We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centers, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs), which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by the use of a bipolar coordinate system, we derive the stream function for the flow as the Reynolds number (Re) tends to zero and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behavior of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon the known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with the existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix R in a number of situations in which positivity is destroyed for multipole/perturbative methods.
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June 2020
Research Article|
June 03 2020
The singular hydrodynamic interactions between two spheres in Stokes flow
B. D. Goddard
;
B. D. Goddard
a)
1
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh
, Edinburgh EH9 3FD, United Kingdom
a)Author to whom correspondence should be addressed: b.goddard@ed.ac.uk
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R. D. Mills-Williams;
R. D. Mills-Williams
b)
1
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh
, Edinburgh EH9 3FD, United Kingdom
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J. Sun
J. Sun
2
School of Engineering, University of Edinburgh
, Edinburgh EH9 3JL, United Kingdom
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a)Author to whom correspondence should be addressed: b.goddard@ed.ac.uk
b)
Electronic mail: r.mills@ed.ac.uk
Physics of Fluids 32, 062001 (2020)
Article history
Received:
March 27 2020
Accepted:
May 18 2020
Citation
B. D. Goddard, R. D. Mills-Williams, J. Sun; The singular hydrodynamic interactions between two spheres in Stokes flow. Physics of Fluids 1 June 2020; 32 (6): 062001. https://doi.org/10.1063/5.0009053
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