Tecfonophysics, 113 (1985) 163-183

Eisevier Science Publishers B.V., Amsterdam - Printed in The Nethertands 163

THE MOTION OF RIGID ELLIPSOIDAL PARTICLES IN SLOW FLOWS

BRETT FREEMAN

Department of Geology. University Park, Nottingham, NG7 2RD (United Kingdom) *

(Received February 8, 1984; revised version accepted September 24, 1984)

ABSTRACT

Freeman, B., 1985. The motion of rigid ellipsoidal particles in slow flows. Teczon~phy~jc~, 113: 163-183.

Preferred orientations of fabric elements in tectonites are usually interpreted with reference to two

continuum models, namety those of March (1932) and Jeffrey (1922). It is argued that the first of these is

usually inappropriate if the fabric elements are rigid inhomogeneities embedded in a deformable matrix.

Jeffrey’s model is more applicable but until recently its use has been limited to problems involving

axisymmetric particles and simple flow geometries. In this paper Jeffrey’s equations have been solved

numerically in order to investigate the behaviour of more general systems. The motions of two particle

shapes, one a prolate spheroid, the other a triaxiaf ellipsoid, are considered in parallel for four specified

flow geometries which involve pure or simple shear and combinations of both. Any departures from

particle axisymmetry or from simple flow geometries have a significant effect on particle behaviour

compared with that predicted by spheroid/simple 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 upon Tyne, Newcastle upon Tyne,.

NE1 7RU (U.K.).

0040-1951/85/$03.30 0 1985 Elsevier Science Publishers B.V.

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. symme-

try arguments show that if the particle is ellipsoidal, then the shape tensor has only

three non-zero components, and these are easily evaluated given the semi-axial

lengths (Bretherton, 1962). Jeffrey (1922) derived the general equations of motion for

a rigid particle of neutral buoyancy isolated in a slow viscous flow whose streamlines

are steady at large distances from the particle. He also provided explicit solutions for

spheroidal particles in simple shear flows. Of course it is rare that geologists

encounter perfectly ellipsoidal particles but from the work of Eirich 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 (1957) have shown. (For a further discussion on the effects of non-el-

lipsoidal shapes see Ferguson ( 1979).)

Anulytical 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)] (1)

tan /? = C [ cos*a: + ( l/r* ) sin2a] I” (2)

where (Y and p are the polar angles (azimuth and plunge) of the particle long axis

with respect to the plane of undisturbed flow, r is the axial ratio and y is the shear

strain. Equation (1) is clearly periodic and together (1) and (2) predict that the long

axis of a prolate spheroid will precess one of an infinite family of closed, spherical

elliptical orbits described by the orbit constant C, which depends only on the initial

particle orientation and 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 (r- 1) ln(cot aI) = ln(cot ai) + - ~

ln

i 1

3 1’2

2 (r+l) h, (3)

165

and the plunge:

sin 201. ( i l/2

Cot #If = COt pi + & f

where X, and X, are the two principal quadratic extensions (the third being unity).

Subscripts f and i refer to the final and initial angles respectively. Tullis (1976)

presents similar equations for pure flattening strains of orientation distributions of

spheroids.

Although, as in simple shear, the material coordinates of the end of a prolate

spheroid differ from those of a corresponding passive linear marker after deforma-

tion, it is interesting to note that the angular paths swept out by both are identical

for all coaxial irrotational strains. The particle motions differ only in their angular

velocities, passive lines rotating 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; Ferguson

(1979) has addressed this problem in some detail and concludes that the model

should give acceptable approximations even for power law fluids provided that

strains are not too high.

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 approxi-

mated 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 of sidewall ramps and steep zones parallel to nappe transport

directions. Similarly interplay between pure and simple shear is important during the

development of folds, as indicated by Ramberg (1975). Secondly the particles are

required to be axisymmetric which greatly limits the geological applications. In

particular, it precludes any meaningful interpretation of fabrics composed of general

ellipsoidal (or approximately ellipsoidal) particles in the light of hitherto published

multiparticle models.

Recent theoretical work (Gierszewski and Chaffey, 1978; Hinch and Leal, 1979)

has brought attention to the rather more complex behaviour of triaxial particles in

simple shear flows. Both studies use numerical solution of Jeffrey’s (1922) equations,

and the results have a fair correspondence to the experimental results obtained from

Couette flow (Harris et al., 1979). I have used a similar numerical approach to

investigate the behaviour of isolated axi- and non-axisymmetric particles in some

general, geologically interesting, 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 axes of the particle. The relationship between X, and X,’

is given at any instant by the rotation matrix:

cos 4 cos 9 - cos e sin 9 sin 4 , cos I/ sin $ + cos e cos $3 sin I$ , sin 4 sin e

-sin$cos$-cosesin$cos#, -sinJisincp+cosecosqbcosrC,, cos+sinB

sin e sin 9 - sin e cos + 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 product of the rotation matrices

for each of the three operations. In this work the rotations are made in sequence

about Xi, X, then X, following the “x ” convention of Goldstein (1980, p. 147)

(though they are somewhat arbitrary).

At large distances from the particle the undisturbed flow is specified by the

velocity gradients tensor (spatial description):

(6)

167

which satisfies the constant volume condition:

( Lri, i =j are the rates of natural strain and L:j, i #j, are the shear strain rates.)

The rate-of-deformation and vorticity tensors of the undisturbed flow in the fixed

.coordinate system are the symmetric and antisymmetric 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 are ai the only non-zero components of

Bretherton’s (1962) shape tensor are:

Jeffrey’s (1922) equations of motion (p. 169, eqn. 37) are now easily rewritten as:

where w, are the angular velocities of the particle about its own axes X,. If the euler

angles change with velocities b,& 4, then it is straightforward to formulate the w, in

those terms (Goldstein, 1980, p, 176):

w1 =BsinBsin\I,+~cosIC, /

w2 = $2 sin t? cos + - S sin $.J

W) = 4 cos 8 + $L

j (121

I

which after rearrangement gives three differential equations:

B = w,cos +LJ - w,sin 4

~=(w,sin++o,cos~)/sinB

\t = w3 - (b cos 8

(13)

These are solved numerically as an initial value problem. Throughout this analysis

the boundary conditions are set in terms of L,‘,, though it is obviously necessary to

appreciate the actual strains involved. Geologists have tended towards the use of the

deformation gradients tensor:

dx, F,:=dX; 04)

for the computation of deformations (e.g., Flinn, 1978; Sanderson, 1982) and the

quadratic stretch tensor:

D,; = F:, F;p (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:

0 0 0 LI, = 0.05 0 0

[ 1 0 0 0

The amount of shear per ten time units is L;, = y = 0.5 and 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 and Leal. 19’79). Figure 2 summarizes the

results for nine initial orientations and shows the corresponding Jeffrey orbits for the

prolate particle. It appears that the motion is still periodic about the vorticity. but

the periods are not constant in time. Imposed on the primary periodicity is a

secondary drift through families of Jeffrey orbits. 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 Jeffrey orbit is

c‘ = 03, and of these four tjnere are two distinct periods:

8=90, +=90,0. 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 corre-

sponding 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 there is no vorticity and L?,, = 0 (eqn. 9b).

Now, as the particle axes approach parallelism with the fixed axes, ,k; then:

1 0 0

H,, + L 1

0 1 0 0 0 1

and since E,‘, = 0, i f j (eqn. 9a), then E,, + 0 and w, become asymptotically smaller.

When o, = 6 the particle is in a position of stable equilibrium and will remain fixed

in position unless the applied flow is modified.

For axisymmetric particles the components of Bretherton’s shape tensor reduce

further to B, = 0 and B, = -B,. Therefore spin about 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 trajecto-

ries which are the same as those predicted by the March model for passive planes

and lines respectively.

Triaxiality does not have such a drastic effect on the motion as in simple shear

but we should note that all the w, are non-zero so the ellipsoid motion always lies

somewhere between the two extremes of behaviour for spheroids.

(3) Simultaneous pure and simple shear (L;, = - L;,, L;, + 0)

All the flows described in this section are geometrically similar in that the plane

perpendicular to the vorticity has a pure shear superimposed on it. However, the

velocity components producing 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 through such a flow, made in the plane perpendicular

to the vorticity, are identical and that their area remains constant throughout

deformation so the resulting finite strain ellipsoid is k = 1 (Flinn, 1978).

The behaviour of axisymmetric particles 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-

(A)

e=60

30

:rl-

a

I 1 I L=(

p) I

xj

.005

0

.05

-.c10

5

0 0

.o 1

0

.05

-.Ol

0 0

.05

0

.05

-.05

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1.05

1

,500

0

1.10

5

.501

0

0

I[

I.646

0 ,5

21

0 0

0

,951

0 0

,905

0 0

.607

0

0

0 1.

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0 1. 1

0 0 1. 1 (a

>

Fig.

3.

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Path

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sphe

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in

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pure

an

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Fig.

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b, c

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175

-A v

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O z 0 0 0 0 0 0 ;: 0

I’ I’ I’

176

,

0

1” 0 0 G

c ,----l

- -I u+j 0 “, 0

II LL (D .-

0 - :: O c

0 0 0 0 0 0 0 0 0

:: 0 0 0 0 0 0

I’ 0 ; 0

i/‘ / ---.

‘.

i ” I’ / p/ \ \

111

1 I 1 I --

0 0 $ 0 0 ; 0 0

: cl _

;; h

0 ? O 2 Q cn O 0 ; 0

0 0 0

1 1

0 0 0

Q 6 0 0 q 0

178

179

- ._ .-

a3 II

r I

0 0 0

0 : 0 I’

t r

0 0 r

0 0 0

0 0 0

-c 0 0 G -

0 0 0

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 perpendicu- lar 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. This work was

carried out during the receipt of a NERC research studentship at Nottingham

University.

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