J.
Fluid
Mech.
(1975),
vol.
69,
part
2,
pp.
305-337
Printed in
Great
Britain
305
The
slow
motion
of
slender rod-like particles
in
a
second-order fluid
By
L.
G.
LEAL
Chemical Engineering, California Institute
of
Technology, Pasadena
(Received 16 March 1974 and in revised form 10 June 1974)
The motion of
a
slender axisymmetric rod-like particle is investigated theo-
retically for translation through
a
quiescent second-order fluid and for rotation
in
a
simple shear flow of the same material. The analysis consists of an asymptotic
expansion about the limit of rheologically slow flow, coupled with an application
of a generalized form of the reciprocal theorem of Lorentz to calculate the force
and torque on the particle.
It
is shown that an arbitrarily oriented particle with
fore-aft symmetry translates, to a
first
approximation, at the same rate as in
an equivalent Newtonian fluid, but that the motion of particles with no fore-aft
symmetry may be modified at the same level of approximation. In addition,
it
is
found that freely translating particles with fore-aft symmetry exhibit
a
single
stable orientation with the axis of revolution vertical. In simple shear flow at
small and moderate shear rates, the non-Newtonian nature of the suspending
fluid causes a drift through Jeffery orbits to the equilibrium orbit
C
=
0
in which
the particle rotates about
its
axis of revolution. At larger shear rates, the particle
aligns itself in the direction of flow and ceases
to
rot@e. Comparison with the
available experimental data indicates that the measured rate of orbit drift may
be used to determine the second normal stress difference parameter of the second-
order fluid model. Finally,
in
an appendix, some preliminary observations are
reported of the motion of slender rod-like particles falling thmough
a
quiescent
viscoelastic fluid.
1.
Introduction
It
is becoming apparent that the motion of a submerged body
in
a
viscoelastic
ambient fluid is often fundamentally different from its motion in a Newtonian
fluid. Examples of such differences include the lateral migration
of
rigid particles
in nonlinear shear flow at very small Reynolds number (Karnis
&
Mason
1967),
the drift of rigid non-spherical particles towards preferred equilibrium orbits in
simple shear flow (Gauthier, Goldsmith
&
Mason
1971)
and the existence, at
larger shear rates, of an equilibrium orientation with
a
complete lack of rotation
for slender rod-like particles and very flat disks (Bartram
&
Mason
1974).
Very
few of these phenomena have been adequately investigated theoretically. Indeed,
the only external flow solutions at present available are for simple translation
and/or rotation of a rigid sphere under the simplifying approximation of rheo-
logically slow flow (Leslie
1961;
Caswell
&
Schwarz
1962;
Giesekus
1963),
and the
20
FLM 69
306
L.
Q.
Leal
corresponding uniform flow past a circular cylinder, which was recently obtained
by Mena
&
Caswell(l974, private communication). The key result of these latter
investigations is that both the drag and the torque exerted on the body occur at
second order with respect to the ratio
of
the relaxation time scale of the material
to the convective time scale of the motion, in spite of the fact that the velocity
and pressure fields are already altered in a non-trivial way at first order.
The present paper represents an initial study whose purpose is the generaliza-
tion of previous theoretical work to the case of straight slender axisymmetric
particles. The analysis is based on the approximation of rheologically slow flow
and therefore employs the Rivlin-Erickson nth-order fluid model. Two separate
cases are considered: translation (or sedimentation) through a quiescent ambient
fluid and the rotation of a neutrally buoyant particle in simple (linear) shear flow.
The chief practical interest in the sedimentation calculation
is
the possibility that
the non-Newtonian characteristics of the ambient fluid may lead to intrinsically
preferred equilibrium orientations. In addition, the analysis makes possible
a
partial assessment of the role
of
particle geometry in determining the presence
or
absence of first-order contributions to the hydrodynamic force and torque on
the particle. The case of particle rotation in shear flow, which was first discussed
(qualitatively) by Saffman (1956),
is
a logical first step in understanding the
rheological behaviour of a dilute viscoelastic suspension of rod-like particles
which is undergoing simple bulk shear flow.
The analysis is formally carried out as
a
perturbatidn expansion in the small
parameter for rheologically slow flow, which is the ratio of the natural relaxation
time of the ambient fluid to the convective time scale of the motion. At first order,
the problem reduces to the case of Newtonian creeping flow, and in order to main-
tain maximum flexibility with respect to the detailed geometry, the solution is
represented via the approximate slender-body theory
?or
low Reynolds number
flow (cf. Batchelor 1970; Cox
1970a).
At second order, corresponding to thesecond-
order fluid approximation, the velocity and pressure fields may also be calculated
using the slender-body approach. However, in the present work, we obtain only
the force and torque on the particle, using
a
generalized version of the reciprocal
theorem for low Reynolds number flow. As we shall demonstrate, these second-
order contributions to the force and torque can be calculated knowing only
the Newtonian velocity and pressure fields.
The approximation of nearly Newtonian slow flow severely limits the magni-
tude of the instantaneous, non-Newtonian contributions to the particle’s motion.
Nevertheless, these small effects may still have a large accumulative influence
on the particle orientation in sedimentation and on the orbit of rotation in shear
flow. In each instance, the orientation in the Newtonian case
is
fully determined
by the orientation of the particle at some initial time; no intrinsic preference is
shown for any orientation
or
any orbit. In these circumstances, small non-
Newtonian contributions can have a profound influence, ultimately causing the
particle to attain a steady-state sedimentation orientation
or
a steady-state shear
orbit which is completely independent of the initial state.
In
3
2,
we define and set up the general problem in terms suitable for solution
by the slow-flow perturbation expansion. This is followed in
3
3
by a brief review of
Rod-like particles
in
second-order
Jluid
307
the basic Newtonian slender-body solutions for uniform translation and rotation
in simple shear flow. Section
4
is concerned with the general scheme for applica-
tion of the generalized reciprocal theorem to obtain the first (second-order fluid)
non-Newtonian corrections for the force and torque acting on the particle. The
remainder of the paper reports the application of the general formulae to calculate
the hydrodyriamie contributions to the force and torque acting on
a
rigid axi-
symmetric particle in the specific cases of translation with arbitrary orientation
through a quiescent fluid and rotation in a simple shear
flow.
In the latter case,
a detailed comparison is made with the recent experimental observations of
Mason and co-workers (Karnis
&
Mason
1967;
Gauthier
et al.
1971;
Bartram
&
Mason
1974).
Finally, in an appendix we report the results of some simple
observations of the translational and rotational motion
of
slender cylinders and
circular cones falling under the action of gravity through a quiescent viscoelastic
fluid.
2.
The
general
problem
We
consider
a
straight, slender, axisymmetric, rigid body of length
21
which
has a cross-sectional radius
R(x,),
with
x,
measured along the axis of revolution
from
-
1
to
+
1.
The radius
R
is assumed to vary continuously with xland to satisfy
the additional constraints
R(Z)
=
R(
-
1)
=
0
and
R(xl)/21
+
1
for
-
1
<
x,
<
1.
We
shall use Cartesian reference axes
(xl, x,, x3) $xed
in
the purticle
and denote the
undisturbed flow relative to these as
-
U(x,
t).
The time dependence of
U
will
be assumed
to
result in the present circumstances entirely from time-dependent
changes in particle orientation rather than time variations in the flow itself. The
suspending fluid is assumed to be incompressil$e and non-Newtonian, and
is
modelled as a Rivlin-Erickson second-order fluid in order to be consistent
with the basic assumption
of
rheologically slow flow. Hence,
in
dimensionless
terms, the bulk stress
is
given by
T
=
-PI
+
A1
+
h[(A1)2
+glA,]
+
O(h2),
(1)
with
A,
representing the nth-order Rivlin-Erickson tensor. The components of
A,
and
A,,
respectively, are
a:;)
ui,
j
+
uj,
i
a,..
- -
du&/dt
+
ukaz,
+
a&
uk,
+
a!;
uk,
i.
and
23
The dimensionless parameters
h
and
el
are defined as
Q3
U/pZ
and
Q,/Q3,
in which
,LA
is
the
zero-shear viscosity,
Q2
and
Q3
the zero-shear normal stress coefficients
and
U
a characteristic velocity scale based on the undisturbed velocity field.
The dimensionless parameter
h
is a measure of the intrinsic relaxation time for
the fluid relative to the dynamic scale
l/U.
In the present
work
we consider
h
to
be small
so
that the constitutive relationship
(1)
differs only slightly from that
for a Newtonian fluid.
It
is in this sense that we call the bulk motion rheologically
slow. In addition, we assume that the fluid motion
is
also dynamically slow
so
that inertial effects may be neglected. More precisely, we assume that
Re
<
h
+
1
20-2
308
L.
G.
Leal
so
that the creeping-motion velocity and pressure fields are primarily modified
by nonlinear effects associated with non-zero values of
h
rather than dynamic
inertia effects associated with non-zero values of the Reynolds number. Hence
we seek solutions for
u
and
p
in the asymptotic form
as
h+O.
u
=
uo+hu,+
...
p
=
po+hpl+
...
Substituting these expansions into the equations of motion and continuity,
neglecting the inertia terms and equating like powers of
h
in the remaining terms,
we obtain the governing equations
at
O(1)
and
at
O(A):
v2uo-vpo
=
0,
v.uo
=
0,
(3)
V2~1-Vp1
=
-V.[(AJ2+~1A2]u=ao,
V.U,
=
0.
(4)
The symbol
[
in
(4)
signifies that the included expression is to be evaluated
using the zeroth-order velocity field
uo.
It
is
convenient to express
uo
as the sum
of the undisturbed velocity field
-
U(x,
t)
and a disturbance field
+
vCo)(x,
t).
Provided that
U(x,
t)
satisfies
(3),
as we shall assume, the Newtonian contribution
to the disturbance velocity field is obtained as the solution of the problem
with boundary conditions
(5)
VZV(0)
-
vp
=
0,
v
.
$0)
=
0
0
as
r-foo,
+
U(x,
t)
for
x
on the surface
S
of the particle.
3.
The Newtonian solutions
The linear problem posed by
(5)
and
(6)
is solved most conveniently, for the
present purposes, using the standard methods of slender-body theory for low
Reynolds number
flows.
In this approach, an approximation to the disturbance
flow
is
obtained by modelling the effect of the
actual
particle by
a
distribution
of
force, source, force-dipole and higher-order singularities spread along the
particle’s axis of revolution. When the undisturbed velocity field evaluated along
this axis is non-zero, as we shall assume, the disturbance velocity field
is
domin-
ated by the force (Stokeslet) singularities. The unknown distribution
of
Stokeslet
strength is determined by the requirement that
-
V(O)
+
U
be zero at the body
surface. The resulting (dimensionless) zeroth-order velocity field, in terms of
the Cartesian co-ordinates fixed in the particle, can be expressed as
in which
F(x‘,
t)
is
the Iine density
of
Stokeslet strength non-dimensionalized
with respect to
,MU,
X;
=
Silx’
and
1x1
=
{r2
+
(5,
-
x‘)~}*,
with
r
=
(x:
+
xi)&.
As
we have already noted, the dependence on
t
occurs because of the rotation of the
particle, and hence of the Cartesian co-ordinates which are fixed
in
the particle,
with respect to fixed laboratory co-ordinates in which the undisturbed velocity
field
is
assumed to be steady. The specific form of
Fi
depends both on the form of
Rod-like particles
in
second-order fluid 309
FIGURE
1.
A
typical axisymmetric particle sedimepting through
a
quiescent fluid.
the undisturbed field evaluated on the particle axis and on the detailed particle
geometry. However, Batchelor
(1970),
Cox
(19704,
Tillett
(1970)
and others
have shown that the Stokeslet strength distribution can be expressed as an
infinite series in increasing powers of (lne)-l:
=
(In
€)-Iflo)
+
(lne)-2f11)
+
.
. .
,
(8)
where
e
(
=
R,/21)
is the small parameter of slender-body analysis. Here
R,
is
a
length scale representative in some way of the values of
R
over the length of the
body. Higher-order singularities and further corrections to the Stokeslet distribu-
tion occur
at
O(e2).
The functions
fi(O),
fjl)
and
fi2’
have been evaluated by Cox
(1970~)
for a general linear undisturbed velocity distribution. Most significant
is
the result, also given by Batchelor and others, that the dominant term in the
Stolteslet expansion
is
independent
of the detailed particle geometry. Results of
considerable generality with respect to particle geometry are therefore possible.
When
-
U
is
a
vector of uniform direction and constant magnitude along the
particle’s axis of revolution, it is easily shown (cf. Batchelor
1970)
that
fp
=
274,
fp’
=
ad4
(i
=
2,3). (9)
This
is
the case relevant to the sedimentation of an arbitrarily oriented particle
without rotation at
a
velocity
U*
through an otherwise quiescent fluid as sketched