Skip to main content Accessibility help
×
Home

Drag force on spherical particle moving near a plane wall in highly rarefied gas

  • P. Goswami (a1), T. Baier (a1), S. Tiwari (a2), C. Lv (a1) (a3), S. Hardt (a1) and A. Klar (a2)...

Abstract

The drag force on a sphere in tangential and normal motion to a plane wall is evaluated in the limit of large Knudsen number and small Mach (and Strouhal) number assuming isothermal conditions and diffuse reflection of gas molecules on walls. In the limit of free molecular flow, the molecular distribution function of the gas is evaluated using a set of coupled Fredholm integral equations. The results are compared with direct simulation Monte Carlo calculations and extended for finite Knudsen numbers. In all cases stronger dependence of the force on the width of the gap is found for normal compared to tangential motion. When the flow within the gap can be considered as essentially collisionless in nature, a similar dependence of the force on the gap width is observed at finite Knudsen numbers as in the free molecular case.

Copyright

Corresponding author

Email address for correspondence: hardt@nmf.tu-darmstadt.de

References

Hide All
Babovsky, H. & Illner, R. 1989 A convergence proof for Nanbu’s simulation method for the full Boltzmann equation. SIAM J. Numer. Anal. 26 (1), 4565.
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press.
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16 (3–4), 242251.
Cercignani, C. & Daneri, A. 1963 Flow of a rarefied gas between two parallel plates. J. Appl. Phys. 34 (12), 35093513.
Cercignani, C., Illner, R. & Pulvirenti, M. 2013 The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106. Springer.
Cox, R. G. & Hsu, S. K. 1977 The lateral migration of solid particles in a laminar flow near a plane. Intl J. Multiphase Flow 3 (3), 201222.
Epstein, P. S. 1924 On the resistance experienced by spheres in their motion through gases. Phys. Rev. 23 (6), 710733.
Faxen, H. 1923 Die Bewegung einer starren Kugel längs der Achse eines mit zäher Flüssigkeit gefüllten Rohres. Ark. Mat. Astron. Fys. 17, 128.
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall – I Motion through a quiescent fluid. Chem. Engng Sci. 22 (4), 637651.
Gopinath, A. & Koch, D. L. 1997 A method for calculating hydrodynamic interactions between two bodies in low Mach number free-molecular flows with application to the resistivity functions for two aligned cylinders. Phys. Fluids 9 (11), 35503565.
Gopinath, A. & Koch, D. L. 1999 Hydrodynamic interactions between two equal spheres in a highly rarefied gas. Phys. Fluids 11 (9), 27722787.
Goren, S. L. 1973 The hydrodynamic force resisting the approach of a sphere to a plane wall in slip flow. J. Colloid Interface Sci. 44 (2), 356360.
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.
Hocking, L. M. 1973 The effect of slip on the motion of a sphere close to a wall and of two adjacent spheres. J. Engng Maths 7 (3), 207221.
Lorentz, H. A. 1907 Ein allgemeiner Satz, die Bewegung einer reibenden Flüssigkeit betreffend, nebst einigen Anwendungen desselben. In Abhandlungen über theoretische Physik Erster Band (ed. Lorentz, H. A.), pp. 2342. B. G. Teubner.
Loussaief, H., Pasol, L. & Feuillebois, F. 2015 Motion of a spherical particle in a viscous fluid along a slip wall. Q. J. Mech. Appl. Maths 68 (2), 115144.
Luo, H. & Pozrikidis, C. 2008 Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall. J. Engng Maths 62 (1), 121.
Maude, A. D. 1961 End effects in a falling-sphere viscometer. Brit. J. Appl. Phys. 12 (6), 293295.
Nanbu, K. 1980 Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent gases. J. Phys. Soc. Japan 49 (5), 20422049.
Neunzert, H. & Struckmeier, J. 1995 Particle methods for the Boltzmann equation. Acta Numer. 4, 417457.
O’Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11 (1), 6774.
O’Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27 (4), 705724.
Ramanathan, S., Koch, D. L. & Bhiladvala, R. B. 2010 Noncontinuum drag force on a nanowire vibrating normal to a wall: simulations and theory. Phys. Fluids 22 (10), 103101.
Shrestha, S., Tiwari, S., Klar, A. & Hardt, S. 2015 Numerical simulation of a moving rigid body in a rarefied gas. J. Comput. Phys. 292, 239252.
Sone, Y. 2007 Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkhäuser.
Sone, Y. & Onishi, Y. 1978 Kinetic theory of evaporation and condensation – hydrodynamic equation and slip boundary condition. J. Phys. Soc. Japan 44 (6), 19811994.
Sundararajakumar, R. R. & Koch, D. L. 1996 Non-continuum lubrication flows between particles colliding in a gas. J. Fluid Mech. 313, 283308.
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), 716737.
Vasseur, P. & Cox, R. G. 1977 The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech. 80 (3), 561591.
Vinogradova, O. I. 1995 Drainage of a thin liquid film confined between hydrophobic surfaces. Langmuir 11 (6), 22132220.
Wang, C.-T. 1972 Free molecular flow over a rotating sphere. AIAA J. 10 (5), 713714.
Ying, R. & Peters, M. H. 1989 Hydrodynamic interaction of two unequal-sized spheres in a slightly rarefied gas: resistance and mobility functions. J. Fluid Mech. 207, 353378.
Ying, R. & Peters, M. H. 1991 Interparticle and particle surface gas dynamic interactions. Aerosol Sci. Technol. 14 (4), 418433.
Zeng, L., Balachandar, S. & Fischer, P. 2005 Wall-induced forces on a rigid sphere at finite Reynolds number. J. Fluid Mech. 536, 125.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Drag force on spherical particle moving near a plane wall in highly rarefied gas

  • P. Goswami (a1), T. Baier (a1), S. Tiwari (a2), C. Lv (a1) (a3), S. Hardt (a1) and A. Klar (a2)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed