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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]
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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