Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-01T02:20:03.505Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice Boltzmann simulation of electromechanical resonators in gaseous media

Published online by Cambridge University Press:  30 March 2010

CARLOS E. COLOSQUI ,
DEVREZ M. KARABACAK ,
KAMIL L. EKINCI  and
VICTOR YAKHOT
CARLOS E. COLOSQUI*
Affiliation:
Department of Mechanical Engineering, Boston University, Boston, MA 02215, USA
DEVREZ M. KARABACAK
Affiliation:
IMEC Holst Centre, Eindhoven, 5605 KN, The Netherlands
KAMIL L. EKINCI
Affiliation:
Department of Mechanical Engineering, Boston University, Boston, MA 02215, USA
VICTOR YAKHOT
Affiliation:
Department of Mechanical Engineering, Boston University, Boston, MA 02215, USA
*
†Present address: Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA. Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work, we employ a kinetic-theory-based approach to predict the hydrodynamic forces on electromechanical resonators operating in gaseous media. Using the Boltzmann–BGK equation, we investigate the influence of the resonator geometry on the fluid resistance in the entire range of non-dimensional frequency variation 0 ≤ τω ≤ ∞; here the fluid relaxation time τ = μ/p is determined by the gas viscosity μ and pressure p at thermodynamic equilibrium, and ω is the (angular) oscillation frequency. Our results here capture two important aspects of recent experimental measurements that covered a broad range of experimental parameters. First, the experimentally observed transition from viscous to viscoelastic flow in simple gases at τω ≈ 1 emerges naturally in the numerical data. Second, the calculated effects of resonator geometry are in agreement with experimental observations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 241 - 257
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Bhiladvala, R. B. & Wang, Z. J. 2003 Effect of fluids on the Q factor and resonance frequency of oscillating micrometre and nanometre scale beams. Phys. Rev. E 69, 036307.CrossRefGoogle Scholar
Binnig, G., Quate, C. F. & Gerber, C. 1982 Atomic force microscope. Phys. Rev. Lett. 56, 930.CrossRefGoogle Scholar
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2002 Transport Phenomena. Wiley & Sons.Google Scholar
Cercignani, C. 1969 Mathematical Methods in Kinetic Theory. Plenum.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Chen, H., Orszag, S. A. & Staroselsky, I. 2007 Macroscopic description of arbitrary Knudsen number flow using Boltzmann–BGK kinetic theory. J. Fluid Mech. 574, 495–505.CrossRefGoogle Scholar
Cleland, A. N. & Roukes, M. L. 1998 A nanometre-scale mechanical electrometer. Nature 392, 161.CrossRefGoogle Scholar
Colosqui, C. E., Chen, H., Shan, X., Staroselsky, I. & Yakhot, V. 2009 Propagating high-frequency shear waves in simple fluids. Phys. Fluids 21, 013105.CrossRefGoogle Scholar
Colosqui, C. E. & Yakhot, V. 2007 Lattice Boltzmann simulation of a non-Newtonian oscillating flow in a high-frequency limit. Intl J. Mod. Phys. C 18 (4), 473–482.CrossRefGoogle Scholar
Ekinci, K. L., Huang, X. M. & Roukes, M. L. 2004 Ultrasensitive nanoelectromechanical mass detection. Appl. Phys. Lett. 84, 4469.CrossRefGoogle Scholar
Ekinci, K. L., Karabacak, D. M. & Yakhot, V. 2008 Universality in Oscillating Flows. Phys. Rev. Lett. 101, 26.CrossRefGoogle ScholarPubMed
Evans, D. & Morriss, G. 2008 Statistical Mechanics of Nonequilibrium Liquids. Cambridge University Press.CrossRefGoogle Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2 (4), 331–407.CrossRefGoogle Scholar
Hadjiconstantinou, N. G. 2005 Oscillatory shear-driven gas flows in the transition and free-molecular-flow regimes. Phys. Fluids 17, 100611.CrossRefGoogle Scholar
Karabacak, D. M., Yakhot, V. & Ekinci, K. L. 2007 High-frequency nanofluidics: an experimental study using nanomechanical resonators. Phys. Rev. Lett. 98, 254505.CrossRefGoogle ScholarPubMed
Kouh, T., Karabacak, D., Kim, D. H. & Ekinci, K. L. 2005 Diffraction effects in optical interferometric displacement detection in nanoelectromechanical systems. Appl. Phys. Lett. 86, 013106.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Lauga, E., Brenner, M. P. & Stone, H. A. 2007 Microfluidics: the no-slip boundary condition. In Handbook of Experimental Fluid Mechanics (ed. Foss, J., Tropea, C. & Yarin, A.), Chap. 19, pp. 1219–1240. Springer.Google Scholar
Park, J. H., Bahukudumbi, P. & Beskok, A. 2004 Rarefaction effects on shear driven oscillatory gas flows: a direct simulation Monte Carlo study in the entire Knudsen regime. Phys. Fluids 16, 317.CrossRefGoogle Scholar
Paul, M. R. & Cross, M. C. 2004 Stochastic dynamics of nanoscale mechanical oscillators immersed in a viscous fluid. Phys. Rev. Lett. 92 (23), 235501–235501.CrossRefGoogle Scholar
Sader, J. E. 1998 Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J. Appl. Phys. 84, 64.CrossRefGoogle Scholar
Shan, X. & Chen, H. 2007 A general Multiple-Relaxation Boltzmann Collision Model. Intl J. Mod. Phys. C 18 (04), 635–643.CrossRefGoogle Scholar
Shan, X., Yuan, X. F. & Chen, H. 2006 Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation. J. Fluid Mech. 550, 413–441.CrossRefGoogle Scholar
Tuck, E. O. 1969 Calculation of unsteady flows due to small motions of cylinders in a viscous fluid. J. Engng Math. 3 (1), 29–44.CrossRefGoogle Scholar
Weng, H. C. 2008 A challenge in Navier–Stokes-based continuum modelling: Maxwell–Burnett slip law. Phys. Fluids 20, 106101.CrossRefGoogle Scholar
Yakhot, V. & Colosqui, C. E. 2007 Stokes' second flow problem in a high-frequency limit: application to nanomechanical resonators. J. Fluid Mech. 586, 249–258.CrossRefGoogle Scholar
Zhang, R., Shan, X. & Chen, H. 2006 Efficient kinetic method for fluid simulation beyond the Navier–Stokes equation. Phys. Rev. E 74 (4), 046703.CrossRefGoogle ScholarPubMed
Zou, Q. & He, X. 1997 On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids 9 (6), 1591–1598.CrossRefGoogle Scholar