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The slow unsteady settling of two fluid spheres along their line of centres

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Abstract

The relative motion of two spherical drops along their line of centres is considered and a general solution for creeping flow presented. The exact solution agrees with solutions previously published to certain limiting conditions and expressions for the drag force for these cases determined. Asymptotic solutions for large particle separations are also presented which may be used to predict the fractional achievement of terminal velocity of the particles.

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Abbreviations

a :

characteristic diameter

a11, a12, a21, a22:

constants (see (50))

a n , b n , c n , d n :

constant of integration (see (8))

A n , B n , C n , D n :

constant of integration=K(n)(a n , b n , c n , d n )

A * n :

constant defined by (22)

c :

constant=r 1 sinh α=−r 2 sinh β

C 1/2n+1 (μ):

Gegenbauer polynomial of order (n+1) and degree −1/2 with argument μ

d :

equivalent spherical diameter

d L :

Laplace drop diameter

E:

differential operator (see (5))

E n (ξ):

function defined by (24)

Fα, Fβ :

forces on spheres ξ=α, ξ=β respectively

F n (ξ1, ξ2):

function defined by (24)

G n (ξ):

function defined by (24)

h :

characteristic distance in a direction parallel to flow

h1, h2:

distance of centres of spheres ξ=ξ 1=α and ξ=ξ 2=β from z=0

i :

subscript denoting fluid to which solution applies

K(n):

= (2n-1)(2n+3)/(2n(+1)✓2c 2)

n :

summation index

p :

pressure

P n (ξ), Q n (ξ):

functions defined by (24)

r :

fluid sphere radius

r1, r2:

radii of spheres ξ=ξ 1 and ξ=ξ 2 respectively

R, Z, φ :

cylindrical co-ordinates

S n (ξ):

function defined by (24)

t :

time

T n (ξ):

function defined by (24)

U :

characteristic velocity

U1, U2:

terminal velocities of spheres ξ=ξ 1 and ξ=ξ 2 respectively

U f :

terminal settling velocity for a fluid sphere

U n (ξ):

mathematical function (see (8))

U * n (ξ):

function defined by (19)

U s :

particle terminal settling velocity as predicted by Stokes' Law

v :

vector fluid velocity

\(\tilde V\) n (ξ):

=K(n) U n (ξ)

V n (ξ):

function defined by (24)

V * n (ξ):

=K(n) U n *(ξ)

v R , v z , vφ:

velocity vector components in cylindrical co-ordinates

v ξ , v η , vφ:

velocity vector components in bi-polar co-ordinates

X :

=(n+1/2)α

X n (ξ):

function defined by (24)

Y :

=(n+1/2)β

Y n (ξ), Z n (ξ):

functions defined by (24)

α :

=cosh−1 (h 1/r 1)

β :

=cosh−1 (h 2/r 2)

δ :

constant defined by (32)

Δ⋆, Δ ⋆1 , Δ ⋆2 , Δ ⋆33 :

functions defined by (24)

ε :

coefficient equation (25) equals +1 when U 1=U, and −1 when U 1=−U.

λ :

=λ±α, dimensionless unsteady reciprocal particle velocity =U s/U

λ n :

= (λ n )±α (see (27))

λ∞ :

dimensionless reciprocal terminal settling velocity for a fluid sphere =U s/U f

λ r :

dimensionless reciprocal relative particle velocity =U f/U

Γ :

=e−α

μ :

=cos η

μ i :

fluid viscosity of region i

μ⋆:

dimensionless viscosity group =(3μ 1+2μ 3)(3μ 2+2μ 3)/(4(μ 1+μ 3)(μ 2+ μ 3))

ξ, η, φ :

bi-polar co-ordinates

ξ1, ξ2:

=α, β, respectively

ρ :

density

ν :

kinematic viscosity

τξ, τηφ, τηφ :

components of shear stress

σ :

interfacial tension

ψ :

stream function

ψ n :

\(\Psi = \sum\limits_{n = 0}^\infty {\Psi n} \)

References

  1. Rybczynski, W., Bull. Acad. Sci. Cracovie Ser. A., (1911) 40

  2. Hadamard, J. S., C.r. hebd. Séanc Acad. Sci., 152 (1911) 1735 and 154 (1912) 109.

    MATH  Google Scholar 

  3. Taylor, T. D. and Acrivos, A., J. Fluid Mech., 18 (1964) 466.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Boussinesq, J., C.r. hebd. Séanc Acad. Sci., 156 (1913) 124.

    Google Scholar 

  5. Bart, E., Chem. Engng. Sci., 23 (1968) 193.

    Article  Google Scholar 

  6. Brenner, H., Chem. Engng. Sci., 16 (1961) 242.

    Article  Google Scholar 

  7. Maude, A. D., Br. J. Appl. Phys., 12 (1961) 293.

    Article  ADS  Google Scholar 

  8. Lorentz, H. A., Abh. Theoret. Phys., 1 (1907) 23.

    Google Scholar 

  9. Faxén, H. and Dahl, H., Ark. Mat. Astron. Fys., 19A (1925) 13.

    MATH  Google Scholar 

  10. Wakiya, S. J., Res. Rep. Fac. Engng. Niigata Univ., 9 (1960) 31.

    Google Scholar 

  11. Stimson, M. and Jeffery, G. B., Proc. Roy. Soc., A111 (1926) 110.

    ADS  Google Scholar 

  12. Pshenay-Severin, S. V., Bull. Acad. Sci. USSR Geophys. Ser. 10 (1958) 724. (English translation).

    Google Scholar 

  13. Mackay, G. D. M., Suzuki, M. and Mason, S. G., J. Colloid Sci., 18 (1963) 103.

    Article  Google Scholar 

  14. Steinberger, E. H., Pruppacher, H. R. and Nerbuiger, M., J. Fluid. Mech., 34 (1968) 809.

    Article  ADS  Google Scholar 

  15. Isaakyan, S. M. and Gasparyan, A. M., A. M. Int. Chem. Eng., 6 (1966) 74.

    Google Scholar 

  16. Cox, R. G. and Brenner, H., Chem. Engng. Sci., 22 (1967) 1753.

    Article  Google Scholar 

  17. Love, A. E. H., The Mathematical Theory of Elasticity, (1927).

  18. Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamics (Prentice-Hall, 1965).

  19. Rushton, E., Ph. D. Thesis, University of Manchester.

  20. Cooley, M. D. A. and O'Neill, M. E., Mathematika, 16 (1969) 37.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. of Chemical Engineering, University of Manchester, England

    E. Rushton & G. A. Davies

  2. Institute of Science and Technology, Manchester 1, England

    E. Rushton & G. A. Davies

Authors
  1. E. Rushton
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  2. G. A. Davies
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Rushton, E., Davies, G.A. The slow unsteady settling of two fluid spheres along their line of centres. Appl. Sci. Res. 28, 37–61 (1973). https://doi.org/10.1007/BF00413056

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  • DOI: https://doi.org/10.1007/BF00413056

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