Abstract
A theoretical treatment is presented for the determination of the drag upon a sphere settling along the axis of a long square duct under the condition that the creeping motion equations are applicable. In order to obtain the second reflection velocity field, it was necessary to develop a new general solution in cartesian coordinates to the creeping motion equation, applicable within the domain of a long square duct. Using the second reflection velocity field solution an Faxen's law, a third reflection (first correction to the Stokes' value) drag correction is obtained.
The results show that the drag correction for a square container is quite close to (but smaller than) the drag correction produced by a cylinder whose diameter is the same as the duct width.
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Abbreviations
- a :
-
sphere radius
- F :
-
drag force vector
- F (0), F (1), ..., F (n) :
-
drag force vectors resulting from the superscript numbered reflection
- F z :
-
z component of the drag force vector
- g :
-
local acceleration of gravity
- i z :
-
unit vector along z-axis
- k :
-
a universal constant which is required to obtain the first correction for the container upon the Stokes drag
- l :
-
half-width of a square duct
- L :
-
length of a duct
- m, n :
-
integer indices
- p :
-
hydrodynamic pressure
- p (1) :
-
first reflection hydrodynamic pressure field
- Q :
-
volumetric flow rate
- R 0 :
-
radius of a cylinder
- t :
-
dimensionless parameter, equal to y/l
- U :
-
speed of a settling sphere
- U s :
-
Stokes' settling speed
- v :
-
fluid velocity vector
- v (0), v (1), ..., v (n) :
-
velocity field vectors arising from the superscript numbered reflections
- v (2)*:
-
the non-homogeneous portion of the second reflection velocity field
- v (2)**:
-
the homogeneous portion of the second reflection velocity field
- x, y, z :
-
cartesian directions
- β n :
-
the non-dimensionalized separation constant of index n, equal to α nl
- λ :
-
separation constant in Laplace's equation which is continuously variable from 0 to ∞
- μ :
-
fluid viscosity
- ρ :
-
fluid density
- ρ s :
-
sphere density
- τ :
-
a dimensionless separation constant, equal to λl
- φ :
-
void fraction of solids in cubic array
- x, y, z :
-
x, y, and zcomponents of vectors
References
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Faxen, H., Arkiv. Mat. Astron. Fys. 17 (1923) No. 27; dissertation, Uppsala University 1921.
Haberman, W. L. and R. M. Sayre, David Taylor Model Basin Reports Nos. 1143 (1958), 1563 (1961), 1578 (1961), Washington, D.C., U.S. Navy Dept.
Bohlin, T., Trans. Roy. Inst. Technol. (Stockholm) 155 (1960).
Happel, J. and P. A. Ast, Chem. Engng. Sci. 11 (1960) 286.
Happel, J. and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall Inc., Englewood Cliffs, 1965, out of print; 2nd revised ed., Noordhoff Publ. Co, Leijden.
Hasimoto, H., J. Fluid Mech. 5 (1959) 317.
Bart, E. N., Ph.D. dissertation, New York University 1971.
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Happel, J., Bart, E. The settling of a sphere along the axis of a long square duct at low Reynolds' number. Appl. Sci. Res. 29, 241–258 (1974). https://doi.org/10.1007/BF00384149
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DOI: https://doi.org/10.1007/BF00384149