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Exact solutions of the equations of slow viscous flow generated by the asymmetrical motion of two equal spheres

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Abstract

Exact solutions are presented for the asymmetrical slow viscous flows of an infinite fluid caused by either the rotation of two spheres each of radius a with equal uniform angular velocities about diameters perpendicular to their line of centres or by the translation of the spheres with equal and opposite velocities along directions perpendicular to their line of centres. The technique of solution is applicable for any value of the distance 2d between the centres of the spheres. The asymptotic forms of the solutions are discussed for the cases when the ratio d/a is large or small.

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Abbreviations

a :

radius of the spheres

A n , B n , D n , F m :

coefficients of series

c :

a constant length

d :

distance between centres of spheres

f 1, f 2 :

dimensionless force coefficients

g 1, g 2 :

dimensionless couple coefficients

k k :

coth(n + 1/2)α coth α

p :

hydrodynamic fluid pressure

P :

pressure function defined in (3.1)

P mn (σ):

Legendre function of the first kind of order n and degree m

r, z :

dimensionless cylindrical polar coordinates

R, Z :

strained coordinates defined in (7.5)

u, v, w :

cylindrical components of velocity

U, V, W :

functions of r, z

V :

fluid velocity

U :

speed of sphere

α :

particular value of coordinate ξ

ε :

dimensionless minimum clearance between spheres

θ :

cylindrical polar coordinate

μ :

coefficient of dynamic viscosity

ξ, η :

coordinates defined in (3.7)

ρ :

density of fluid

ρ s :

density of either sphere

σ :

cos η

φ, ξ, ψ :

velocity functions defined in (3.1) and (3.2)

Ω :

magnitude of angular velocity of a sphere

References

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Authors and Affiliations

  1. Department of Mathematics, University College, London, UK

    M. E. O'Neill

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  1. M. E. O'Neill
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O'Neill, M.E. Exact solutions of the equations of slow viscous flow generated by the asymmetrical motion of two equal spheres. Appl. sci. Res. 21, 452–466 (1969). https://doi.org/10.1007/BF00411627

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  • DOI: https://doi.org/10.1007/BF00411627

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