A finite-difference scheme is used to solve the Navier-Stokes equations for the steady flow inside and outside viscous spheres in a fluid of different properties. Hence, the hydrodynamic force and the steady-state drag coefficient of the spheres are obtained. The Reynolds numbers of the computations range between 0.5 and 1000 and the viscosity ratio ranges between 0 (inviscid bubble) and infinity (solid particle). Unlike the numerical schemes previously implemented in similar studies (uniform grid in a stretched coordinate system) the present method introduces a two-layer concept for the computational domain outside the sphere. The first layer is a very thin one $[ORe−1/2]$ and is positioned at the interface of the sphere. The second layer is based on an exponential function and covers the rest of the domain. The need for such a double-layered domain arises from the observation that at intermediate and large Reynolds numbers a very thin boundary layer appears at the fluid-fluid interface. The computations yield the friction and the form drag of the sphere. It is found that with the present scheme, one is able to obtain results for the drag coefficient up to 1000 with relatively low computational power. It is also observed that both the Reynolds number and the viscosity ratio play a major role on the value of the hydrodynamic force and the drag coefficient. The results show that, if all other conditions are the same, there is a negligible effect of the density ratio on the drag coefficient of viscous spheres.

1.
Lovalenti
,
P. M.
, and
,
J. F.
,
1995
, “
Force on a Body in Response to an Abrupt Change in Velocity at Small but Finite Reynolds Number
,”
J. Fluid Mech.
,
293
, pp.
35
46
.
2.
Mei
,
R.
,
Lawrence
,
C. J.
, and
,
R. J.
,
1991
, “
Unsteady Drag on a Sphere at Finite Reynolds Number with Small Fluctuations in the Free-Stream Velocity
,”
J. Fluid Mech.
,
233
, pp.
613
631
.
3.
Mei
,
R.
, and
,
R. J.
,
1992
, “
Flow past a Sphere with an Oscillation in the Free-Stream and unsteady Drag at Finite Reynolds Number
,”
J. Fluid Mech.
,
237
, pp.
323
341
.
4.
Feng, Z.-G., 1996, “Heat Transfer from Small Particles at Low Reynolds Numbers,” Sc. D. Dissertation, Tulane Univ.
5.
Feng
,
Z. G.
, and
Michaelides
,
E. E.
,
1998
, “
Transient Heat Transfer from a Particle with Arbitrary Shape and Motion
,”
ASME J. Heat Transfer
,
120
, pp.
674
681
.
6.
Michaelides
,
E. E.
, and
Feng
,
Z.-G.
,
1995
, “
The Equation of Motion of a Small Viscous Sphere in an Unsteady Flow with Interface Slip
,”
Int. J. Multiphase Flow
,
21
, pp.
315
321
.
7.
Sirignano, W. A., 1999, Fluid Dynamics and Transport of Droplets and Sprays, Cambridge Univ. Press, Cambridge.
8.
Leal, L. G., 1992, Laminar Flow and Convective Transport Processes, Butterworth-Heineman, Boston.
9.
Kim, S., and Karrila, S. J., 1991, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heineman, Boston.
10.
Clift, R., Grace, J. R., and Weber, M. E., 1978, Bubbles, Drops and Particles, Academic Press, New York.
11.
Happel
,
J.
, and
Moore
,
D. W.
,
1968
, “
The motion of a spherical liquid drop at high Reynolds number
,”
J. Fluid Mech.
,
32
, part 2, pp.
367
391
.
12.
Le Clair
,
B. P.
, and
Hamielec
,
A. E.
,
1972
, “
A theoretical and experimental study of the internal circulation in water drops falling at terminal velocity in air
,”
J. Atmos. Sci.
,
29
, No.
2
, pp.
728
740
.
13.
Rivkind
,
V. Y.
,
Ryskin
,
G. M.
, and
Fishbein
,
G. A.
,
1976
, “
Flow around a spherical drop in a fluid medium at intermediate Reynolds numbers
,”
Appl. Math. Mech.
,
40
, pp.
687
691
.
14.
Oliver
,
D. L.
, and
Chung
,
J. N.
,
1987
, “
Flow about a fluid sphere at low to moderate Reynolds numbers
,”
J. Fluid Mech.
,
177
, pp.
1
18
.
15.
El-Shaarawi
,
M. A. I.
,
Al-Farayedhi
,
A.
, and
Antar
,
M. A.
,
1997
, “
Boundary layer flow about and inside a liquid sphere
,”
ASME J. Fluids Eng.
,
119
, pp.
42
49
.
16.
Briley
,
W. R.
,
1971
, “
A numerical study of laminar separation bubbles using the Navier-Stokes equations
,”
J. Fluid Mech.
,
47
, pp.
713
736
.
17.
Rivkind
,
V. Y.
, and
Ryskin
,
G. M.
,
1976
, “
Flow structure in motion of a spherical drop at intermediate Reynolds numbers,” Translated from Russian
,
Fluid Mechanics
,
11
, No.
1
, pp.
5
12
.
18.
Elzinga
,
E. R.
, Jr.
, and
Banchero
,
J. T.
,
1961
, “
Some Observations on the Mechanics of Drops in Liquid-Liquid Systems
,”
AIChE J.
,
7
, No.
3
, pp.
394
399
.
19.
Brabston
,
D. C.
, and
Keller
,
H. B.
,
1975
, “
Viscous flows past spherical gas bubbles
,”
J. Fluid Mech.
,
69
, No.
1
, pp.
179
189
.
20.
Winnikow
,
S.
, and
Chao
,
B. T.
,
1966
, “
Droplet Motion in Purified Systems
,”
Phys. Fluids
,
9
, pp.
50
61
.
21.
Harper
,
J. F.
,
1972
, “
The Motion of Bubbles and Drops through Liquids
,”