A model of turbulent cylindrical particle suspensions is proposed to predict the orientation distribution of particles. The fluctuating equation for the orientation distribution function (ODF) of cylindrical particles is theoretically solved using the method of characteristics. The orientation-correlated terms in the mean equation for the ODF due to the random motion of cylindrical particles are related to the correlations of the mean ODF and the fluid velocity gradient. Thus, the evolution of the mean ODF is described by a modified convection-dispersion equation. The model and modified equation are used to calculate the ODF in a pipe flow numerically. The results compare qualitatively with the experimental data and show that the turbulent dispersion makes cylindrical particles have a broad orientation distribution, while the velocity gradient plays an opposite role. The increase of the particle aspect ratio leads to a less aligned distribution in the vicinity of the axis and a narrower orientation distribution at positions far from the axis.

Topics
Velocity gradient tensor, Euler angles, Liquid crystals, Optical fibers, Fluid flows, Laminar flows, Turbulent flows, Vortex dynamics, Stochastic processes, Brownian motion
1.
E. S.G.
 
Shaqfeh
 and
G. H.
 
Fredrickson
, “
The hydrodynamic stress in a suspension of rods
,”
Phys. Fluids A
https://doi.org/10.1063/1.857683
 
2
,
7
(
1990
).
Google Scholar
Crossref
Search ADS
 
2.
M. D.
 
Rahnama
,
L.
 
Koch
, and
E. S.G.
 
Shaqfeh
, “
The effect of hydrodynamic interactions on the orientation distribution in a fiber suspension subject to simple shear flow
,”
Phys. Fluids
https://doi.org/10.1063/1.868647
 
7
,
487
(
1995
).
Google Scholar
Crossref
Search ADS
 
3.
L. G.
 
Leal
 and
E. J.
 
Hinch
, “
The effect of weak Brownian rotation on particles in shear flow
,”
J. Fluid Mech.
 
46
,
685
(
1972
).
Google Scholar
Crossref
Search ADS
 
4.
E. J.
 
Hinch
 and
L. G.
 
Leal
, “
Time-dependent shear flows of a suspension of particles with weak Brownian rotations
,”
J. Fluid Mech.
 
57
,
753
(
1973
).
Google Scholar
Crossref
Search ADS
 
5.
G. B.
 
Jeffery
, “
The motion of ellipsoidal particles immersed in a viscous fluid
,”
Proc. R. Soc. London, Ser. A
 
102
,
161
(
1922
).
Google Scholar
 
6.
E.
 
Anczurowski
 and
S. G.
 
Mason
, “
The kinetics of flowing dispersions: III Equilibrium orientations of rods and discs (experimental)
,”
J. Colloid Interface Sci.
https://doi.org/10.1016/0021-9797(67)90200-7
 
23
,
533
(
1967
).
Google Scholar
Crossref
Search ADS
 
7.
E. M.
 
Krushkal
 and
I.
 
Gallily
, “
On the orientation distribution function of non-spherical aerosol particles in a general shear flow-I. the laminar case
,”
J. Colloid Interface Sci.
 
99
,
141
(
1984
).
Google Scholar
Crossref
Search ADS
 
8.
J. A.
 
Olson
, “
The motion of fibres in turbulent flow, stochastic simulation of isotropic homogeneous turbulence
,”
Int. J. Multiphase Flow
 
27
,
2083
(
2001
).
Google Scholar
Crossref
Search ADS
 
9.
J. A.
 
Olson
,
I.
 
Frigaard
,
C.
 
Chan
, and
J. P.
 
Hamalainen
, “
Modeling a turbulent fibre suspension flowing in a planar contraction: the one-dimensional headbox
,”
Int. J. Multiphase Flow
 
30
,
51
(
2004
).
Google Scholar
Crossref
Search ADS
 
10.
J. Z.
 
Lin
,
W. F.
 
Zhang
, and
Z. S.
 
Yu
, “
Numerical research on the orientation distribution of fibers immersed in laminar and turbulent pipe flows
,”
J. Aerosol Sci.
 
35
,
63
(
2004
).
Google Scholar
Crossref
Search ADS
 
11.
E. M.
 
Krushkal
 and
I.
 
Gallily
, “
On the orientation distribution function of non-spherical aerosol particles in a general shear flow-II. The turbulent case
,”
J. Aerosol Sci.
 
19
,
197
(
1988
).
Google Scholar
Crossref
Search ADS
 
12.
J. A.
 
Olson
 and
R. J.
 
Kerekes
, “
The motion of fibres in turbulent flow
,”
J. Fluid Mech.
 
377
,
47
(
1988
).
Google Scholar
Crossref
Search ADS
 
13.
J.
 
Feng
 and
L. G.
 
Leal
, “
Pressure-driven channel flows of a model liquid-crystalline polymer
,”
Phys. Fluids
https://doi.org/10.1063/1.870141
 
11
,
2821
(
2000
).
Google Scholar
Crossref
Search ADS
 
14.
M. K.
 
Lyon
,
D. W.
 
Mead
,
R. E.
 
Elliott
, and
L. G.
 
Leal
, “
Structure formation in moderately concentrated viscoelastic suspensions in simple shear flow
,”
J. Rheol.
https://doi.org/10.1122/1.1381008
 
45
,
881
(
2001
).
Google Scholar
Crossref
Search ADS
 
15.
G.
 
Sgalari
,
L. G.
 
Leal
, and
E.
 
Meiburg
 “
Texture evolution of sheared liquid crystalline polymers: Numerical predictions of roll-cells instability, director turbulence, and striped texture with a molecular model
,”
J. Rheol.
https://doi.org/10.1122/1.1621420
 
47
,
1417
(
2003
).
Google Scholar
Crossref
Search ADS
 
16.
O.
 
Bernstein
 and
M.
 
Shapiro
, “
Direct determination of the orientation distribution function of cylindrical particles immersed in laminar and turbulent shear flows
,”
J. Aerosol Sci.
 
25
,
113
(
1994
).
Google Scholar
Crossref
Search ADS
 
17.
G. K.
 
Batchelor
, “
The stress system in a suspension of force-free particles
,”
J. Fluid Mech.
 
41
,
545
(
1970
).
Google Scholar
Crossref
Search ADS
 
18.
S. M.
 
Dinh
 and
R. C.
 
Armstrong
, “
A rheological equation of state for semiconcentrated fiber suspensions
,”
J. Rheol.
https://doi.org/10.1122/1.549748
 
28
,
207
(
1984
).
Google Scholar
Crossref
Search ADS
 
19.
S. G.
 
Advani
 and
C. L
 
Tucker
, “
The use of tensors to describe and predict fiber orientation in short fiber composites
,”
J. Rheol.
https://doi.org/10.1122/1.549945
 
31
,
751
(
1987
).
Google Scholar
Crossref
Search ADS
 
20.
J. C.H.
 
Fung
,
J. C.R.
 
Hunt
,
N. A.
 
Malik
, and
R. J.
 
Perkins
, “
Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes
,”
J. Fluid Mech.
 
236
,
281
(
1992
).
Google Scholar
Crossref
Search ADS
 
21.
L. P.
 
Wang
 and
D. E.
 
Stock
, “
Numerical simulation of heavy particle dispersion-scale ratio and flow decay considerations
,”
ASME J. Fluids Eng.
 
116
,
154
(
1994
).
Google Scholar
Crossref
Search ADS
 
© 2005 American Institute of Physics.
2005
American Institute of Physics
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