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Rigid suspensions in viscous fluid
Leu Appl Engng Sci Vol. 23, No. 4, pp. 491-495. Printed in Great Britain
@ItO-7225/S $3.W t .X.l Pergamon Press Ltd.
1985
Letter Section RIGID SUSPENSIONS IN VISCOUS FLUID
A. Cemsl Eringen
Princeton University Princeton N.J. 08544
HRSTRACT
In a previous paperfi]I presented a continuum theory of rigid suspensions. In this work the effect of the torque (due to surrounding viscous fluid) on the rotations of fibers was ignored. Here I consider this effect.
I. INTRODUCTION
Recently I developed a theory of rigid suspensions based on micropolar continuum theory. For thermo-mechanical fiber suspensions the constitutive equations were obtained for the stress and heat vector. Here I quote the stress constitutive equations for the isothermal, incompressible viscous fIuids only
t = - p I + X,(i. dj) I + 2~ d + -" ?I 'L% '1, "u 21
(1) where
p
$
is the deformation rate tensor and
is the pressure
io
,
u
,
Xz
and
h2(j. dj) j 0 %%%'L
i
13
j + X,(j @dj + dj@ % % vu 11%
is the director with magnitude
j
0
j) ?I , i.e.
are the viscosity coefficients. We employ
1
This paper was written during my association with the University of Paris VI (Pierre et Marie Curie) as a visiting Professor, in spring 1985. I express my appreciation to the Department of Theoretical Mechanics and in particular to Prof. G.A. Maugin for kind hospitality
491
ERINCEN
A. CEMAL
492 the notation (% @
&jij
When (I) taining
X,
absence
of
and 13
= aibj
is h3
-term
compared with are absent
of rigid
suspensions
(3)
Pk
In some flow
situations
=
effect
of fluid
brings
the modified
the Batchelor’s
of
Pk
’
this
effect
equations.
(due to viscous
to an equation
terms con-
We believe
that
the
of
friction)
on the
rota-
the form
.. JoJk
=
By taking
can be considerable.
on the micro-inertia,1
modify
in agreement with
(3)
led
CZ, 31 we see that
situations.
the torque
This
was neglected.
theory
constitutive
in some flow
the effect
WklPl
eq.
($,_j,ji = dikj,
in Batchrlor’s
can be serious
In my development tions
,
(3)
to include
this
the corresponding
into
consideration
effect.
As we shall
equation
of Batchelor.
the see
this
2. MOTIONOF FIBERS
As discussed fibers
in Ref.
I,
for
rigid
fibers
we may approximate
the micro-inertia
of
by
(4)
jkl
=
.2 JO 6kl - jkjI
where (5)
Here
jk= “K/r
JK
directors nates
is a constant between
\.
kK
director
JK
goes
From the equation
--’djkl dt
where
“kl
=
-
‘km “1,
?k
is
=
representing
the rectangular
The vector
(6)
..2 Jo
,
the rigid
coordinates into
of
jk
XK
with
conservation
- ’ lm “km =
. . JkJk suspension
in the natural
are the cosine
and state
and the spatial
the motion. of micro-inertia
we have
0
the microgyration
tensor.
From (6) we see that
coordi-
493
Rigid suspensions in viscous fluid (7)
jlk
In terms of
j,
djkl r=a this
reads
dj, (8)
=
Jk dt
which also
follows
0
from (5)2. the effect
In the form (5)1 taken into
account.
(9) where
j,
To include
=
is a function
fk
From (9)
dj, -=
(10)
(11)
it
with
+
(5)1
to the suspensions
as
that
=
wkl
vector
made in Ref.
$ (v,
=
For an isotropic i=
fk
the assumption
the form of
function
trg
of
31 - Vl I k)
expression
of
+ oz c2 $
the invariants
, tr % d3
tr g2
,
1, we have
we have the rigourous
tiO$ + al q 4
are functions
c(. I
variables.
sticking
fk
follows
We now determine
where
the constitutive
+
of
‘kljl
Vkl
(13)
we modify
fluid
.
dt
In accordance
effect,
JK /YkK
this
of viscous
, i.i
=
ji
(14) i.cj, Neglecting
the terms higher l
(15) The condition
f
4.e’
>
= (8)
order
(aoltr$ states
+ that
i than the first r( j. o2
$ j,)i
power of ‘“iC4
$
we have
f
asi [4]:
are not
494
A. CEMAL
ERINGEN
Consequently
Eq.
2 al = - a02 i.
agi = 0
(17)
(IO) now reads
dj,
-= dt
This equation
(19) where
coincides
a1 r
is
=
with
2 r2 +
-2 (dkrj, - j. dmni,i,jk)
+ al
Wkl jl
that
of Batchelor
if
we set
jk = j0 pk
and identify
al
by
I
---
1
the slenderness
of fibers.
ratio
3. EFFECTOF THERMALGRADIENT
For thermo-viscous of
the temperature
gradient.
Generators the linear
term in
C
of and
(21)
&=
(alo
+a,lj,2
Consequently
Eq.
10 reads
suspensions In this
case
sotropic
the
WC have
x8
+
Eq.
(18)
Eq. (12)
function
will is
C
have to be modified
replaced
is
by the inclusion
by
read from Ref.
4. Considering
only
Rigid suspensions in viscous fluid
495 2
(23)
!&k
=
wkljl + +
“1 Cdklj,
-
lo *-*
dmnj,jnjk~
*
a2ce
-2 ax(d klB,l - jo jmdmll8,njk) +
,k - j:
i,e,,jk)
o4(jIe idkmjm - _imdmnjn~,k)
,
where we set
cl1 =
(24)
2 -oi3i0
Constitution equations for the stress tensor
t De
and heat vector
%
remain
valid as given in Ref. 1.
Summarizing : constitutive equations (1);
(i)
The present theory differ from that of Batchelor in the stress
(ii)
Our result (18) coincide with the corresponding equation of
Batchelor for slender bars ; (iii)
Constitutive equations for the thermo-viscous suspensions
given in Ref.1 and the equation of evolution (23) of directors are entirely new.
Acknowledgement I am indebted to Professor Ifoward Brenner of MIT and Prof. G.A. Kaugin of the University of Paris VI for some illuminating discussions.
REFERENCES
I.1
I
ERINGEN, A. C., -i-Int. j. Engng. Sei. (Letters section)
22 , No I1/12, p. 1373, 1984
E21
BATC~LOR,
[31
BATCHELOR, G.K., -J. Fluid e.,
L41
ERINGEN; A.C., Mechanics -___ of Continua (second edition) Appendix B, Robert E. Krieger Co, 1980
G.K., --.I. Fluid Mech.. 2 66
, 545, 1970
, 813, 1971
@ItO-7225/S $3.W t .X.l Pergamon Press Ltd.
1985
Letter Section RIGID SUSPENSIONS IN VISCOUS FLUID
A. Cemsl Eringen
Princeton University Princeton N.J. 08544
HRSTRACT
In a previous paperfi]I presented a continuum theory of rigid suspensions. In this work the effect of the torque (due to surrounding viscous fluid) on the rotations of fibers was ignored. Here I consider this effect.
I. INTRODUCTION
Recently I developed a theory of rigid suspensions based on micropolar continuum theory. For thermo-mechanical fiber suspensions the constitutive equations were obtained for the stress and heat vector. Here I quote the stress constitutive equations for the isothermal, incompressible viscous fIuids only
t = - p I + X,(i. dj) I + 2~ d + -" ?I 'L% '1, "u 21
(1) where
p
$
is the deformation rate tensor and
is the pressure
io
,
u
,
Xz
and
h2(j. dj) j 0 %%%'L
i
13
j + X,(j @dj + dj@ % % vu 11%
is the director with magnitude
j
0
j) ?I , i.e.
are the viscosity coefficients. We employ
1
This paper was written during my association with the University of Paris VI (Pierre et Marie Curie) as a visiting Professor, in spring 1985. I express my appreciation to the Department of Theoretical Mechanics and in particular to Prof. G.A. Maugin for kind hospitality
491
ERINCEN
A. CEMAL
492 the notation (% @
&jij
When (I) taining
X,
absence
of
and 13
= aibj
is h3
-term
compared with are absent
of rigid
suspensions
(3)
Pk
In some flow
situations
=
effect
of fluid
brings
the modified
the Batchelor’s
of
Pk
’
this
effect
equations.
(due to viscous
to an equation
terms con-
We believe
that
the
of
friction)
on the
rota-
the form
.. JoJk
=
By taking
can be considerable.
on the micro-inertia,1
modify
in agreement with
(3)
led
CZ, 31 we see that
situations.
the torque
This
was neglected.
theory
constitutive
in some flow
the effect
WklPl
eq.
($,_j,ji = dikj,
in Batchrlor’s
can be serious
In my development tions
,
(3)
to include
this
the corresponding
into
consideration
effect.
As we shall
equation
of Batchelor.
the see
this
2. MOTIONOF FIBERS
As discussed fibers
in Ref.
I,
for
rigid
fibers
we may approximate
the micro-inertia
of
by
(4)
jkl
=
.2 JO 6kl - jkjI
where (5)
Here
jk= “K/r
JK
directors nates
is a constant between
\.
kK
director
JK
goes
From the equation
--’djkl dt
where
“kl
=
-
‘km “1,
?k
is
=
representing
the rectangular
The vector
(6)
..2 Jo
,
the rigid
coordinates into
of
jk
XK
with
conservation
- ’ lm “km =
. . JkJk suspension
in the natural
are the cosine
and state
and the spatial
the motion. of micro-inertia
we have
0
the microgyration
tensor.
From (6) we see that
coordi-
493
Rigid suspensions in viscous fluid (7)
jlk
In terms of
j,
djkl r=a this
reads
dj, (8)
=
Jk dt
which also
follows
0
from (5)2. the effect
In the form (5)1 taken into
account.
(9) where
j,
To include
=
is a function
fk
From (9)
dj, -=
(10)
(11)
it
with
+
(5)1
to the suspensions
as
that
=
wkl
vector
made in Ref.
$ (v,
=
For an isotropic i=
fk
the assumption
the form of
function
trg
of
31 - Vl I k)
expression
of
+ oz c2 $
the invariants
, tr % d3
tr g2
,
1, we have
we have the rigourous
tiO$ + al q 4
are functions
c(. I
variables.
sticking
fk
follows
We now determine
where
the constitutive
+
of
‘kljl
Vkl
(13)
we modify
fluid
.
dt
In accordance
effect,
JK /YkK
this
of viscous
, i.i
=
ji
(14) i.cj, Neglecting
the terms higher l
(15) The condition
f
4.e’
>
= (8)
order
(aoltr$ states
+ that
i than the first r( j. o2
$ j,)i
power of ‘“iC4
$
we have
f
asi [4]:
are not
494
A. CEMAL
ERINGEN
Consequently
Eq.
2 al = - a02 i.
agi = 0
(17)
(IO) now reads
dj,
-= dt
This equation
(19) where
coincides
a1 r
is
=
with
2 r2 +
-2 (dkrj, - j. dmni,i,jk)
+ al
Wkl jl
that
of Batchelor
if
we set
jk = j0 pk
and identify
al
by
I
---
1
the slenderness
of fibers.
ratio
3. EFFECTOF THERMALGRADIENT
For thermo-viscous of
the temperature
gradient.
Generators the linear
term in
C
of and
(21)
&=
(alo
+a,lj,2
Consequently
Eq.
10 reads
suspensions In this
case
sotropic
the
WC have
x8
+
Eq.
(18)
Eq. (12)
function
will is
C
have to be modified
replaced
is
by the inclusion
by
read from Ref.
4. Considering
only
Rigid suspensions in viscous fluid
495 2
(23)
!&k
=
wkljl + +
“1 Cdklj,
-
lo *-*
dmnj,jnjk~
*
a2ce
-2 ax(d klB,l - jo jmdmll8,njk) +
,k - j:
i,e,,jk)
o4(jIe idkmjm - _imdmnjn~,k)
,
where we set
cl1 =
(24)
2 -oi3i0
Constitution equations for the stress tensor
t De
and heat vector
%
remain
valid as given in Ref. 1.
Summarizing : constitutive equations (1);
(i)
The present theory differ from that of Batchelor in the stress
(ii)
Our result (18) coincide with the corresponding equation of
Batchelor for slender bars ; (iii)
Constitutive equations for the thermo-viscous suspensions
given in Ref.1 and the equation of evolution (23) of directors are entirely new.
Acknowledgement I am indebted to Professor Ifoward Brenner of MIT and Prof. G.A. Kaugin of the University of Paris VI for some illuminating discussions.
REFERENCES
I.1
I
ERINGEN, A. C., -i-Int. j. Engng. Sei. (Letters section)
22 , No I1/12, p. 1373, 1984
E21
BATC~LOR,
[31
BATCHELOR, G.K., -J. Fluid e.,
L41
ERINGEN; A.C., Mechanics -___ of Continua (second edition) Appendix B, Robert E. Krieger Co, 1980
G.K., --.I. Fluid Mech.. 2 66
, 545, 1970
, 813, 1971