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Near-contact approach of two permeable spheres

Published online by Cambridge University Press:  19 August 2021

Rodrigo B. Reboucas Open the ORCID record for Rodrigo B. Reboucas [Opens in a new window]  and
Michael Loewenberg Open the ORCID record for Michael Loewenberg [Opens in a new window]
Rodrigo B. Reboucas
Affiliation:
Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06520-8286, USA
Michael Loewenberg*
Affiliation:
Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06520-8286, USA
*
†Email address for correspondence: michael.loewenberg@yale.edu
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Abstract

An analysis is presented for the axisymmetric lubrication resistance between permeable spherical particles. Darcy's law is used to describe the flow in the permeable medium and a slip boundary condition is applied at the interface. The pressure in the near-contact region is governed by a non-local integral equation. The asymptotic limit $K=k/a^{2} \ll 1$ is considered, where $k$ is the arithmetic mean permeability, and $a^{-1}=a^{-1}_{1}+a^{-1}_{2}$ is the reduced radius, and $a_1$ and $a_2$ are the particle radii. The formulation allows for particles with distinct particle radii, permeabilities and slip coefficients, including permeable and impermeable particles and spherical drops. Non-zero particle permeability qualitatively affects the axisymmetric near-contact motion, removing the classical lubrication singularity for impermeable particles, resulting in finite contact times under the action of a constant force. The lubrication resistance becomes independent of gap and attains a maximum value at contact $F=6{\rm \pi} \mu a W K^{-2/5}\tilde {f}_c$, where $\mu$ is the fluid viscosity, $W$ is the relative velocity and $\tilde {f}_c$ depends on slip coefficients and weakly on permeabilities; for two permeable particles with no-slip boundary conditions, $\tilde {f}_c=0.7507$; for a permeable particle in near contact with a spherical drop, $\tilde {f}_c$ is reduced by a factor of $2^{-6/5}$.

JFM classification

Low-Reynolds-number flows: Porous media Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 925 , 25 October 2021 , A1
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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