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The motion of rigid ellipsoidal particles in slow flows
163
113 (1985) 163-183
Tecfonophysics,
Eisevier Science Publishers
THE MOTION
B.V., Amsterdam
in The Nethertands
- Printed
OF RIGID ELLIPSOIDAL
PARTICLES IN SLOW FLOWS
BRETTFREEMAN Department
(Received
of Geology.
February
University
Park, Nottingham,
NG7 2RD (United
8, 1984; revised version accepted
September
Kingdom)
*
24, 1984)
ABSTRACT Freeman,
B., 1985. The motion of rigid ellipsoidal
Preferred continuum usually
orientations
inappropriate
Jeffrey’s
model
axisymmetric numerically shapes,
if the fabric elements
is more
particles
one a prolate
applicable and simple
spheroid,
which
axisymmetry
compared
elements
in tectonites
in slow flows. Teczon~phy~jc~, 113: 163-183. are usually
interpreted
with reference
those of March (1932) and Jeffrey (1922). It is argued
in order to investigate
flow geometries particle
of fabric
models, namety
particles
involve or from
with that predicted
are rigid inhomogeneities
but until
recently
flow geometries. the behaviour
the other a triaxiaf pure or simple simple
in a deformable
its use has been limited
to problems
In this paper
Jeffrey’s
equations
of more general
systems.
The motions
ellipsoid,
are considered
shear and combinations
flow geometries
by spheroid/simple
embedded
in parallel of both.
have a significant
effect
to two
that the first of these is matrix. involving
have been solved of two particle for four specified
Any departures on particle
from
behaviour
flow solutions.
INTRODUCTION Orientation distributions of planar and linear fabric elements in deformed rocks, such as micas in slates (Oertel, 1970; Tullis, 1976), prismatic crystals in igneous rocks (Bhattacha~a, 1966) and elongate pebbles in glacial tills (Glen et al., 1957; Allen, 1982, p. 199) are often invoked as indicators of the magnitude and geometry of the bulk deformation. The theoretical basis for this is found in the contrasting works of March (1932) and Jeffrey (1922). In the March model it is assumed that fabric elements behave as passive material markers, i.e. they rotate with angular velocities equal to those of lines and planes of equivalent position in a homogeneous and isotropic deforming medium. This is usually an inappropriate treatment for problems involving rigid i~omogeneities (such as porphyroblasts in a rnet~o~~c
* Present
address:
Department
of Geology,
University
of Newcastle
NE1 7RU (U.K.).
0040-1951/85/$03.30
0 1985 Elsevier Science Publishers
B.V.
upon Tyne, Newcastle
upon Tyne,.
164
tectonite) since their rates of rotation are controlled through a tensor of third rank (Bretherton, 1962) which is entirely dependent on particle shape. However. symmetry arguments three
show that if the particle
non-zero
components,
lengths (Bretherton,
and
is ellipsoidal,
these are easily
then the shape tensor has only evaluated
given
the semi-axial
1962). Jeffrey (1922) derived the general equations
a rigid particle of neutral
buoyancy
are steady at large distances spheroidal
particles
encounter
perfectly
of motion
for
isolated in a slow viscous flow whose streamlines
from the particle.
in simple
shear
ellipsoidal
particles
flows. but
He also provided Of course from
explicit solutions
it is rare
the work
that
of Eirich
for
geologists and
Mark
(1937) it seems that the effects of minor surface imperfections on particle motions may be neglected. This is borne out experimentally even in extreme cases as Bartok and Mason lipsoidal
(1957) have shown.
shapes see Ferguson
Anulytical
(For a further
discussion
on the effects of non-el-
( 1979).)
solutions of Jeffrey;; equutions
Provided
that the volume concentration
of particles
is sufficiently
small such that
the disturbed velocity field around one particle does not interfere with that around any other, then Jeffrey’s (1922) equations constitute an appropriate framework for modelling multi-particle behaviour. Within these constraints the existing limitations of the model are due chiefly to the complexity of the governing equations. Analytical solutions have been found only for axisymmetric particles in simple flows. The well known solution for simple shear is given by Jeffrey (1922, eqns. 48 and 49): tana=rtan[y/(r+
l/r)]
tan /? = C [ cos*a: + ( l/r*
(1) ) sin2a] I”
(2)
where (Yand p are the polar angles (azimuth with respect to the plane of undisturbed strain.
Equation
axis of a prolate
(1) is clearly periodic spheroid
elliptical orbits described particle orientation and
and plunge)
of the particle
long axis
flow, r is the axial ratio and y is the shear and together
(1) and (2) predict
will precess one of an infinite
that the long
family of closed, spherical
by the orbit constant C, which depends only on the initial axis ratio. This motion is quite distinct from that of a
passive material line (as in the March model): during the course of the rotation the particle always decelerates towards the shear plane, passes through the plane and accelerates away from it. In contrast the end of a passive material line remains in the same plane moving at constant velocity in the direction of shear. For flows involving no vorticity Jeffrey’s equations have been solved for pure shear (Gay, 1966, 1968). the azimuth of the long axis being given by: 1’2 1 (r- 1) ln 3 ln(cot aI) = ln(cot ai) + - ~ 2 (r+l) i h, 1
(3)
165
and the plunge: sin 201.
Cot
#If
=
COt
pi
+
l/2
( i &
f
where X, and X, are the two principal Subscripts presents
f and i refer to the final similar
spheroids. Although, spheroid
equations
as in simple
quadratic
for pure flattening shear,
for all coaxial velocities,
to note that the angular
irrotational
strains.
passive lines rotating
angles
strains
(the third being unity).
respectively.
Tullis
of orientation
(1976)
distributions
of
the material coordinates of the end of a prolate
differ from those of a corresponding
tion, it is interesting
extensions
and initial
The particle
passive linear
marker
after deforma-
paths swept out by both are identical motions
differ only in their angular
faster than any spheroid.
Experimental verification of Jeffrey’s model Several workers have attempted to assess the validity of Jeffrey’s model by investigating the correspondence between the theoretically predicted behaviour and observations obtained experimenta~y for simpie shear systems. Trevelyan and Mason (1951), using a Couette apparatus, found an excellent agreement between the calculated and observed rates of rotation of spheres, and verified the nature of the orbits for prolate cylinders, though the actual periods are always less than the predicted
values for spheroids.
This may not be too important
geologically
since the
shear strains necessary to complete an orbit are very large compared to anticipated shear strains in rocks. More restricted shear box type experiments also show a fair correspondence between observed and predicted behaviour for geologically realistic strains
(Ghosh
and Ramberg,
1976, fig. 5), but all observations
and initial
orienta-
tions are restricted to the plane perpendicular to the vorticity, i.e. they are all C = co orbits. A further consideration, with regard to the interpretation of tectonites, comes from the notion that the matrix rheology is unlikely to be Newtonian; (1979) has addressed this problem in some detail and concludes that should give acceptable approximations strains are not too high.
even
for power
law fluids
Ferguson the model
provided
that
Previous applications of Jeffrey ‘f model Until
recently
the effects of strain
on multiparticulate
systems
have been mod-
elled using the analytical solutions to Jeffrey’s (1922) equations (e.g., Reed and Tryggvason, 1974; Tullis, 1976; Harvey and Ferguson, 1978). There are two major restrictions here. Firstly discussion must be limited to plane strains of either pure or simple shear although, clearly, many geological deformations cannot be approximated by such a simple treatment. For example, Sanderson’s (1982) differential. transport model for strain variations in thrust sheets combines pure and simple shear
in its interpretation directions. Similarly development required
of sidewall ramps and steep zones parallel to nappe transport interplay between pure and simple shear is important during the
of folds, as indicated
to be axisymmetric
particular,
it precludes
ellipsoidal
(or approximately
multiparticle Recent
which
greatly
any meaningful ellipsoidal)
particles
attention
work (Gierszewski to the rather
flow (Harris
general,
the geological
the particles
are
applications.
In
of fabrics composed
of general
in the light of hitherto
and Chaffey,
more complex
simple shear flows. Both studies use numerical
investigate
limits
interpretation
and the results have a fair correspondence Couette
(1975). Secondly
published
models. theoretical
has brought
by Ramberg
solution
of isolated
interesting,
and Leal, 1979)
of triaxial
particles
in
of Jeffrey’s (1922) equations,
to the experimental
et al., 1979). I have used a similar
the behaviour
geologically
1978; Hinch
behaviour
results obtained numerical
axi- and non-axisymmetric
from
approach
particles
to
in some
flows.
THEORY
Consider a set of right handed orthogonal Cartesian axes X,’ fixed in orientation but able to move so that the origin of the basis is always coincident with the centre of gravity of an ellipsoidal particle which is suspended in a slowly deforming fluid. A second coordinate system X, is also centred at the origin of X,’ and is instantaneously coincident
with the principal
is given at any instant
axes of the particle.
by the rotation
cos 4 cos 9 - cos e sin 9 sin 4 , -sin$cos$-cosesin$cos#, sin e sin 9
The relationship
between
X, and X,’
matrix: cos I/ sin $ + cos e cos $3sin I$ , sin 4 sin e -sinJisincp+cosecosqbcosrC,, - sin e cos +
cos+sinB cos e (5)
so that X, = R,,X,‘. 8, + and + are the three euler angles and are the minimum number of parameters needed to describe the orientation of the particle. They are defined from three successive
rotations
(Fig. l), R,, being the matrix
for each of the three operations. about Xi, X, then X, following
product
of the rotation
matrices
In this work the rotations are made in sequence of Goldstein (1980, p. 147) the “x ” convention
(though they are somewhat arbitrary). At large distances from the particle the undisturbed velocity gradients tensor (spatial description):
flow is specified
by the
(6)
167
which satisfies the constant
volume condition:
( Lri, i =j are the rates of natural The rate-of-deformation .coordinate
strain and L:j, i #j,
and vorticity
system are the symmetric
are the shear strain rates.)
tensors of the undisturbed
and antisymmetric
flow in the fixed
parts respectively
of L:,:
(84
W (for example, see Malvern, 1969). To find the rate of deformation
and vorticity
tensors
with respect
to the rotating
Line of. nodes
Fig. 1. Eufer angle definitions. A’: refer to the fried axes and X,
to tile rotating particle axes.
X, coordinates
we employ
the usual rules for transformation
of tensors:
E!, = R<,R,&,
(9a)
and: Q,, = R,,R,&,
(9b)
Now, if the semi axial lengths of the particle Bretherton’s
are ai the only non-zero
components
of
(1962) shape tensor are:
Jeffrey’s (1922) equations
where w, are the angular
of motion
velocities
angles change with velocities those terms (Goldstein,
(p. 169, eqn. 37) are now easily rewritten
of the particle
as:
about its own axes X,. If the euler
b,& 4, then it is straightforward
to formulate
the w, in
1980, p, 176):
w1 =BsinBsin\I,+~cosIC,
/
w2 = $2sin t? cos + - S sin $.J
j
W) = 4 cos 8 + $L
I
which after rearrangement
gives three differential
(121
equations:
B = w,cos +LJ - w,sin 4 ~=(w,sin++o,cos~)/sinB
(13)
\t = w3 - (b cos 8 These are solved numerically as an initial value problem. Throughout the boundary conditions are set in terms of L,‘,, though it is obviously appreciate
the actual strains involved.
deformation
gradients
Geologists
have tended
towards
the use of the
tensor:
dx, F,:=dX; for the computation of deformations quadratic stretch tensor: D,; = F:, F;p
this analysis necessary to
04) (e.g., Flinn,
1978; Sanderson,
1982) and the (15)
for the calculation of finite strains (e.g., Sanderson, 1982; De Paor, 1983). The same convention is used here, but because we are dealing with velocities, t;lIt is time dependent with a rate of change: c; = Lip FiJ
(16)
169
(Malvern, 1969, p. 163). Equation (16) are thus nine differential equations which can be solved over an interval, t - rO, to give ‘;;;. All systems of differential equations used here have been solved to approximately eight significant figures accuracy using a variable-order, variable-step Adams method (Numerical Algorithm Group routine W02CBF). ISOLATED
PARTICLES
IN SLOW FLOC’S
In this study two particle shapes have been considered, a prolate spheroid of axis ratio 3 : 1: 1 and an ellipsoid of axes 3 : 1”1 : f . They have been chosen specifically to have the same major axis length and identical volumes so their motions can be sensibly compared for equivalent boundary conditions. For most of the following discussion we consider the motion from nine starting orientations. These are represented as the plunge of the long axis (X,) on a stereographic projection and the subsequent motion is depicted by a smooth curve with dots at ten time unit intervals. To calculate the finite strain between any number of dots, n, we simply evaluate the quadratic stretch tensor:
The quadratic extensions are then the eigenvalues of O/j and the eigenvectors give their directions in space.
Here we use the velocity gradients: LI, =
0 0.05 [0
0 0 0
0 0 0
1
The amount of shear per ten time units is L;,= y = 0.5and ej is simply:
As noted previously the motion of an axisymmet~c particle is periodic with the ends of the spheroid describing closed spherical elliptical orbits about the pole to the plane of the undisturbed flow, i.e. the vorticity. The period is constant in time if y is constant, but is always proportional to y so that for one half rotation: y = Iz(T + l/r)
(18)
Further it appears that the particle will precess the same orbit for all time depending on C, the orbit constant, which is determined by the initial orientation, and is equal to tan a for a when /3 = 0. These are referred to as Jeffrey orbits.
170
Any departure from axisymmetry has profound effects on the nature of the orbit compared to that of a spheroid starting from the same long axis orientation (see also Gierszewski
and Chaffey,
1978; Hinch
results for nine initial orientations prolate
particle.
the periods secondary
It appears
and shows the corresponding
that the motion
are not constant drift through
and Leal. 19’79). Figure
in time.
families
is still periodic
Imposed
of Jeffrey orbits.
2 summarizes
the
Jeffrey orbits for the about
on the primary
the vorticity.
but
periodicity
is a
This is a fundamental
difference
between axi- and non-axisymmetric particle motions; the latter can drift through the plane of the undisturbed flow. If the time spent in the flow is large then the motion reveals itself to be doubly periodic; drift through the Jeffrey orbits eventually returns the particle to its original position. To assess the general behaviour in simple shear we first consider the degenerate cases for which the rotations are singly periodic. Dealing only with the long axis we can identify four initial orientations for which the corresponding c‘ = 03, and of these four tjnere are two distinct periods: 8=90,
+=90,0.
Jeffrey
orbit
is
l&=0,0
and: e = 90,
Q,= 90.0
li, = 90,90
(there are of course symmetrically equivalent positions outside the range 8, (p, + = 0 - 90). Secondly there are an infinite set of orientations for which the corresponding orbit constant is c’ = 0, and which have identical periods: B=O,
$X=0-90,
+=o-90
Now it is possible to consider the general behaviour in terms of the relationship of the initial orientation to that of one of the degenerate cases. Figure 2 shows that for the first few increments of shear (up to about y = 2) the orbits of the two particle geometries are qualitatively similar. As initial orientations approach degeneracy the similarity between the two orbits persists for greater shear magnitudes. Conversely as the symmetry of the position with respect to the plane of the undisturbed flow decreases,
the orbits
become
less like the equivalent
closed Jeffrey
orbit.
We note,
however, that for ail non-degenerate boundary conditions orbital drift is significant by shears of y = 4. Thus approximations for triaxial particle motions based on prolate
particle
theory are always likely to be misleading.
(2) Pure shear and other coaxial strains The equations of Gay (1966, 1968) (eqns. (2) and (3) here) indicate an important distinction between coaxial irrotational pure shear and flattening and simple shear for axisymmetric particles in that the former are non-periodic. Indeed this can easily be shown to be true for all coaxial strains involving any ellipsoidal particle geometry. For coaxial strains the only non-zero components of the velocity gradients tensor are
L
,
-
172
the diagonals,
i.e. L:, f 0, i =.j. Hence
Now, as the particle
H,, +
1 0 L0
0 1 0
axes approach
there is no vorticity
parallelism
and L?,, = 0 (eqn. 9b).
with the fixed axes, ,k; then:
0 0 11
and since E,‘, = 0, i f j (eqn. 9a), then E,, + 0 and w, become asymptotically smaller. is in a position of stable equilibrium and will remain fixed
When o, = 6 the particle
in position unless the applied flow is modified. For axisymmetric particles the components further
to B, = 0 and B, = -B,.
Therefore
of Bretherton’s
spin about
shape tensor
reduce
the long axis is zero and o2
and w3 are proportional to E3, and E,, respectively and are scaled by constants of equal modulii. Their velocities, which are clearly dependent on the axis ratio of the particle,
become maximum
as B, = -B,
-+ 1 and at the limit ~,/a,
= m. w2 = -E,,
and w3 = E,, which are the angular velocities of a passive material line. Therefore the axisymmetric particle motion for oblate and prolate spheroids follows trajectories which are the same as those predicted
by the March
and lines respectively. Triaxiality does not have such a drastic
model for passive
effect on the motion
as in simple shear
but we should note that all the w, are non-zero so the ellipsoid motion somewhere between the two extremes of behaviour for spheroids. (3) Simultaneous All the
flows
pure and simple shear (L;, = - L;,, described
always lies
L;, + 0)
in this section are geometrically
perpendicular to the vorticity velocity components producing
planes
similar
in that the plane
has a pure shear superimposed on it. However, the the pure shear are varied over an order of magnitude
so the ratio L;,/L;, ranges from 0.1 to 1.0, and the velocity producing simple shear is held constant. Particle paths for this type of flow are given by Ramberg (1975, fig. 3). We note that cross sections to the vorticity, deformation
so the resulting
The behaviour
through
are identical
and
such a flow, made in the plane perpendicular that
their
finite strain ellipsoid
of axisymmetric
particles
area
remains
constant
is k = 1 (Flinn,
throughout
1978).
is shown in Fig. 3a. Drift through
the
Jeffrey orbits occurs even for small amounts of pure shear and as this component is increased the periodic motion becomes subordinate to the asymptotic behaviour described in the previous section. When the velocities for pure and simple shear are equal the particle rotates from any starting position towards x’. This is an important observation because it implies that any multiparticulate fabric will tend to be prolate even though the finite strain ellipsoid is plane. Qualitatively similar results apply to the triaxial particles in as much as the gross departures from the double periods become more profound as the ratio of pure to simple shear velocities approach unity (Fig. 3). More subtle modifications occur in accordance with the symmetry of the initial orientation; in low symmetry orientations, and particularly when 6 is small, depar-
See text for discussion.
b, c, d. Paths of an ellipsoid
Fig. 3. a. Paths of a spheroid
(A) e=60
I 1
0
.05
.005
0
0
in simultaneous
in simultaneous
pure and simple shear for three initial orientations
with Fig. 2.
0
.607
1 1.
0
about
the long axis.
1 (a> 0
1.
0
0
1.
0
0
of the long axis and three initial rotations
Compare
0
0
points.
0
0
-.05
.05
pure and simple shear for two starting
a
,521
0
I.646 0
0
,905
0
.501
0
I[
0
0
0
,500
1.105
,951
0
0
-.Ol
0
0
1.051
.05
I
0
.05
.o 1
xj
-.c105
L=(p)I
:rl-
30
. .
‘..
\
-\
175
-A 0
0
O
zI’ 0
;
0
0
0
r
00,:
0
0
0
0
I’
v -
;:
I’
0
176
, 0
1”
0
0
G
0
“,
0
c ,----l -
u+j -I
II
LL
(D
0 -
c
0
0
0
:: 0
.-
::
O
0
0
0
0
0
0
0
0
0
0
0
0
;
0
I’
---.
/i/‘ i \
‘.
” / p/ I’
\
111
I
1
I
1 0
0
0
;; ?
$
O
0
0
Q
2 cn O
;
1 0
0
0
-:
0
0
0
;
0
0
q
0
h
1
0
0
0
Q
6
0
cl
_
178
179
r
t
r
0
0
r
II
0
0
G
0
0
0
I
0
0
0
0
0
0
0
:
0
0
0
0
- ._ .-
a3
-c
I’
-
180
181
tures from the simple shear case become less predictable, cusps appear in the particle trajectory and the time spent in orientations close to the Xi/X; plane becomes relatively large. (4) Simple shear on twoperpendicularplunes with simultaneous pure shear (L;, = - L;,, L;, = L;, f 0)
This is essentially a more general version of the previous flow. The two simple shear components are equivalent to a single shear direction on the plane whose pole is the vector [co545 co545 01, thus the pure shear defo~ation plane is at 45’ to the simple shear plane, and the vorticity of the flow is along the vector [ -cos45 cos45 01. Without the pure shear component we would expect the particle to precess about the vorticity as in previous examples. This is the case, but even when Z&i,is small, drift through the Jeffrey orbits is spectacular (Fig. 4a) for initial orientations which are close to the vorticity. However, if the long axis lies close to the plane perpendicular to the vorticity then the particle rotates in quasi-stable orbits with constants fluctuating about C = 00. Similar behaviour is observed for L;, = L;,/5 but when L;, = L;, all initial orientations eventually rotate to a position a few degrees away from Xi. Periodicity is retained for all initial orientations if the particle is triaxial (at least up to L;, = L;,). The rate of secondary drift through the orbits (Fig. 4a, b, c) is, as usual, dependent on the ratio L&/L;,. When this ratio is large the particle motion is similar to the axisymmetric one, but when L;, = L;, a rapid movement towards the X;/X; plane is followed by singly periodic rotation in a slightly elliptical spherical orbit which crosses the plane perpendicular to the vorticity twice every period. SUMMARY
Numerical solution of Jeffrey’s (1922) equations governing the motion of a rigid ellipsoidal particle suspended in a creeping fluid facilitates the study of spheroidal and ellipsoidal particles in plane and non-plane strain flows, with or without vorticity. The treatment here covers some strain geometries which are geologically important, and deals with magnitudes of strains up to and beyond those anticipated in most geological situations. It is by no means exhaustive, but demonstrates that departures from axisymmetry and/or plane strain have important consequences for the nature of the particle motion. Of particular importance is the motion of particles in flows involving pure and simple shear. The suggestion is that multiparticle fabrics will become prolate even though the finite strain ellipsoid is plane. Computer simulation experiments have been carried out in order to investigate multiparticle behaviour in the flows already described here and will be published in a later paper,
182
ACKNOWLEDGEMENTS
I thank Cohn Ferguson for reading and improving the manuscript. carried out during the receipt of a NERC research studentship
This work was at Nottingham
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of a slaty, lapillar
tuff in the lake district,
England.
Geol. Sot. Am. Bull.,
81: 1173-1187. Ramberg.
H., 1975. Particle paths, displacement
and progressive
strain applicable
to rocks. Tectonophysics,
28: l-37. Reed, L.J. and Tryggvason,
E., 1974. Preferred
orientations
by Pure shear and simple shear. Tectonophysics, Sanderson,
D.J., 1982. Models in strain variation
of rigid particles
in a viscous matrix deformed
24: 85-98. in nappes
and thrust
sheets: a review. Tectonophysics,
88: 201-233. Trevelyan,
B.J. and Mason,
S.G., 1952, Particle
motions
in sheared
suspensions.
I. Rotations.
J. Colloid
Sci., 5: 345-367. Tullis, T.E., 1976. Experiments 745-753.
on the origin of slatey cleavage
and schistosity.
Geol. Sot. Am. Bull., 87:
113 (1985) 163-183
Tecfonophysics,
Eisevier Science Publishers
THE MOTION
B.V., Amsterdam
in The Nethertands
- Printed
OF RIGID ELLIPSOIDAL
PARTICLES IN SLOW FLOWS
BRETTFREEMAN Department
(Received
of Geology.
February
University
Park, Nottingham,
NG7 2RD (United
8, 1984; revised version accepted
September
Kingdom)
*
24, 1984)
ABSTRACT Freeman,
B., 1985. The motion of rigid ellipsoidal
Preferred continuum usually
orientations
inappropriate
Jeffrey’s
model
axisymmetric numerically shapes,
if the fabric elements
is more
particles
one a prolate
applicable and simple
spheroid,
which
axisymmetry
compared
elements
in tectonites
in slow flows. Teczon~phy~jc~, 113: 163-183. are usually
interpreted
with reference
those of March (1932) and Jeffrey (1922). It is argued
in order to investigate
flow geometries particle
of fabric
models, namety
particles
involve or from
with that predicted
are rigid inhomogeneities
but until
recently
flow geometries. the behaviour
the other a triaxiaf pure or simple simple
in a deformable
its use has been limited
to problems
In this paper
Jeffrey’s
equations
of more general
systems.
The motions
ellipsoid,
are considered
shear and combinations
flow geometries
by spheroid/simple
embedded
in parallel of both.
have a significant
effect
to two
that the first of these is matrix. involving
have been solved of two particle for four specified
Any departures on particle
from
behaviour
flow solutions.
INTRODUCTION Orientation distributions of planar and linear fabric elements in deformed rocks, such as micas in slates (Oertel, 1970; Tullis, 1976), prismatic crystals in igneous rocks (Bhattacha~a, 1966) and elongate pebbles in glacial tills (Glen et al., 1957; Allen, 1982, p. 199) are often invoked as indicators of the magnitude and geometry of the bulk deformation. The theoretical basis for this is found in the contrasting works of March (1932) and Jeffrey (1922). In the March model it is assumed that fabric elements behave as passive material markers, i.e. they rotate with angular velocities equal to those of lines and planes of equivalent position in a homogeneous and isotropic deforming medium. This is usually an inappropriate treatment for problems involving rigid i~omogeneities (such as porphyroblasts in a rnet~o~~c
* Present
address:
Department
of Geology,
University
of Newcastle
NE1 7RU (U.K.).
0040-1951/85/$03.30
0 1985 Elsevier Science Publishers
B.V.
upon Tyne, Newcastle
upon Tyne,.
164
tectonite) since their rates of rotation are controlled through a tensor of third rank (Bretherton, 1962) which is entirely dependent on particle shape. However. symmetry arguments three
show that if the particle
non-zero
components,
lengths (Bretherton,
and
is ellipsoidal,
these are easily
then the shape tensor has only evaluated
given
the semi-axial
1962). Jeffrey (1922) derived the general equations
a rigid particle of neutral
buoyancy
are steady at large distances spheroidal
particles
encounter
perfectly
of motion
for
isolated in a slow viscous flow whose streamlines
from the particle.
in simple
shear
ellipsoidal
particles
flows. but
He also provided Of course from
explicit solutions
it is rare
the work
that
of Eirich
for
geologists and
Mark
(1937) it seems that the effects of minor surface imperfections on particle motions may be neglected. This is borne out experimentally even in extreme cases as Bartok and Mason lipsoidal
(1957) have shown.
shapes see Ferguson
Anulytical
(For a further
discussion
on the effects of non-el-
( 1979).)
solutions of Jeffrey;; equutions
Provided
that the volume concentration
of particles
is sufficiently
small such that
the disturbed velocity field around one particle does not interfere with that around any other, then Jeffrey’s (1922) equations constitute an appropriate framework for modelling multi-particle behaviour. Within these constraints the existing limitations of the model are due chiefly to the complexity of the governing equations. Analytical solutions have been found only for axisymmetric particles in simple flows. The well known solution for simple shear is given by Jeffrey (1922, eqns. 48 and 49): tana=rtan[y/(r+
l/r)]
tan /? = C [ cos*a: + ( l/r*
(1) ) sin2a] I”
(2)
where (Yand p are the polar angles (azimuth with respect to the plane of undisturbed strain.
Equation
axis of a prolate
(1) is clearly periodic spheroid
elliptical orbits described particle orientation and
and plunge)
of the particle
long axis
flow, r is the axial ratio and y is the shear and together
(1) and (2) predict
will precess one of an infinite
that the long
family of closed, spherical
by the orbit constant C, which depends only on the initial axis ratio. This motion is quite distinct from that of a
passive material line (as in the March model): during the course of the rotation the particle always decelerates towards the shear plane, passes through the plane and accelerates away from it. In contrast the end of a passive material line remains in the same plane moving at constant velocity in the direction of shear. For flows involving no vorticity Jeffrey’s equations have been solved for pure shear (Gay, 1966, 1968). the azimuth of the long axis being given by: 1’2 1 (r- 1) ln 3 ln(cot aI) = ln(cot ai) + - ~ 2 (r+l) i h, 1
(3)
165
and the plunge: sin 201.
Cot
#If
=
COt
pi
+
l/2
( i &
f
where X, and X, are the two principal Subscripts presents
f and i refer to the final similar
spheroids. Although, spheroid
equations
as in simple
quadratic
for pure flattening shear,
for all coaxial velocities,
to note that the angular
irrotational
strains.
passive lines rotating
angles
strains
(the third being unity).
respectively.
Tullis
of orientation
(1976)
distributions
of
the material coordinates of the end of a prolate
differ from those of a corresponding
tion, it is interesting
extensions
and initial
The particle
passive linear
marker
after deforma-
paths swept out by both are identical motions
differ only in their angular
faster than any spheroid.
Experimental verification of Jeffrey’s model Several workers have attempted to assess the validity of Jeffrey’s model by investigating the correspondence between the theoretically predicted behaviour and observations obtained experimenta~y for simpie shear systems. Trevelyan and Mason (1951), using a Couette apparatus, found an excellent agreement between the calculated and observed rates of rotation of spheres, and verified the nature of the orbits for prolate cylinders, though the actual periods are always less than the predicted
values for spheroids.
This may not be too important
geologically
since the
shear strains necessary to complete an orbit are very large compared to anticipated shear strains in rocks. More restricted shear box type experiments also show a fair correspondence between observed and predicted behaviour for geologically realistic strains
(Ghosh
and Ramberg,
1976, fig. 5), but all observations
and initial
orienta-
tions are restricted to the plane perpendicular to the vorticity, i.e. they are all C = co orbits. A further consideration, with regard to the interpretation of tectonites, comes from the notion that the matrix rheology is unlikely to be Newtonian; (1979) has addressed this problem in some detail and concludes that should give acceptable approximations strains are not too high.
even
for power
law fluids
Ferguson the model
provided
that
Previous applications of Jeffrey ‘f model Until
recently
the effects of strain
on multiparticulate
systems
have been mod-
elled using the analytical solutions to Jeffrey’s (1922) equations (e.g., Reed and Tryggvason, 1974; Tullis, 1976; Harvey and Ferguson, 1978). There are two major restrictions here. Firstly discussion must be limited to plane strains of either pure or simple shear although, clearly, many geological deformations cannot be approximated by such a simple treatment. For example, Sanderson’s (1982) differential. transport model for strain variations in thrust sheets combines pure and simple shear
in its interpretation directions. Similarly development required
of sidewall ramps and steep zones parallel to nappe transport interplay between pure and simple shear is important during the
of folds, as indicated
to be axisymmetric
particular,
it precludes
ellipsoidal
(or approximately
multiparticle Recent
which
greatly
any meaningful ellipsoidal)
particles
attention
work (Gierszewski to the rather
flow (Harris
general,
the geological
the particles
are
applications.
In
of fabrics composed
of general
in the light of hitherto
and Chaffey,
more complex
simple shear flows. Both studies use numerical
investigate
limits
interpretation
and the results have a fair correspondence Couette
(1975). Secondly
published
models. theoretical
has brought
by Ramberg
solution
of isolated
interesting,
and Leal, 1979)
of triaxial
particles
in
of Jeffrey’s (1922) equations,
to the experimental
et al., 1979). I have used a similar
the behaviour
geologically
1978; Hinch
behaviour
results obtained numerical
axi- and non-axisymmetric
from
approach
particles
to
in some
flows.
THEORY
Consider a set of right handed orthogonal Cartesian axes X,’ fixed in orientation but able to move so that the origin of the basis is always coincident with the centre of gravity of an ellipsoidal particle which is suspended in a slowly deforming fluid. A second coordinate system X, is also centred at the origin of X,’ and is instantaneously coincident
with the principal
is given at any instant
axes of the particle.
by the rotation
cos 4 cos 9 - cos e sin 9 sin 4 , -sin$cos$-cosesin$cos#, sin e sin 9
The relationship
between
X, and X,’
matrix: cos I/ sin $ + cos e cos $3sin I$ , sin 4 sin e -sinJisincp+cosecosqbcosrC,, - sin e cos +
cos+sinB cos e (5)
so that X, = R,,X,‘. 8, + and + are the three euler angles and are the minimum number of parameters needed to describe the orientation of the particle. They are defined from three successive
rotations
(Fig. l), R,, being the matrix
for each of the three operations. about Xi, X, then X, following
product
of the rotation
matrices
In this work the rotations are made in sequence of Goldstein (1980, p. 147) the “x ” convention
(though they are somewhat arbitrary). At large distances from the particle the undisturbed velocity gradients tensor (spatial description):
flow is specified
by the
(6)
167
which satisfies the constant
volume condition:
( Lri, i =j are the rates of natural The rate-of-deformation .coordinate
strain and L:j, i #j,
and vorticity
system are the symmetric
are the shear strain rates.)
tensors of the undisturbed
and antisymmetric
flow in the fixed
parts respectively
of L:,:
(84
W (for example, see Malvern, 1969). To find the rate of deformation
and vorticity
tensors
with respect
to the rotating
Line of. nodes
Fig. 1. Eufer angle definitions. A’: refer to the fried axes and X,
to tile rotating particle axes.
X, coordinates
we employ
the usual rules for transformation
of tensors:
E!, = R<,R,&,
(9a)
and: Q,, = R,,R,&,
(9b)
Now, if the semi axial lengths of the particle Bretherton’s
are ai the only non-zero
components
of
(1962) shape tensor are:
Jeffrey’s (1922) equations
where w, are the angular
of motion
velocities
angles change with velocities those terms (Goldstein,
(p. 169, eqn. 37) are now easily rewritten
of the particle
as:
about its own axes X,. If the euler
b,& 4, then it is straightforward
to formulate
the w, in
1980, p, 176):
w1 =BsinBsin\I,+~cosIC,
/
w2 = $2sin t? cos + - S sin $.J
j
W) = 4 cos 8 + $L
I
which after rearrangement
gives three differential
(121
equations:
B = w,cos +LJ - w,sin 4 ~=(w,sin++o,cos~)/sinB
(13)
\t = w3 - (b cos 8 These are solved numerically as an initial value problem. Throughout the boundary conditions are set in terms of L,‘,, though it is obviously appreciate
the actual strains involved.
deformation
gradients
Geologists
have tended
towards
the use of the
tensor:
dx, F,:=dX; for the computation of deformations quadratic stretch tensor: D,; = F:, F;p
this analysis necessary to
04) (e.g., Flinn,
1978; Sanderson,
1982) and the (15)
for the calculation of finite strains (e.g., Sanderson, 1982; De Paor, 1983). The same convention is used here, but because we are dealing with velocities, t;lIt is time dependent with a rate of change: c; = Lip FiJ
(16)
169
(Malvern, 1969, p. 163). Equation (16) are thus nine differential equations which can be solved over an interval, t - rO, to give ‘;;;. All systems of differential equations used here have been solved to approximately eight significant figures accuracy using a variable-order, variable-step Adams method (Numerical Algorithm Group routine W02CBF). ISOLATED
PARTICLES
IN SLOW FLOC’S
In this study two particle shapes have been considered, a prolate spheroid of axis ratio 3 : 1: 1 and an ellipsoid of axes 3 : 1”1 : f . They have been chosen specifically to have the same major axis length and identical volumes so their motions can be sensibly compared for equivalent boundary conditions. For most of the following discussion we consider the motion from nine starting orientations. These are represented as the plunge of the long axis (X,) on a stereographic projection and the subsequent motion is depicted by a smooth curve with dots at ten time unit intervals. To calculate the finite strain between any number of dots, n, we simply evaluate the quadratic stretch tensor:
The quadratic extensions are then the eigenvalues of O/j and the eigenvectors give their directions in space.
Here we use the velocity gradients: LI, =
0 0.05 [0
0 0 0
0 0 0
1
The amount of shear per ten time units is L;,= y = 0.5and ej is simply:
As noted previously the motion of an axisymmet~c particle is periodic with the ends of the spheroid describing closed spherical elliptical orbits about the pole to the plane of the undisturbed flow, i.e. the vorticity. The period is constant in time if y is constant, but is always proportional to y so that for one half rotation: y = Iz(T + l/r)
(18)
Further it appears that the particle will precess the same orbit for all time depending on C, the orbit constant, which is determined by the initial orientation, and is equal to tan a for a when /3 = 0. These are referred to as Jeffrey orbits.
170
Any departure from axisymmetry has profound effects on the nature of the orbit compared to that of a spheroid starting from the same long axis orientation (see also Gierszewski
and Chaffey,
1978; Hinch
results for nine initial orientations prolate
particle.
the periods secondary
It appears
and shows the corresponding
that the motion
are not constant drift through
and Leal. 19’79). Figure
in time.
families
is still periodic
Imposed
of Jeffrey orbits.
2 summarizes
the
Jeffrey orbits for the about
on the primary
the vorticity.
but
periodicity
is a
This is a fundamental
difference
between axi- and non-axisymmetric particle motions; the latter can drift through the plane of the undisturbed flow. If the time spent in the flow is large then the motion reveals itself to be doubly periodic; drift through the Jeffrey orbits eventually returns the particle to its original position. To assess the general behaviour in simple shear we first consider the degenerate cases for which the rotations are singly periodic. Dealing only with the long axis we can identify four initial orientations for which the corresponding c‘ = 03, and of these four tjnere are two distinct periods: 8=90,
+=90,0.
Jeffrey
orbit
is
l&=0,0
and: e = 90,
Q,= 90.0
li, = 90,90
(there are of course symmetrically equivalent positions outside the range 8, (p, + = 0 - 90). Secondly there are an infinite set of orientations for which the corresponding orbit constant is c’ = 0, and which have identical periods: B=O,
$X=0-90,
+=o-90
Now it is possible to consider the general behaviour in terms of the relationship of the initial orientation to that of one of the degenerate cases. Figure 2 shows that for the first few increments of shear (up to about y = 2) the orbits of the two particle geometries are qualitatively similar. As initial orientations approach degeneracy the similarity between the two orbits persists for greater shear magnitudes. Conversely as the symmetry of the position with respect to the plane of the undisturbed flow decreases,
the orbits
become
less like the equivalent
closed Jeffrey
orbit.
We note,
however, that for ail non-degenerate boundary conditions orbital drift is significant by shears of y = 4. Thus approximations for triaxial particle motions based on prolate
particle
theory are always likely to be misleading.
(2) Pure shear and other coaxial strains The equations of Gay (1966, 1968) (eqns. (2) and (3) here) indicate an important distinction between coaxial irrotational pure shear and flattening and simple shear for axisymmetric particles in that the former are non-periodic. Indeed this can easily be shown to be true for all coaxial strains involving any ellipsoidal particle geometry. For coaxial strains the only non-zero components of the velocity gradients tensor are
L
,
-
172
the diagonals,
i.e. L:, f 0, i =.j. Hence
Now, as the particle
H,, +
1 0 L0
0 1 0
axes approach
there is no vorticity
parallelism
and L?,, = 0 (eqn. 9b).
with the fixed axes, ,k; then:
0 0 11
and since E,‘, = 0, i f j (eqn. 9a), then E,, + 0 and w, become asymptotically smaller. is in a position of stable equilibrium and will remain fixed
When o, = 6 the particle
in position unless the applied flow is modified. For axisymmetric particles the components further
to B, = 0 and B, = -B,.
Therefore
of Bretherton’s
spin about
shape tensor
reduce
the long axis is zero and o2
and w3 are proportional to E3, and E,, respectively and are scaled by constants of equal modulii. Their velocities, which are clearly dependent on the axis ratio of the particle,
become maximum
as B, = -B,
-+ 1 and at the limit ~,/a,
= m. w2 = -E,,
and w3 = E,, which are the angular velocities of a passive material line. Therefore the axisymmetric particle motion for oblate and prolate spheroids follows trajectories which are the same as those predicted
by the March
and lines respectively. Triaxiality does not have such a drastic
model for passive
effect on the motion
as in simple shear
but we should note that all the w, are non-zero so the ellipsoid motion somewhere between the two extremes of behaviour for spheroids. (3) Simultaneous All the
flows
pure and simple shear (L;, = - L;,, described
always lies
L;, + 0)
in this section are geometrically
perpendicular to the vorticity velocity components producing
planes
similar
in that the plane
has a pure shear superimposed on it. However, the the pure shear are varied over an order of magnitude
so the ratio L;,/L;, ranges from 0.1 to 1.0, and the velocity producing simple shear is held constant. Particle paths for this type of flow are given by Ramberg (1975, fig. 3). We note that cross sections to the vorticity, deformation
so the resulting
The behaviour
through
are identical
and
such a flow, made in the plane perpendicular that
their
finite strain ellipsoid
of axisymmetric
particles
area
remains
constant
is k = 1 (Flinn,
throughout
1978).
is shown in Fig. 3a. Drift through
the
Jeffrey orbits occurs even for small amounts of pure shear and as this component is increased the periodic motion becomes subordinate to the asymptotic behaviour described in the previous section. When the velocities for pure and simple shear are equal the particle rotates from any starting position towards x’. This is an important observation because it implies that any multiparticulate fabric will tend to be prolate even though the finite strain ellipsoid is plane. Qualitatively similar results apply to the triaxial particles in as much as the gross departures from the double periods become more profound as the ratio of pure to simple shear velocities approach unity (Fig. 3). More subtle modifications occur in accordance with the symmetry of the initial orientation; in low symmetry orientations, and particularly when 6 is small, depar-
See text for discussion.
b, c, d. Paths of an ellipsoid
Fig. 3. a. Paths of a spheroid
(A) e=60
I 1
0
.05
.005
0
0
in simultaneous
in simultaneous
pure and simple shear for three initial orientations
with Fig. 2.
0
.607
1 1.
0
about
the long axis.
1 (a> 0
1.
0
0
1.
0
0
of the long axis and three initial rotations
Compare
0
0
points.
0
0
-.05
.05
pure and simple shear for two starting
a
,521
0
I.646 0
0
,905
0
.501
0
I[
0
0
0
,500
1.105
,951
0
0
-.Ol
0
0
1.051
.05
I
0
.05
.o 1
xj
-.c105
L=(p)I
:rl-
30
. .
‘..
\
-\
175
-A 0
0
O
zI’ 0
;
0
0
0
r
00,:
0
0
0
0
I’
v -
;:
I’
0
176
, 0
1”
0
0
G
0
“,
0
c ,----l -
u+j -I
II
LL
(D
0 -
c
0
0
0
:: 0
.-
::
O
0
0
0
0
0
0
0
0
0
0
0
0
;
0
I’
---.
/i/‘ i \
‘.
” / p/ I’
\
111
I
1
I
1 0
0
0
;; ?
$
O
0
0
Q
2 cn O
;
1 0
0
0
-:
0
0
0
;
0
0
q
0
h
1
0
0
0
Q
6
0
cl
_
178
179
r
t
r
0
0
r
II
0
0
G
0
0
0
I
0
0
0
0
0
0
0
:
0
0
0
0
- ._ .-
a3
-c
I’
-
180
181
tures from the simple shear case become less predictable, cusps appear in the particle trajectory and the time spent in orientations close to the Xi/X; plane becomes relatively large. (4) Simple shear on twoperpendicularplunes with simultaneous pure shear (L;, = - L;,, L;, = L;, f 0)
This is essentially a more general version of the previous flow. The two simple shear components are equivalent to a single shear direction on the plane whose pole is the vector [co545 co545 01, thus the pure shear defo~ation plane is at 45’ to the simple shear plane, and the vorticity of the flow is along the vector [ -cos45 cos45 01. Without the pure shear component we would expect the particle to precess about the vorticity as in previous examples. This is the case, but even when Z&i,is small, drift through the Jeffrey orbits is spectacular (Fig. 4a) for initial orientations which are close to the vorticity. However, if the long axis lies close to the plane perpendicular to the vorticity then the particle rotates in quasi-stable orbits with constants fluctuating about C = 00. Similar behaviour is observed for L;, = L;,/5 but when L;, = L;, all initial orientations eventually rotate to a position a few degrees away from Xi. Periodicity is retained for all initial orientations if the particle is triaxial (at least up to L;, = L;,). The rate of secondary drift through the orbits (Fig. 4a, b, c) is, as usual, dependent on the ratio L&/L;,. When this ratio is large the particle motion is similar to the axisymmetric one, but when L;, = L;, a rapid movement towards the X;/X; plane is followed by singly periodic rotation in a slightly elliptical spherical orbit which crosses the plane perpendicular to the vorticity twice every period. SUMMARY
Numerical solution of Jeffrey’s (1922) equations governing the motion of a rigid ellipsoidal particle suspended in a creeping fluid facilitates the study of spheroidal and ellipsoidal particles in plane and non-plane strain flows, with or without vorticity. The treatment here covers some strain geometries which are geologically important, and deals with magnitudes of strains up to and beyond those anticipated in most geological situations. It is by no means exhaustive, but demonstrates that departures from axisymmetry and/or plane strain have important consequences for the nature of the particle motion. Of particular importance is the motion of particles in flows involving pure and simple shear. The suggestion is that multiparticle fabrics will become prolate even though the finite strain ellipsoid is plane. Computer simulation experiments have been carried out in order to investigate multiparticle behaviour in the flows already described here and will be published in a later paper,
182
ACKNOWLEDGEMENTS
I thank Cohn Ferguson for reading and improving the manuscript. carried out during the receipt of a NERC research studentship
This work was at Nottingham
University. REFERENCES
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