Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-02-11T12:01:51.949Z Has data issue: false hasContentIssue false
  • English
  • Français

Stokes' second flow problem in a high-frequency limit: application to nanomechanical resonators

Published online by Cambridge University Press:  14 August 2007

VICTOR YAKHOT  and
CARLOS COLOSQUI
VICTOR YAKHOT
Affiliation:
Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215, USA
CARLOS COLOSQUI
Affiliation:
Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Solving the Boltzmann–BGK equation, we investigate a flow generated by an infinite plate oscillating with frequency ω. The geometrical simplicity of the problem allows a solution in the entire range of dimensionless frequency variation 0 ≤ ωτ ≤ ∞, where τ is a properly defined relaxation time. A transition from viscoelastic behaviour of a Newtonian fluid (ωτ → 0) to purely elastic dynamics in the limit ωτ → ∞ is discovered. The relation of the derived solutions to nanofluidics is demonstrated on a solvable example of a ‘plane oscillator’. The results from the derived formulae compare well with experimental data on various nanoresonators operating in a wide range of both frequency and pressure variation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 586 , 10 September 2007 , pp. 249 - 258
Copyright
Copyright © Cambridge University Press 2007

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.)

References

Benzi, R., Succi, S. & Vergassola, M. 1992 The lattice Boltzmann equation: theory and applications. Phys. Rep. 222, 145.Google Scholar
Bhatnagar, P. L., Gross, E. & Krook, M. 1954 A model for collisions in gases I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525.Google Scholar
Binning, C., Quate, C. F. & Gerber, C. 1986 Atomic force microscope. Phys. Rev. Lett. 56, 930–933.CrossRefGoogle Scholar
Bird, R. B., Stewart, W. E. & Lightfoot, E. R. E. 2002 Transport Phenomena, 2nd Edn. Wiley and Sons.Google Scholar
Brodkey, R. S. 1967 The Phenomena of Fluid Motions. Dover.Google Scholar
Chen, H., Kandasamy, S., Orszag, S. A., Shock, R., Succi, S. & Yakhot, V. 2003 Extended Boltzmann kinetic equation for turbulent flows. Science 301, 633–636.CrossRefGoogle ScholarPubMed
Chen, H., Orszag, S. A., Staroselsky, I. & Succi, S. 2004 Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech. 519, 301–313.Google Scholar
Chen, S. Y. & Doolen, G. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–xxx.CrossRefGoogle Scholar
Cleland, A. N. & Roukes, M. L. 1998 A nanometer-scale mechanical electrometer. Nature 392, 160–162.CrossRefGoogle Scholar
Colosqui, C. & Yakhot, V. 2007 Lattice Boltzmann simulations of second Stokes flow problem. J. Mod. Phys. (in press).Google Scholar
Ekinci, K. L., Huang, X. M. & Roukes, M. L. 2004 Ultrasensitive mass detection. Appl. Phys. Lett. 84, 4469.CrossRefGoogle Scholar
Karabacak, D. M., Yakhot, V. & Ekinci, K. L. 2007 High frequency nanofluidics: experimental study with nano-mechanical resonators. Phys. Rev. Lett. (in press) (ArXiv, cond-mat/0703230).CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1981 Physical Kinetics. Butterworth-Heinemann.Google Scholar
Rosenau, P. 1993 Random walker and the telegrapher's equation: A paradigm of generalized hydrodynamics. Phys. Rev. E 48, R655–657.CrossRefGoogle Scholar
Service, R. B. 2006 Tipping scales - just barely. Science 312, 683.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction in fluids on the motion of pendulums. Camb. Phil. Trans. IX, 8.Google Scholar
Verbridge, S. S., Bellan, L. M., Parpia, J. M. & Craighead, H. G. 2006 Optically driven resonance of nanoscale flexural oscillators in fluids. Nano Lett. 6, 2109–2114.CrossRefGoogle Scholar