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Controllable flows of ideal fibre-reinforced fluids
MECHANICS RESEARCH COMMUNICATIONS Vol. 19(6), 527-534, 1992. Printed in the USA 0093-6413/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press Ltd.
CONTROLLABLE FLOWS OF IDEAL FIBRE-REINFORCED FLUIDS
B . D . H u l i , T.G.Rogers and A . J . M . S p e n c e r Department o f T h e o r e t i c a l Mechanics, U n i v e r s i t y o f N o t t i n g h a m , N o t t i n g h a m , E n g l a n d . NG7 2RD
(Received 3 February 1992; accepted for print 3 March 1992)
Introduction
The use of thermoplastic composite materials provides an efficient processing route for components since the composite can be heated until molten and then quickly pressed into the required shape. The development of such manufacturing techniques has led to increasing interest in the rheologicai properties of fibre-reinforced fluids. Some experimental results have been obtained [1] using torsional flows between parallel plate viscometers [2]. For isotropic fluids the classical approach for determining viscometric functions has been the use of controllable (or partially controllable) flows. All such flows have been listed for isotropic fluids [3] but the investigation of controllable flows in anisotropic fluids has received little attention. Ideal fibre-reinforced fluids are an example of anisotropic fluids and they admit controllable flows in addition to those found in isotropic fluids. This paper is concerned with one such additional flow viz. flow between concentric cylinders. The concept of ideal fibre-reinforced materials [4] which are incompressible and contain inextensible fibres has proved useful in analysing many problems in composite materials. This continuum model and its governing equations are outlined and solutions are then derived for a general ideal fibre-reinforced viscous fluid and an ideal, linear, fibre-reinforced viscous fluid flowing between concentric cylinders.
Governin~ Equations for Fibre-Reinforced Viscous Fluids
The governing equations are formulated in the cylindrical polar coordinates r,~o and z and in this system the components of the velocity vector u are denoted by Ur,U_ and uz. The fibre-reinforced fluid is modelled as a continuum consisting of an incompressible viscous fluid reinforced by a single family of fibres. The constraint of incompressibility requires that the divergence of the velocity field is zero so that 1 a(rur) V.u -
1 auso +
r
Or
au z +
r
O~a
az
527
0
(1)
528
B.D. HULL, T.G. ROGERS, and A.J.M. SPENCER
The reinforcing fibres are assumed to be continuously distributed and locally parallel and their direction can in general vary with both position and time t. Their direction is denoted by a unit vector a with components a r, a9 and a z which satisfy a r 2 + a~ 2 + az2 - 1
(2)
The inextensibility of the fibres may be expressed mathematically as aTda = 0
(3)
where a is the column matrix containing the components cylindrical polar coordinates. The 3x3 matrix d contains tensor d in cylindrical polar coordinates and the superscript matrix. The components of the strain-rate tensor d may components of the velocity gradient tensor L as d
=
of the direction vector a in components of the strain-rate T denotes the transpose of a be expressed in terms of the
~(L + L T)
(4)
and the components of L in cylindrical polar form are
L
[ 0Ur/0r 0u~/0r
=
(0Ur/09 - u~,)/r (0u9/09 + Ur)/r
au~/ar
auz/r~
OUr/0z 0u_/0z
]
au~z/az
(5)
During flows of the molten composite the fibres are assumed to convect with the fluid and thus [4]
o.
[?]
_ _
+
u
Dt
=
r
(6)
La
r
where the convected derivative D/Dt is defined as D/Dt
=
0/0t
+ ur 0/0r
+ (ug/r)O/~p
+ Uz0/0z
The constitutive equation relating the stress tensor ff and the strain rate tensor d for ideal unidirectionally reinforced fluids is given by Rogers [5] as -
-pi
+ TA_ + vld_ +
v2d2~ +
v3(Ad._ + dA) . . .+ . v4dAd .
(7)
The identity tensor is represented by [ and the tensor A is defined as the dyadic product A
ffi
a®a
.
(8)
The f i ~ o m e t r i c functions v l , v 2, v s and v 4 are functions of the three invariants t r ( d 2)
,
t r ( d 3)
,
t r ( A d ~)
(9)
where tr denotes the trace of a tensor. The hydrostatic pressure p and the fibre tension T represent the reactions to the imposed kinematic constraints of incompressibility and fibre inextensibility. The occurrence of two arbitrary functions in the constitutive equation rather than the single arbitrary pressure
FIBER-REINFORCED FLUIDS
529
found in incompressible isotropic fluids has implications for viscometric flows and these are discussed in the following section.
the
controllability
of
The constitutive equation (7) is the equivalent of the Reiner-Riviin fluid used for isotropic fluids. It provides a model for the behaviour of fluids reinforced by a single family of continuous inextensible fibres and also for fluids reinforced by unidirectional chopped strand fibres.
Controllable Flows of Ideal Fibre-Reinforced Fluids Betwcgn Cylinders
In this section the flow between two concentric cylinders is examined. The inner cylinder of radius r 0 rotates with constant angular velocity [20 while the outer cylinder of radius r 1 rotates at a constant angular velocity ill. The cylinders are positioned so that their common axis (the z-axis) is in the vertical direction as shown in FIG 1.
]Z~~r
FIG 1 Flow Between Concentric Cylinders
The velocity field u
-
B
-
(0, B r l n [ C r ] ,
0)
(lo)
with
o0,
['~0 -- [~1 ,
ln(r0)
- In(r1)
C
t
fl0 - fll
n(rl )']
satisfies the incompressibility constraint and the boundary conditions. The slip surfaces of this viscometric flow take the form of cylinders concentric with the bounding cylinders. The fibres are assumed to lie on these slip cylinders and so the fibre direction vector a may be written in the form
530
B.D. HULL, T.G. ROGERS, and A.J.M. SPENCER
a
(0,
-
cosO,
sine)
~11
w h e r e e is the inclination of the fibre to the z-axis. T h e s e velocity and fibre direction vectors satisfy the r e q u i r e m e n t s of fibre inextensibility and convection as expressed b) equations (3) and (6) respectively.
In cylindrical polar coordinates the stress c o m p o n e n t s are ~rr
=
-P
ff~t~p
=
-P + Tc°s20
azz
=
-P + Tsin2O
~r~
=
~B(~l+r3c°s~O)
arz
=
~Bv3cos0
(r~pz
=
+
¼ B 2 ( ~ 2 + ~ 4 e ° $ 2 0 )
)
¼B2p2
+
P
, ( 12
sine
, ,
TcosO s i n 0
T h e invariants listed in equation (9) are t r ( d 2)
=
~B
,
~
t r ( d 3)
0
=
,
t r ( A d 2)
¼B2cos2O
=
(13)
T h e s e invariants are i n d e p e n d e n t of position and so for given velocity boundary conditions the viscometric functions are constants. H e n c e the equations of motion b e c o m e
--
op cos20[
- -
-]-
-
~P4 B2 --
-
Or
r
10p
c o s 2 O OT +cosO
0p - -- + sin2e Oz
T
1
=
-pB2r
ln[Cr]
(]4
,
OT B - - + - (vl+VaCOS~O) az r
sine
0T cosO sine -- + az r
[12
[ OT ] - - + ~Be 3 Ov~
- pg
=
-
0
0
(15)
(1(~
w h e r e p d e n o t e s the density of the composite and g is the gravitional acceleration. "I~ ensure that the flow can be sustained in a full annular gap the pressure and tension must be s i n g l e - v a l u e d functions for all positions and hence they are assumed to be i n d e p e n d e n t o f the ~o-coordinate. T h e n provided that the fibres are not aligned either in circles c o n c e n t r i c with the bounding cylinders or parallel to the z - a x i s , viz. O ;~ 0 or 0 ~ 7r/2, the s e c o n d equation of m o t i o n may be integrated to give -B(p l+e 3cos2O)z T
=
+ f(r) rcosO
(17
sine
w h e r e f(r) is an arbitrary function. H e n c e the fibre tension varies linearly with z in such a flow. Similar flows c a n n o t be sustained in isotropic fluids since such fluids do not have the fibres to provide the tension required to balance the equation of motion.
FIBER-REINFORCED FLUIDS
531
Integration of the third equation of motion (16) yields Bv3cos0 stn0 crz2
-
-p + Tstn20
-
pg -
] z + g(r)
(18)
where g(r) is an arbitrary function of integration. This function may be determined by applying an appropriate boundary condition. For example, if it is assumed that °zz
-
-P0
at
z
-
0
then g(r)
-
(19)
-Po
T h e n by measuring the fibre inclination angle 0 and the stress c o m p o n e n t ~zz at say z = - H the value of the viscometric function v 3 can be calculated from (18).
Combining
equations (17) and (18) yields
+ B(Vl+V3COS20)tan0
_
Bv3cosO s i n 0
]z + g(r)
-p r
- f(r)sin20
2r (20)
Using these resul~ in the f i ~ t equation of motion (14) gives {vscosO sin0
- 2(v1+vscos20)tan0
+ 2(p1+v3cos20)cot0}Bz/(2r2)
+ (21)
dg/dr
- sin20
df/dr
+
cos20{~B2v
4
-
f(r)}/r
-
-pB2r[lnCr] 2
Equating the ~ r m in z ~ zero gives 2cot20
-
1 +
(22)
{vl/(Vl+V3COS20)}
Thus for fluids with viscometric functions such that the right hand side is positive the flow is controllable. In principle the relation (22) provides a means of determining v 1 with the value of v 3 obtained as above.
With the choice of g(r) given in equation (19), integration of the terms independent of z in equation (21) yields f(r)
-
~B2v 4 + D r - C ° t 2 0 + B 2 0 r - c ° t 2 0
cosec20I(ln[Cr])2rl+C°t20dr
(23)
where D is an arbitrary constant, so completing the solution (17) for the fibre tension.
In the special cases when the fibres are aligned either transverse or parallel to the z-axis (0 = 0 or 0 = ~/2) it is not in general possible to have solutions in which the pressure p and the tension T are independent of ~o. The solutions for the two cases can be written in the following form :
(i) a
-
(0, 1, 0)
,
532
B.D. HULL, T.G. ROGERS, and A.J.M. SPENCER
p
=
B(v,+ra) ~ + h(r,~)
T
=
h(r,~)
+ k(r)
- pgz
(24)
,
,
where h(r,~) (ii)a
=
¼B2v 4 0,
(1/r)Ik(r)dr
=
(0,
1)
p
-
Bvl¢
T
-
p + 0gz + m(r,~)
(oB2/r)Ir2(ln[Cr])2dr
+
Q(~)/r
,
pB2Ir~(ln[Cr])2dr
+
+
-
n(z)
,
(25)
,
where Q(~), n(z) and m(r,~) are arbitrary functions. Since for these special flows the stress is not single-valued in a complete cylinder they are only sustainable in sectors. Thus they have limited practical importance and are unsuitable for experimental evaluation of the viscometric functions. Controllable flows of this type are also possible in isotropic fluids and also suffer the same restrictions [6].
Controllable Flows of Ideal Linear Fibre-Reinforced Fluids Between Cylinders
T h e r e is some experimental evidence [7] to suggest that the stress in fibre-reinforced fluids is linearly dependent on the strain-rate after an initial yield stress has been overcome and the appropriate constitutive relationship above this yield stress is [5] =
- p l.
+ .T A
+.
2 ~ T.d
+. 2 ( ~.L - ~ T ).( A d
+
dA)
(26)
This constitutive relationship involves only two viscometric functions 7/L and ~T and these are both constants. The longitudinal viscosity T/L is associated with shear along the fibre direction and the transverse viscosity 7/T is associated with shearing perpendicular to the flow direction. For such fluids the results obtained above remain valid with v 1 and u 3 replaced by 27/T and 2(T/L - ~ r ) respectively and u 2 and r 4 set to zero. With these values of u l and u s the expression on the right hand side of equation (22) becomes 1 + {v,/(v,+~acos20
)}
=
1 + { ~ T / ( ~ T s i n 2 O + ~LCOS20)} >
0
and as discussed after equation (22) this implies that F u r t h e r m o r e equation (22) may be rewritten in the form ~L ~T
tan20(2tan20
-
the
flow is always controllable.
1) (27)
(2
-
tan20)
It is interesting to note that for these fluids the angle of inclination 0 does not change with variation of the angular velocities of the bounding cylinders. The relationship in equation (27) is shown in graphical form in FIG 2.
FIBER-REINFORCED FLUIDS
533
1.0'
2,5
2.0
1.5"
i.O
O.
~.0
20'
~v~ ' ' ' ~
f 40'
50'
60'
10'
$0'
#
90'
-0.5
FIG 2 C r a p h o f V i s c o s i t y R a t i o V/L/7/T v e r s u s Fibre Inclination Angle 0
Results obtained by Groves [1 ] using a parallel plate viscometer suggest that the relative size of the two viscosities may depend on the concentration of fibres in the fluid. T h e controllable flow discussed above provides a simple method for examining these variations in viscosities with fibre concentration by simply measuring the fibre inclination angle 0.
For fluids in which the concentration of fibres is low it is reasonable to assume that the the two viscosities are equal and hence v 3 = 0. In this case equation (22) becomes tan20
=
1
(28)
and for such fluids the fibres should lie at 45" to the z-axis.
Conclusions
It has been shown that ideal fibre-reinforced fluids can admit controllable flows which are inadmissible in isotropic fluids. In particular, flows between concentric cylinders have been examined. Such flows can be used to determine two of the viscometric functions for
534
B.D. HULL, T.G. ROGERS, and A.J.M. SPENCER
fibre-reinforced fluids. If the stress exhibits a linear dependence on the strain-rate tensor then flows between rotating concentric cylinders are always controllable and they provide a convenient method for determining the relative sizes of the transverse and longitudinal viscosities.
Acknowledgements
One of the authors (B.D.H.) is supported by a postgraduate award from the Department of Education for N.Ireland with I.C.I. plc as the industrial sponsor. Their financial assistance is gratefully acknowledged.
References
[1] D.J.Groves,D.M.Stocks and A.M.Bellamy, Proceedings of the 3rd European Conference on Rheology, Edinburgh, (1990), ed. D.R.Oliver, Elsevier, p.190. [2] T.G.Rogers, Composites, 20, 21 (1989). [3] W.L.Yin and A.C.Pipkin, Arch. Rat. Mech. Anal., 37, 111 (1970). [4] A.J.M.Spencer, Deformations of Fibre Reinforced materials, Clarendon Press, Oxford, 1972. [5] T.G.Rogers, In Inelastic Deformations of Composite Materials (ed. G.J.Dvorak), Springer-Verlag, New-York,1991. [6] R.I.Tanner, Engineering Rheolog¥, Clarendon Press, Oxford, 1985. [7] F.N.Cogswell, Intl. Polymer Processing, 1, 157 (1987).
CONTROLLABLE FLOWS OF IDEAL FIBRE-REINFORCED FLUIDS
B . D . H u l i , T.G.Rogers and A . J . M . S p e n c e r Department o f T h e o r e t i c a l Mechanics, U n i v e r s i t y o f N o t t i n g h a m , N o t t i n g h a m , E n g l a n d . NG7 2RD
(Received 3 February 1992; accepted for print 3 March 1992)
Introduction
The use of thermoplastic composite materials provides an efficient processing route for components since the composite can be heated until molten and then quickly pressed into the required shape. The development of such manufacturing techniques has led to increasing interest in the rheologicai properties of fibre-reinforced fluids. Some experimental results have been obtained [1] using torsional flows between parallel plate viscometers [2]. For isotropic fluids the classical approach for determining viscometric functions has been the use of controllable (or partially controllable) flows. All such flows have been listed for isotropic fluids [3] but the investigation of controllable flows in anisotropic fluids has received little attention. Ideal fibre-reinforced fluids are an example of anisotropic fluids and they admit controllable flows in addition to those found in isotropic fluids. This paper is concerned with one such additional flow viz. flow between concentric cylinders. The concept of ideal fibre-reinforced materials [4] which are incompressible and contain inextensible fibres has proved useful in analysing many problems in composite materials. This continuum model and its governing equations are outlined and solutions are then derived for a general ideal fibre-reinforced viscous fluid and an ideal, linear, fibre-reinforced viscous fluid flowing between concentric cylinders.
Governin~ Equations for Fibre-Reinforced Viscous Fluids
The governing equations are formulated in the cylindrical polar coordinates r,~o and z and in this system the components of the velocity vector u are denoted by Ur,U_ and uz. The fibre-reinforced fluid is modelled as a continuum consisting of an incompressible viscous fluid reinforced by a single family of fibres. The constraint of incompressibility requires that the divergence of the velocity field is zero so that 1 a(rur) V.u -
1 auso +
r
Or
au z +
r
O~a
az
527
0
(1)
528
B.D. HULL, T.G. ROGERS, and A.J.M. SPENCER
The reinforcing fibres are assumed to be continuously distributed and locally parallel and their direction can in general vary with both position and time t. Their direction is denoted by a unit vector a with components a r, a9 and a z which satisfy a r 2 + a~ 2 + az2 - 1
(2)
The inextensibility of the fibres may be expressed mathematically as aTda = 0
(3)
where a is the column matrix containing the components cylindrical polar coordinates. The 3x3 matrix d contains tensor d in cylindrical polar coordinates and the superscript matrix. The components of the strain-rate tensor d may components of the velocity gradient tensor L as d
=
of the direction vector a in components of the strain-rate T denotes the transpose of a be expressed in terms of the
~(L + L T)
(4)
and the components of L in cylindrical polar form are
L
[ 0Ur/0r 0u~/0r
=
(0Ur/09 - u~,)/r (0u9/09 + Ur)/r
au~/ar
auz/r~
OUr/0z 0u_/0z
]
au~z/az
(5)
During flows of the molten composite the fibres are assumed to convect with the fluid and thus [4]
o.
[?]
_ _
+
u
Dt
=
r
(6)
La
r
where the convected derivative D/Dt is defined as D/Dt
=
0/0t
+ ur 0/0r
+ (ug/r)O/~p
+ Uz0/0z
The constitutive equation relating the stress tensor ff and the strain rate tensor d for ideal unidirectionally reinforced fluids is given by Rogers [5] as -
-pi
+ TA_ + vld_ +
v2d2~ +
v3(Ad._ + dA) . . .+ . v4dAd .
(7)
The identity tensor is represented by [ and the tensor A is defined as the dyadic product A
ffi
a®a
.
(8)
The f i ~ o m e t r i c functions v l , v 2, v s and v 4 are functions of the three invariants t r ( d 2)
,
t r ( d 3)
,
t r ( A d ~)
(9)
where tr denotes the trace of a tensor. The hydrostatic pressure p and the fibre tension T represent the reactions to the imposed kinematic constraints of incompressibility and fibre inextensibility. The occurrence of two arbitrary functions in the constitutive equation rather than the single arbitrary pressure
FIBER-REINFORCED FLUIDS
529
found in incompressible isotropic fluids has implications for viscometric flows and these are discussed in the following section.
the
controllability
of
The constitutive equation (7) is the equivalent of the Reiner-Riviin fluid used for isotropic fluids. It provides a model for the behaviour of fluids reinforced by a single family of continuous inextensible fibres and also for fluids reinforced by unidirectional chopped strand fibres.
Controllable Flows of Ideal Fibre-Reinforced Fluids Betwcgn Cylinders
In this section the flow between two concentric cylinders is examined. The inner cylinder of radius r 0 rotates with constant angular velocity [20 while the outer cylinder of radius r 1 rotates at a constant angular velocity ill. The cylinders are positioned so that their common axis (the z-axis) is in the vertical direction as shown in FIG 1.
]Z~~r
FIG 1 Flow Between Concentric Cylinders
The velocity field u
-
B
-
(0, B r l n [ C r ] ,
0)
(lo)
with
o0,
['~0 -- [~1 ,
ln(r0)
- In(r1)
C
t
fl0 - fll
n(rl )']
satisfies the incompressibility constraint and the boundary conditions. The slip surfaces of this viscometric flow take the form of cylinders concentric with the bounding cylinders. The fibres are assumed to lie on these slip cylinders and so the fibre direction vector a may be written in the form
530
B.D. HULL, T.G. ROGERS, and A.J.M. SPENCER
a
(0,
-
cosO,
sine)
~11
w h e r e e is the inclination of the fibre to the z-axis. T h e s e velocity and fibre direction vectors satisfy the r e q u i r e m e n t s of fibre inextensibility and convection as expressed b) equations (3) and (6) respectively.
In cylindrical polar coordinates the stress c o m p o n e n t s are ~rr
=
-P
ff~t~p
=
-P + Tc°s20
azz
=
-P + Tsin2O
~r~
=
~B(~l+r3c°s~O)
arz
=
~Bv3cos0
(r~pz
=
+
¼ B 2 ( ~ 2 + ~ 4 e ° $ 2 0 )
)
¼B2p2
+
P
, ( 12
sine
, ,
TcosO s i n 0
T h e invariants listed in equation (9) are t r ( d 2)
=
~B
,
~
t r ( d 3)
0
=
,
t r ( A d 2)
¼B2cos2O
=
(13)
T h e s e invariants are i n d e p e n d e n t of position and so for given velocity boundary conditions the viscometric functions are constants. H e n c e the equations of motion b e c o m e
--
op cos20[
- -
-]-
-
~P4 B2 --
-
Or
r
10p
c o s 2 O OT +cosO
0p - -- + sin2e Oz
T
1
=
-pB2r
ln[Cr]
(]4
,
OT B - - + - (vl+VaCOS~O) az r
sine
0T cosO sine -- + az r
[12
[ OT ] - - + ~Be 3 Ov~
- pg
=
-
0
0
(15)
(1(~
w h e r e p d e n o t e s the density of the composite and g is the gravitional acceleration. "I~ ensure that the flow can be sustained in a full annular gap the pressure and tension must be s i n g l e - v a l u e d functions for all positions and hence they are assumed to be i n d e p e n d e n t o f the ~o-coordinate. T h e n provided that the fibres are not aligned either in circles c o n c e n t r i c with the bounding cylinders or parallel to the z - a x i s , viz. O ;~ 0 or 0 ~ 7r/2, the s e c o n d equation of m o t i o n may be integrated to give -B(p l+e 3cos2O)z T
=
+ f(r) rcosO
(17
sine
w h e r e f(r) is an arbitrary function. H e n c e the fibre tension varies linearly with z in such a flow. Similar flows c a n n o t be sustained in isotropic fluids since such fluids do not have the fibres to provide the tension required to balance the equation of motion.
FIBER-REINFORCED FLUIDS
531
Integration of the third equation of motion (16) yields Bv3cos0 stn0 crz2
-
-p + Tstn20
-
pg -
] z + g(r)
(18)
where g(r) is an arbitrary function of integration. This function may be determined by applying an appropriate boundary condition. For example, if it is assumed that °zz
-
-P0
at
z
-
0
then g(r)
-
(19)
-Po
T h e n by measuring the fibre inclination angle 0 and the stress c o m p o n e n t ~zz at say z = - H the value of the viscometric function v 3 can be calculated from (18).
Combining
equations (17) and (18) yields
+ B(Vl+V3COS20)tan0
_
Bv3cosO s i n 0
]z + g(r)
-p r
- f(r)sin20
2r (20)
Using these resul~ in the f i ~ t equation of motion (14) gives {vscosO sin0
- 2(v1+vscos20)tan0
+ 2(p1+v3cos20)cot0}Bz/(2r2)
+ (21)
dg/dr
- sin20
df/dr
+
cos20{~B2v
4
-
f(r)}/r
-
-pB2r[lnCr] 2
Equating the ~ r m in z ~ zero gives 2cot20
-
1 +
(22)
{vl/(Vl+V3COS20)}
Thus for fluids with viscometric functions such that the right hand side is positive the flow is controllable. In principle the relation (22) provides a means of determining v 1 with the value of v 3 obtained as above.
With the choice of g(r) given in equation (19), integration of the terms independent of z in equation (21) yields f(r)
-
~B2v 4 + D r - C ° t 2 0 + B 2 0 r - c ° t 2 0
cosec20I(ln[Cr])2rl+C°t20dr
(23)
where D is an arbitrary constant, so completing the solution (17) for the fibre tension.
In the special cases when the fibres are aligned either transverse or parallel to the z-axis (0 = 0 or 0 = ~/2) it is not in general possible to have solutions in which the pressure p and the tension T are independent of ~o. The solutions for the two cases can be written in the following form :
(i) a
-
(0, 1, 0)
,
532
B.D. HULL, T.G. ROGERS, and A.J.M. SPENCER
p
=
B(v,+ra) ~ + h(r,~)
T
=
h(r,~)
+ k(r)
- pgz
(24)
,
,
where h(r,~) (ii)a
=
¼B2v 4 0,
(1/r)Ik(r)dr
=
(0,
1)
p
-
Bvl¢
T
-
p + 0gz + m(r,~)
(oB2/r)Ir2(ln[Cr])2dr
+
Q(~)/r
,
pB2Ir~(ln[Cr])2dr
+
+
-
n(z)
,
(25)
,
where Q(~), n(z) and m(r,~) are arbitrary functions. Since for these special flows the stress is not single-valued in a complete cylinder they are only sustainable in sectors. Thus they have limited practical importance and are unsuitable for experimental evaluation of the viscometric functions. Controllable flows of this type are also possible in isotropic fluids and also suffer the same restrictions [6].
Controllable Flows of Ideal Linear Fibre-Reinforced Fluids Between Cylinders
T h e r e is some experimental evidence [7] to suggest that the stress in fibre-reinforced fluids is linearly dependent on the strain-rate after an initial yield stress has been overcome and the appropriate constitutive relationship above this yield stress is [5] =
- p l.
+ .T A
+.
2 ~ T.d
+. 2 ( ~.L - ~ T ).( A d
+
dA)
(26)
This constitutive relationship involves only two viscometric functions 7/L and ~T and these are both constants. The longitudinal viscosity T/L is associated with shear along the fibre direction and the transverse viscosity 7/T is associated with shearing perpendicular to the flow direction. For such fluids the results obtained above remain valid with v 1 and u 3 replaced by 27/T and 2(T/L - ~ r ) respectively and u 2 and r 4 set to zero. With these values of u l and u s the expression on the right hand side of equation (22) becomes 1 + {v,/(v,+~acos20
)}
=
1 + { ~ T / ( ~ T s i n 2 O + ~LCOS20)} >
0
and as discussed after equation (22) this implies that F u r t h e r m o r e equation (22) may be rewritten in the form ~L ~T
tan20(2tan20
-
the
flow is always controllable.
1) (27)
(2
-
tan20)
It is interesting to note that for these fluids the angle of inclination 0 does not change with variation of the angular velocities of the bounding cylinders. The relationship in equation (27) is shown in graphical form in FIG 2.
FIBER-REINFORCED FLUIDS
533
1.0'
2,5
2.0
1.5"
i.O
O.
~.0
20'
~v~ ' ' ' ~
f 40'
50'
60'
10'
$0'
#
90'
-0.5
FIG 2 C r a p h o f V i s c o s i t y R a t i o V/L/7/T v e r s u s Fibre Inclination Angle 0
Results obtained by Groves [1 ] using a parallel plate viscometer suggest that the relative size of the two viscosities may depend on the concentration of fibres in the fluid. T h e controllable flow discussed above provides a simple method for examining these variations in viscosities with fibre concentration by simply measuring the fibre inclination angle 0.
For fluids in which the concentration of fibres is low it is reasonable to assume that the the two viscosities are equal and hence v 3 = 0. In this case equation (22) becomes tan20
=
1
(28)
and for such fluids the fibres should lie at 45" to the z-axis.
Conclusions
It has been shown that ideal fibre-reinforced fluids can admit controllable flows which are inadmissible in isotropic fluids. In particular, flows between concentric cylinders have been examined. Such flows can be used to determine two of the viscometric functions for
534
B.D. HULL, T.G. ROGERS, and A.J.M. SPENCER
fibre-reinforced fluids. If the stress exhibits a linear dependence on the strain-rate tensor then flows between rotating concentric cylinders are always controllable and they provide a convenient method for determining the relative sizes of the transverse and longitudinal viscosities.
Acknowledgements
One of the authors (B.D.H.) is supported by a postgraduate award from the Department of Education for N.Ireland with I.C.I. plc as the industrial sponsor. Their financial assistance is gratefully acknowledged.
References
[1] D.J.Groves,D.M.Stocks and A.M.Bellamy, Proceedings of the 3rd European Conference on Rheology, Edinburgh, (1990), ed. D.R.Oliver, Elsevier, p.190. [2] T.G.Rogers, Composites, 20, 21 (1989). [3] W.L.Yin and A.C.Pipkin, Arch. Rat. Mech. Anal., 37, 111 (1970). [4] A.J.M.Spencer, Deformations of Fibre Reinforced materials, Clarendon Press, Oxford, 1972. [5] T.G.Rogers, In Inelastic Deformations of Composite Materials (ed. G.J.Dvorak), Springer-Verlag, New-York,1991. [6] R.I.Tanner, Engineering Rheolog¥, Clarendon Press, Oxford, 1985. [7] F.N.Cogswell, Intl. Polymer Processing, 1, 157 (1987).