Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T10:56:18.094Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-continuum tangential lubrication gas flow between two spheres

Published online by Cambridge University Press:  04 June 2021

Melanie Li Sing How Open the ORCID record for Melanie Li Sing How [Opens in a new window] ,
Donald L. Koch Open the ORCID record for Donald L. Koch [Opens in a new window]  and
Lance R. Collins Open the ORCID record for Lance R. Collins [Opens in a new window]
Melanie Li Sing How
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY14853, USA
Donald L. Koch
Affiliation:
Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY14853, USA
Lance R. Collins*
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY14853, USA Innovation Campus, Virginia Tech, 3000 Potomac Avenue, Suite 101 Alexandria, VA22305, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

As two particles approach each other, the continuum lubrication force diverges, with decreasing separation preventing contact. However, for separations comparable to the mean free path of the gas, $\lambda$, non-continuum effects cause the lubrication force to diverge more slowly with decreasing separation distance, allowing for contact in finite time. The first study of this phenomenon was done by Sundararajakumar & Koch (J. Fluid Mech., vol. 313, 1996, pp. 238–308) for two particles moving along their line of centres. We extend their normal motion study to include tangential motions. For small Knudsen number $Kn=\lambda / a$, where $a$ is the harmonic mean of the two particle radii, we use a matched asymptotic expansion technique to obtain the non-continuum forces and torques for tangential motions of spheres separated by distances within the lubrication regime that are at or below the mean free path of the gas. The hydrodynamic resistivity functions are fitted to provide a uniformly valid approximation that smoothly transitions between the continuum multipole and non-continuum lubrication expressions for the forces and torques as the minimum gap between the particles $h_0$ varies from values of $O(a)$ to values of $O(\lambda )$. These functions, in combination with the result by Sundararajakumar & Koch (J. Fluid Mech., vol. 313, 1996, pp. 238–308) and the classical work by Jeffrey & Onishi (J. Fluid Mech., vol. 139, 1984, pp. 261–290), yield a complete formulation for the hydrodynamic interactions of two spheres at all separations, from non-interacting spheres in the extreme far field through all the transitions that occur up to contact. We apply the new formulation to the classical case of a particle settling parallel to a vertical wall. The continuum Stokes equation predicts a settling speed that decreases with decreasing gap separation and vanishes at contact, whereas the non-continuum model developed herein predicts a finite settling speed at contact.

JFM classification

Drops and Bubbles: Drops Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A2
Check for updates
Copyright
© The Author(s), 2021. Published by 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.)

References

REFERENCES

Batchelor, G.K. 1982 Sedimentation in a dilute polydisperse system of interacting spheres. Part I. General theory. J. Fluid Mech. 119, 379–408.CrossRefGoogle Scholar
Bhatnagar, P.L., Gross, E.P. & Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 (3), 511–525.CrossRefGoogle Scholar
Brenner, H. & O'Neill, M.E. 1972 On the Stokes resistance of a multiparticle systems in a linear shear field. Chem. Engng Sci. 27, 1421–1439.CrossRefGoogle Scholar
Carty, J.J. 1957 Resistance coefficients for spheres on a plane boundary. Master's thesis, MIT.Google Scholar
Cercignani, C. & Daneri, A. 1963 Flow of a rarefied gas between two parallel plates. J. Appl. Phys. 34, 3509–3513.CrossRefGoogle Scholar
Cercignani, C. & Pagani, C.D. 1966 Variational approach to boundary-value problems in kinetic theory. Phys. Fluids 9, 1167–1173.CrossRefGoogle Scholar
Davis, R.H., Schonberg, J.A. & Rallison, J.M. 1989 The lubrication force between two viscous drops. Phys. Fluids A 1, 77–81.CrossRefGoogle Scholar
Devenish, B.J., et al. 2012 Droplet growth in warm turbulent clouds. Q. J. R. Meteorol. Soc. 138, 1401–1429.CrossRefGoogle Scholar
Dhanasekaran, J., Roy, A. & Koch, D.L. 2021 Collision rate of bidisperse spheres settling in a compressional non-continuum gas flow. J. Fluid Mech. 910, A10.CrossRefGoogle Scholar
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, 637–651.CrossRefGoogle Scholar
Gopinath, A., Chen, S.B. & Koch, D.L. 1997 Lubrication flows between spherical particles colliding in a compressible non-continuum gas. J. Fluid Mech. 344, 245–269.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Springer.Google Scholar
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, 207–221.CrossRefGoogle Scholar
Hocking, L.M. & Jonas, P.R. 1970 The collision efficiency of small drops. Q. J. R. Meteorol. Soc. 96, 722–729.CrossRefGoogle Scholar
Jeffrey, D.J. & Onishi, Y. 1984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J. Fluid Mech. 139, 261–290.CrossRefGoogle Scholar
Kim, S. & Karrila, S.J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
Klinzing, G.E. 2001 Pneumatic conveying: transport solutions, pitfalls, and measurements. In Handbook of Powder Technology (ed. A. Levy & H. Kalman), pp. 291–301. Elsevier.CrossRefGoogle Scholar
Kogan, M.N. 1969 Rarefied Gas Dynamics. Plenum.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1980 Statistical Physics. Butterworth-Heinemann.Google Scholar
Lorentz, H.A. 1896 A general theorem concerning the motion of a viscous fluid and a few consequences derived from it. Z. Koninkl. Akad. van Wetensch. Amsterdam 5, 168–175.Google Scholar
Loyalka, S.K. & Ferziger, J.H. 1967 Model dependence of the slip coefficient. Phys. Fluids 10, 1833–1839.CrossRefGoogle Scholar
O'Neill, M.E. & Majumdar, S.R. 1970 Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part II: asymptotic forms of the solutions when the minimum clearance between the spheres approaches zero. Z. Angew. Math. Phys. 21, 180–187.CrossRefGoogle Scholar
O'Neill, M.E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27, 705–724.CrossRefGoogle Scholar
Onsager, L. 1931 a Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426.CrossRefGoogle Scholar
Onsager, L. 1931 b Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265–2279.CrossRefGoogle Scholar
Sharipov, F. 2016 Analytical and numerical calculations of rarefied gas flows. In Handbook of Vacuum Technology (ed. K. Jousten), pp. 167–228. Wiley-vch.CrossRefGoogle Scholar
Shaw, R.A. 2003 Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech. 35, 183–227.CrossRefGoogle Scholar
Sundararajakumar, R.R. & Koch, D.L. 1996 Non-continuum lubrication flows between particles colliding in a gas. J. Fluid Mech. 313, 238–308.CrossRefGoogle Scholar
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, 353–378.CrossRefGoogle Scholar
Zhang, J., Xu, X. & Qian, T. 2015 Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit. Phys. Rev. E 91, 033016.CrossRefGoogle ScholarPubMed