhatnagar–Gross–Krook equation for steady and oscillatory Couette flows. Phys. Fluids 24, 032004.CrossRefGoogle Scholar