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On certain steady flows of anisotropic fluids
Int. J. Engng Sci. Vol. 3,
pp. 515-532. Pergamon Press 1965. Printed in Great Britain
ON CERTAIN STEADY FLOWS OF ANISOTROPIC FLUIDS P. N. KALONI Indian Institute of Technoiogy,
Kharagpur, India
Abstract-The present paper is concerned with the study of some classical steady flows of certain orientable fluids which tend to be unoriented at rest. The analysis is based on a theory of anisotropic fluids developed by Ericksen which seems to be rather general in interpretation. It is shown that the problems formulated are mathematically well set and that at moderate velocity gradients the fluid behaves like a viscoelastic fluid, predicting the behaviour similar to that observed in polymer solutions. Using the notion of material functions it is also pointed out, wherever necessary, that various results obtained are closely related with the corresponding results of Coleman and No11 for ‘simple fluids’.
1. INTRODUCTION
IN A SERIESof papers [l-7], Ericksen has formulated and developed a class of properly invariant theories of anisotropic fluids. Some of the results of these papers suggest that these theories cover a wide class of other theories of isotropic fluids, and describe some of them comparatively in a simpler and better way. Ericksen’s analysis [3,7] of simple shearing flow and homogeneous flows predicts that, at moderate velocity gradients, a class of these theories describes the behaviour similar to that observed in non-Newtonian fluids. For a different sub-class of these theories, which may be called the simplest theory of anisotropic iluids, Ericksen’s [1, 41 and Verma’s [8] analyses illustrate that the behaviour shown resembles that of Bingham materials and that, without using any yield criterion, one set of constitutive equations are sufficient to describe such materials. Further, Hand [9] has shown that his theory of dilute suspensions, which is obtained from Jeffery’s [lo] calculations for the motion of an ellipsoid suspended in a Newtonian Auid, falls as a special case of Ericksen’s theory [I] of anisotropic fluids. The object of the present work is to investigate the behaviour of such fluids in certain classical steady flows, using the general theory of anisotropic fluids [2]. The flow problems considered are: Poiseuille flow, Couette flow, helical flow and flow through pipes of arbitrary cross section.* Our analysis (including that of ffow between rotating cones, which will appear separately) reveals that the predicted behaviour is similar to that observed in viscoelastic fluids. In fact, it is observed that, if we follow the notion of material functions as has been introduced by Markovitz [11] or by Coleman and No11 112, 131,our calculations are simplified and yield results closely similar to those obtained by Coleman and No11 [12, 131for ‘simple fluids’ [14] and by Rivlin [I 51for Rivlin-Ericksen fluids [16]. Because of the vast literature on this subject, the references in this paper are limited to a few which are most closely related to our work. For the general background and an organized account of various isotropic theories we refer the reader to Eringen [17] with its extensive bibliography.
* Simple shearing flow and homogen#us flows with constant velocity gradients have already been analysed by Ericksen 13, 71.
516
P.N.
KALONI
Solutions for some of the flows that we consider here have already been analysed by Ericksen [I, 41 and Verma [8], but their derivations are based upon the simplest theory of anisotropic fluids [l], Recently, Green [18] has pointed out some difficulties with this type of theory on the grounds that it does not give a satisfactory definition of a ‘fluid’. We have therefore reworked these flows, along with others, using a more general theory [2] restricting our attention to fluids for which n+O when they are motionless. This restriction not only clarifies the doubts raised by Green [18] but has some physical precedent for it. In dilute solutions of high polymers at rest, the long-chain flexible molecules are supposed to be randomly coiled, and therefore define no preferred direction. In shear it is believed that they stretch out and align themselves at some definite angle relative to the streamlines. Tn the theory proposed this tendency towards alignment is represented macroscopically by associating with each particle of the fluid, a vector II of variable magnitude, which represents the direction of orientation and whose magnitude measures the amount of stretch. In equilibrium, i.e. at rest, we should therefore have n=O. 2. BASIC
EQUATIONS
The theory considered involves a vector II of variable magnitude which represents a preferred direction in the fluid and is related to the fluid velocity Ri by equations of the form Gi=wiknk+(P1
(2.1)
+Clzd,,n,n,)ni+llJdikn~.
Here dik and Oik are, respectively, the symmetric and anti-symmetric gradient tensor given by 2dij=ii,
j+ij,
i
2Oij=~_i,
j-~j,
i
parts of the velocity
(2.2)
while p’s are functions of n2=nknk to be determined by experiments, and dots denote material derivative. The constitutive equations for such fluids have the form tij = -p6ij + (1, + I,dr,n,n,)ninj + 2Aadrj+ 2J.,(dikn,nj + dikn,ni)l
(2.3)
where tii is the stress tensor, p an arbitrary pressure and A’sare functions of n2. We assume the fluid to be incompressible which means that velocity ii must satisfy the equation . xk,
(2.4)
k- -0.
Finally, we have the equations of motion which in the absence of body forces are
(2.5)
tij, j=pf,
where p is the mass density. In view of what equation (2.3) implies, we have seven equations (2.1), (2.4) and (2.5) to determine seven unknowns p, 1, and n,. The simplest theory of anisotropic fluids has been obtained by imposing the constraint n2 = 1, which implies that 2’s and p’s should then be constant. Also the consistency condition then requires that (2.6) Pl =O, p3=-p2* 3. POISEUILLE
FLOW
We consider the steady flow of the anisotropic fluid through an infinite circular pipe of radius a. We use cylindrical polar coordinates (I, 8, z) and assume the velocity field as
517
On certain steady flows of anisotropic fluids i=o, Assuming
O
i = w(r),
tL0,
(3.1)
that the fluid adheres to the walls of the pipe, we have the boundary condition w(a) = 0.
(3.2)
The physical components of the tensors dii and wii are 0
0
f/2-
0
0
0
f/2 -
0
0
I [dij]=
COijl = -
0
0
-fi
0
0
0
f/2
0
0
(3.3)
-
where f stdnds for doldr. On using (3.3) equation (2.3) becomes t11= -p+ab:+2&n,n,f t2*=
-p+aw:
133 = -p+ml:+21,n,n,f tl,=~flln,+[~3+~4(n:.+n:)]f
(3.4)
t,2=@)111112+L&n3f t,,=@n*n,+A,n,n,f
where @=(1,+Qz,n,)J
(3.5)
On assuming n=n(r, t) equation (2.1) gives
(3.6)
Equations (3.6) can be satisfied by taking n i=O and in that case the theory reduces to that of Newtonian fluid with viscosity A,(O). However, as is discussed by Ericksen [3, 71, such solutions are of less significance because of their unstable nature, in comparison to those with ni*O. In the analysis that follows we restrict our attention to fluids for which 1140 when there is no motion. Following Ericksen [7] we find that all components of n, tend to zero when the fluid is at rest and take steady values for Iarge times when the fluid is in motion, only if Pl(O)
&+
P3’ 19
llftan
$>O
(3.7)
where tan24=k.
-1
/13+1
(3.8)
518
P. N. KALONI
Assuming that the above conditions hold, we then have a critical rate of shear given by
(3.9) where ~~(0) and ~~(0) are the values of pi and jt3 evaluated at a2 =O. It then follows that (cf. Ericksen [7]) n+O when,f’,j’z
(3.11)
where N denotes that value of n which corresponds to stable steady-state orientation. In short, all the above indications are that at low rates of shear the fluid will behave like a Newtonian fluid, but will behave differentIy above the critical shear rate. We now solve equations (3.6) for the components nits for steady-state flow by putting &z,/&=O and assume that aforesaid conditions hold true. It is then a simple matter to verify that these equations have the solutions ltl = N sin 4,
112=o,
n,=Ncos
(3.12)
n,‘S
in the first equation of (3.6)
1) cot 4+pUziV2 sin 2tp]f+2p1=0
(3.13)
where 4 is determined by (3.8). On setting above values of we arrive at the equation [(yj-
cf,
which along with (3.8) determines the value of N in terms of motion. equation is solvable. Using (3.12) we then obtain t11=
-p+a,N’
(3.14)
sin2 ++Af
(3.15)
f22 = -p tj3=
-p+a&v cos2l#+Bf
(3.16) (3.17)
t13=C+Df t,,=t,,-
We assume that this
-0
(3.18)
where A, B, C and D are given by 2A=N2
sin 2c$[A,??2 sin’c#+&]
(3.19)
2B= N2 sin 2$[A,N2 COS~~+~~~]
(3.20)
2C= A,N 2 sin 24
(3.21)
4D=4(~~+~~~2)~~2N4sin2 and where N 2 is given by (3.13).
24.
(3.22)
On certain steady flows of anisotropicfluids
519
We now introduce the material functions defined as (C+Df)=cU) (&IV sin24 + Af) = PI(f)
(3.23)
(&IV2 cos2~+Bf)=p2(f) where LYand /3’s are functions of ,f’ only, as is evident from equations (3.19) to (3.22) and (3.13). Following the time honoured terminology we call, a(f )/f‘shear dependent viscosity’ and PI(f), P2(f) ‘normal stress functions’. We shall, however, for the present, postpone the discussion of these functions to a subsequent section, where we shall try to establish the relationship between these and others introduced in [II] and [12]. The equations of motion (2.5) in the absence of body forces reduce to
(3.24)
Assuming that a and /I’s are independent of temperature of (3.24) : p= -a’z+h(r),
&f)=
with
we have the following solutions
a’r b - yf +r
s
“(f)&.+L
W=Bl(f)+
r
(3.25)
(3.26)
where a’, b and L are constants, of which the middle one, i.e. b should be taken equal to zero, since at r=O, f must be bounded. The second equation of (3.25) is a differential equation for the velocity w(r) to be solved subject to (3.2). However, if we assume, which we shall justify later, that an inverse of the function cl(f) exists, then (3.25) can be written as (3.27) so that with the help of (3.2), o(r) turns out to be (I o(r)=
cl-l $
s ,
dr.
(3.28)
0
The total mass flux across the tube is given by Q=27rp “m(r)dr s
0
(3.29)
which on using (3.2) and (3.27) gives (3.30)
520
P.N.
KALONI
Thus from equations (3.28) and (3.30) it is clear to observe that the determination of both velocity and mass flux simply requires the knowledge of material function g(f). In other words, if the mass flux Q can be determined as a function of pressure head a’, we can calculate the material function ix(S) very easily. To complete the solution we now calculate the normal stresses. From equations (3.14) to (3.16), (3.25) and (3.26), we have (3.31)
(3.32)
(3.33) wherefis
given by (3.27)
It is interesting to observe from the above equations that both radial and downstream normal stresses are always in excesss over the classical values a’z of these quantities. It therefore follows that the well-known swelling phenomenon at the exit section of the tube, often referred to as the Merrington effect [ 191in the literature, will be accounted for qualitatively by this theory. At this stage, we now recall some interesting allied experimental results that have been reported in a series of papers 120-231by Middleman and Gavis in capillary jets of Newtonian and viscoelastic fluids. They have found that not only viscoelastic fluids, but also the laminar jets of Newtonian liquids, expand or contract from a capillary nozzle depending upon the fluid properties and jet velocity. These authors suggest that, besides the effect of surface tension, there are three main sources of normal stresses which can contribute to the observed changes in the jet diameter; viz. (i) a non linear normal stress developed in the tube (ii) a viscous normal stress due to the velocity profile relaxation outside the capillary tube (this exists in Newtonian fluids also) and (iii) a normal stress caused by elastic response to the velocity profile changes. Merrington 1191,on the other hand, attributes this phenomenon to the elastic recovery of the fluid stretched in the capillary tube, meaning thereby that only those class of fluids which possess elastic properties would be capable of predicting such changes. In the view of Middleman and Gavis, this conclusion is, therefore, not perfectly satisfactory as there are other equally important sources which are responsible for such changes. In fact, it has been pointed out by Middleman and Gavis [22] that at high ejection velocities and for highly viscous fluids, the internal normal stress is mainly responsible in comparison to these of the profile relaxation and external normal stresses. The effect of surface tension being, comparatively, negligible [23]. However, since the quantitative estimation of various normal stresses, which have been quoted above, is still incomplete, it is therefore difficult to assess the relative importance of these factors, till the question is resolved completely. To fix the ideas further, we assume, following Middleman and Gavis, that internal nonlinear stress is chiefly responsible for the swelling of the fluid. We suppose that the fluid is issued from the pipe into an atmosphere at pressure pO. The force exerted on the output cross section of the pipe, neglecting the effect of surface tension, is, therefore, equal to nazpo. At the exit section we must. therefore, have the following force balance system
521
On certain steadyflows of anisotropicfluids a
J
7ra2po=-27t
t,;rdr
0
and, which on employing the value of t,, from (3.33) gives xa'pO=-7z s
(3.34)
"12p,(f.)-Bl(f)frd~+na2~. 0
Eliminating the constant L. between (3.31) and (3.34) we get (3.35) Let again P = (-t,,)y =a, i.e. the force per unit area that the fluid exerts on the pipe. Then from (3.35) we get
J
P-P,=$
(3.36)
a @P2V)--1303>rdr. 0
Equation (3.36) demands that for a stream of fluid to swell at the exit section, we must have 2Mf+
(3.37)
~~(~j
where f is given by (3.27). Hence, we conclude that for a fluid under consideration to show the swelling effect the material functions (3.23) must satisfy the condition (3.37). 4. COUETTE
FLOW
We consider the steady flow of the fluid contained between two infinite coaxial cylinders of radii rl and r2(r2>rl) which are rotating about the common axis with constant angular velocities R, and C& respectively. We use cylindrical polar coordinates (r, 8, z) taking zaxis to be the common axis of the cylinders and assume the velocity field as
$=O,
~=~~(r),
i=o
(4.1)
which satisfy (2.4). Assuming that the fluid adheres to the walls we have the boundary conditions as w(r*) = R,. 0@1)=Q*, (4.2) The physical components of the tensors d, and wU are 0 Cdij]=
A2
0 .-
.f/2
0
0-
O
O
0
0
,
-i_ (f-t-20)
[Oij]= i(f+2W)
0 -
0
0
0
0
0
(4.3)
!
where f now stands for r &D/C%. On using (4.3) the stress components given by (2.3) reduce to the same expressions as given by equations (3.4) with suffixes 1, 2, 3 now replaced by r, z, 8 in that order. Also on assuming n=n(r, t), the equations determining the physical components of n,‘s reduce to the same equations as given by (3.6) with the suffixes 1, 2, 3, now replaced by r, z, 8, in that order. For the stable steady-state values of ni)s, corresponding to N+ 0, we therefore have the same restrictions as given by equations (3.7) to (3.11), where now _f denotes r do/dr. The values of Ri’S then turn out to be
522
P.N.
n,=N
KALONI
n,=Ncos
sin t$
q5
n,=O
(4.4)
where 4 and N2 are given by equations (3.8) and(3.13) respectively. The stress components then reduce to the same expressions as given by (3.14) to (3.18) with the suffixes replaced in the order stated above. The dynamical equations (2.5) now take the form (4.5)
(4.6)
(4.7) where p is the density of the fluid and g is the acceleration due to gravity. An elementary analysis shows that the symmetrical solutions of the above equations are - t,, = p(r,z) = pgz + h(r)
(4.8)
u,=$ W=Pdf)+
s
(4.9)
'Bl(fw32(f)dr+
r
pro2
f
(4.10)
where K is the torque per unit height required to maintain the relative motion of the cylinders. For a known value of K, equation (4.9) along with (4.2) is a differential equation to determine the velocity profile o(r). However, as before, if we assume the existence of the inverse of a(f), then we get (4.11) which on integration and using (4.2) gives (4.12) where K is determined from the equation (4.13) Thus, if the material function LX(~)is known, equation (4.12) determines the velocity profile, and (4.13) determines the differences of angular velocities of the cylinders as a function of torque per unit height K. Amongst other expressions, which also occur in the case of Newtonian fluid, equation (4.8) shows the additional variation of the vertical normal pressure which is given by
523
On certain steady flows of anisotropic fluids
P’w=Bl(f>+
r~1(f)-~2(f)dr+COnStallt .r
(4.14)
I
wherefis given by (4.11). This expression is therefore responsible for the so-called Weissenberg climbing effects [24]. It follows, therefore, that pressures must be applied perpendicular to the axis of rotation in order to keep the upper surface horizontal. If such pressures are not present the fluid will tend to rise or sink, the exact shape of which will depend upon the function p’(r). 5. HELICAL
FLOW
The problem of helical flow for simple fluids [14] and for Rivlin-Ericksen fluids [16] has been studied, respectively, by Coleman and No11 [ 131and Rivlin [ 151. Our aim here is to analyse this problem for anisotropic fluids and to show that similar results can be obtained by much simpler calculations. We consider the steady motion of the fluid contained between two coaxial cylinders of radii rl and rz(rz > I~) which are rotating about their common axis with angular velocities R, and Sz,, respectively, and a pressure gradient and/or a gravitational force is acting on the fluid parallel to the axis of rotation. We again use cylindrical polar coordinates (r, 8, z) taking the z-axis to be the common axis of the cylinders and assume the velocity field as i=o, i=u(r). 4=rw(r), (5.1) which satisfy (2.4). Assuming that the fluid does not slip at the solid boundaries, the boundary conditions take the form u(rr) = u(rJ = 0.
W(Tz)= %,
c-01)=%
(5.2)
The physical components of the tensors d, and oy are 0
_.
--I
Yl
Yz
2
2
0
Yl
Cdijl = 2
0
0
0
0
9
Ewijl =
(?+CO> -72 (1)
Y2 2
(5.3) 0
0
_
_I
where du do . y1 = r dr and y2 = do’
(5.4)
On using (5.3) the stress components reduce to t,, = -p+cDn;+2L&y, tzz= -p+@r$+2L,n,n,y, t,,=
++yz)n,
-p+@r;+2a,n,n,y,
tl,=~,y,+~nln,+~4(n2-n32)yl+;l,n2n3y, t13=a3~2+cDnln3+~,~n2-n~~~2+~,n2n3~l t23=~'n2n3+~4(yln3+Y2n2)nl
(5.5)
524
P.N.
KALONI
where @=
(4 +J,(n,y, +n,y,)i.
(5.6)
Assuming n = n(r,t), equations (2.1) give
an1 dt=
dn, -= at
( 1 cL3-1
-
c
2
P3+1 -
2
(Yrn,+y,n,)+[C1l+~zfYtn,+y,n,)n,ln,
(5.7)
n~y~+[~~+~,(y,n,+y~n3)n,ln,
(5.8)
>
an, Cr,+l
( >
at= - 2
nly2 +
[k
+~2(hn2fy2n3hln3.
(5.9)
We again restrict to fluids for which n-0 at rest and assume N to denote that value of n which corresponds to stable steady-state orientation. To determine the stabte values of N, we write (cf. Ericksen 171) ni=Ni+mi
(5.10)
and linearize equations (5.7) to (5.9) with respect to mi and then determine the stability conditions. We thus observe that all solutions of the above equations will approach N=O in time, only if /p3(0)l > I and y <
-2P1(o) MO)- 1>*
(5.11)
where (5.12)
We notice that conditions (5.11) are satisfied for small values of y. If its value exceeds the critical value yCgiven by 4&(O) (5.13) y,z= @L:(O)- 1) the associated configuration will be unstable. It therefore follows that if conditions (5.11) hold true, then the ffuid will behave like a Newtonian fluid and when (5.1 I) fails then N=O will represent unstable con~guration. We are, however, interested, in those solutions which correspond to stable steady state N+O when (5.11) is violated. For this we linearize equations (5.8) and (5.9) with respect to mi, which reduce to am,
-_=
(5.14)
at
am3 P3fl -= --7j-~l~2+[IJil~1?2~~I~2+~2~3)~llm3 at
( 1
(5.15)
where bars over p’s denote their values at PZ’= N2. On multiplying (5.14) by yZ and (5.15) by y1 and then subtracting we get 84 at=C~,+~2(~1N2+~2N3)N115
(5.16)
525
On certain steady flowsof anisotropicfluids
where <=(m,y, -m,y,)+O. This equation gives one of the necessary conditions, showing that for a stable value of N we must have (5.17) Combining (5.17) with (5.8) (after setting l&/at (5.17) as
=0
in the latter) we get an alternative to (5.18)
So long as above conditions hold, we can solve equations (5.7) to (5.9) for stable values of ili’S by putting h,/lJt=O. We thus get n, = N sin 4,
n,=Nsin
n,=Ncosecos&
0 cos 4
(5.19)
where tan e=e
(5.20) Yl
and I#Jis determined by (3.8). Using (5.19) equation (5.8) then gives [(pa+ 1) tan 4+p2NZ sin 24]y+2p, =O.
(5.21)
which along with (3.8) and (5.12) determines N2 in terms of motion. To summarize what has been said above, we notice from equation (5.21) that if the conditions (5.11) hold and when y < y. then N does not take the real values until y reaches its critical value yc given by (5.13) and that in the latter case it tends to zero. However,if y 2 yc (y not exceeding too much from yc cf. Ericksen [7]) then N turns out to be a real vector having the components given by (5.19). Assuming that the above conditions for real values of N hold true and that yr and yZ are well-behaved inside the annulus, the stress components on using (5.19) become t11=
-P+P,(Y)
(5.22)
*,,=
_p+fi’oyf
(5.23)
_,+!&,:
t2,=
Y
f12=-Y,
t23
44 Y
7
_b2(y) -2YlY2 Y
U(Y)
t13 =-Y2
(5.25)
Y
(5.26)
where a and p’s are the material functions introduced earlier and y’s are given by equations (5.4) and (5.12). It is clear from the above equations that stresses are functions of r only, except for a dependence on z occurring in p. Hence the dynamical equations (2.5) when transferred to cylindrical polar coordinates take the form
526
P. N.KALONI
t,r-
--
be
= _pro2
I‘
dl,e 24, - +-=o dr
(5.27)
r
dr,,
t,, dr,__ -& + ;+z f&7,=0
where g, is the z-component of the gravitational field. An elementary inspection of the above equations suggests that these are satisfied only if p=p(r,
(5.28)
z)=Pz+h(r)
4Y)
trs=~ y1=
K
(5.29)
g2
(5.30)
and (5.31) where K is the torque per unit height required to maintain the motion of the Q and K, are constant while P= (+/a~ - pgZ)is a constant which may be called the pressure gradient’. Equations (5.29) and (5.30) are a pair of simultaneous non-linear differential to determine w(r) and u(r) subject to the boundary conditions (5.2). From these we observe that
cylinders, ‘frictional equations equations
(5.32) so that a(y)=&r)
(5.33)
or y =a-'[G(r)]
where 6(r) is purely a function of r only. It is thus possible to write (5.29) and (5.30) in the form do K cc-‘[G(r)] -=(5.34) dr 2nr3 a(r) du
(5.35)
-= dr
These equations along with the boundary conditions (5.2) give
s
w(r)=$rI, ~-llfi(r)ldr+R r36(r)
(5.36)
u(r)=
(5.37)
j:,
{(~+~)‘-~~l)dr
1
527
On certain steady flows of anisotropic fluids
where the constants involved in these equations are to be evaluated with the help of following equations rz ~-‘C4’>1,, &-R,=K 27rs I, r36(r) (5.38)
The volume discharge rate per unit time is given by (5.39)
Q=27rnp ‘*m(r)dr s r, which on using (5.35) and (5.38) turns out to be
(5.40) From the above equations it therefore follows that velocity profile and volume discharge rate, once again, can be determined by the simple knowledge of the material function c+J). The pressure distribution at any point is given by
Ypr~2+Kl
p(r, 2)= Pz + PI(Y)+
(5.41)
where w(r,), yl, y2 and y have the values already obtained earlier. Introducing this value of p in equations (5.22) to (5.24), it is straight forward to determine the normal stresses and their differences without any difficulty. 6. FLOW
THROUGH
A NON-CIRCULAR
PIPE
We now consider the flow of anisotropic fluids through tubes of arbitrarycross section under uniform pressure gradient. Several years ago Ericksen [25] considered this problem for Reiner-Rivlin fluids [26, 271 and showed that rectilinear flow of such fluids is possible only for special shapes of the tube and that in general (except when the coefficients that occur in the constitutive equation satisfy certain definite conditions), it is accompanied with a secondary flow. The significance of this conclusion has been examined by a number of authors [28-311 for various other theories and have obtained the results in agreement to the findings of Ericksen [25]. Our interest here is to examine the possibility of rectilinear flow in the case of anisotropic fluids.
We use Cartesian coordinates (x, y, z) and assume the velocity field as k=O,
I;=o,
i =f(x, y).
(6.1)
which satisfy (2.4). The physical components of the symmetric and antisymmetric velocity gradient tensors are 0 0 0 -Y2/2 [dij]
=
0 Y2P
0
hiI =
0 Y2P
0
-Y1/2
Y1/2
0
(6.2)
P. N. KALONI
528
where now (6.3)
On using (6.2) the stress components given by (2.3) reduce to the same expressions as given by equations (5.5) with suffixes 1,2, 3, now replaced by z, y, x in that order, and where now yt and yz are given by (6.3) Assuming n=n(x, y, t) equation (2.1) gives (6.4) dn y=
( ) c1&
at
2
an,_ dt
-
!
‘+
Yllrz+ [cl, +C(2(Y2n,$yln,)n,ln,,
(6.5)
(~2n,+yl~~,)+[~l
(6.6)
+~2(Y2~x+Ylny)n,]nZ.
>
Stability conditions for stable steady state N=i=Oin the present case can also be worked out in the previous section. We find that the conditions (5.11) to (5.13), (5.18) and the discussion presented these still hold true except that now y stands for (6.7) It is thus possible to obtain the steady-state solution of equations (6.4) to (6.6) as n,=Nsinflsin4,
n,=NcosUsincp,
nz=
Ncos&
(6.8)
where C#Iis given by (3.8) and 0 is determined by (5.20) with y1 and y2 now given by (6.3). With the above values of ni’s, the equation determining N in terms of motion again reduces to (5.21) where now y has the value given by (6.7). On substituting the values of n,‘s from equation (6.8) in the equation determining the stress components we obtain the same expressions as given by (5.22) to (5.26) with the suffixes 1,2,3, now replaced by z, y, x in that order, where y’s have the new values defined in this section. The equations of motion in the absence of body forces are
“,‘; atxY-() 1
aY
3&c, i)x+-=O
ai,,
(6.9)
JY
%z at,, z+-=O ay where it is evident from (6.1) that such motions are accelerationless. expressions for stress components in the above equations we get
Employing the
On certain steady flows of anisotropic fluids
529
(6.10)
where ,=!z!$) Y (6.11)
and @=a(y) -. Y
Equations precisely of the same form, as above, have been analysed by Ericksen [25]. From his analysis it can therefore be concluded that for a solution of equations (6.10) to exist, we must have a constant q and a function R(f) such that
a a 4=&w,)+-m)
(6.12)
aY
(6.13) and that, then P = N_fl +.kW2)
(6.14)
+qz.
Equations (6.12) and (6.13) are two partial differential equations to determine a single unknown functionf(x, y). Ericksen [25] has shown that when x and 0 are independent functions, the rectilinear motion is not physically possible except for rigid motions and for motions in which the curves of constant speed are circles or straight lines. For a special value of 0, @=m/y where m is a constant, Stone [32] has shown that curve~=constant are straight lines if q=O and circles of radius l//mq] if q=l=O. However if x and 0 are not independent but are linearly related such that x=MO
(6.15)
where M is a constant, then all solutions of (6.12) satisfy equation (6.13) with R(f)=Mq..f, so that the ffow in such cases is rectilinear. In our present analysis under general conditions, neither (6.15) holds nor 0 has the form ~~~ unless or(y)is constant. It therefore follows that for each choice of q, the solutions of (6.12) will not satisfy (6.13) for any choice of R(f) and that the flow cannot be rectilinear. Although it is rather difficult to conclude at this stage, that secondary flows will always be observed in the flow of anisotropic fluids through the tubes of arbitrary cross section, but the results discussed here and those of isotropic fluids suggest that such inferences are likely to be correct. 7. SOME REMARKS
ON THE MATERIAL
FUNCTIONS
In the course of our present study we have seen that the three material functions introduced by us through equations (3.23), determine velocity profile, stress and pressure distribution completely. In the present section we shall compare these, making some observations, with those introduced by Coleman and No11 [12] in the theory of ‘simple
530
P. N. KALONI
fluids’ [14], which is thought to be the most general fluids reported
hitherto
and appropriate
theory
of isotropic
[33].
We begin our discussion with the remark that the existence of our material functions, which are merely the differences of the normal stress functions, ascertains that, in general, neither the famous conjecture due to the Weissenberg [34] nor the symmetry relation predicted by the Reiner-Rivlin theory, regarding the normal stress function, hold true. This is in agreement with the findings of Markovitz and Brown [35]. We also observe that even for complex flows, such as those which have been discussed in sections 5 and 6, our three material functions are sufficient to determine the stresses and velocity profiles completely. It follows, therefore, that if experimental measurements are performed with any one of the simpler solutions to determine these functions, then complete velocity profiles and stresses can be predicted for other complicated flows. From equations (3.13) to (3.23), and remembering the fact that n-+0 when the fluid is at rest, we find that all material functions vanish iff=O, i.e. a(0) = PI(O) =&(O) = 0.
(7.1)
Also from the same equations it is noticed that if we substitute f= -f in the equations determining these material functions, then LX(~)changes its sign whereas pi(f) and /IX(f) remain unaffected. The second law of thermodynamics requires the energy dissipation to be positive. This means that f&(f)>0 for f* 0. (7.2) Since c$f) is an odd function 0f.h (7.2) implies that f and g(f) must have the same sign. All the above results are in agreement with [ 121. In the foregoing analyses we have not emphasized the fact that derivatives of a(f) and c1-‘(f) will be discontinuous of f=f,,the discontinuity generally being finite. This rather weak discontinuity does not invalidate the solutions. Whether it occurs naturally depends on, whether singular value f,is attained in the flow regime. That such a singular value does occur in some of the flows discussed can be accounted for in the following manner. In the case of analysis of section 3, for example, we have from equation (3.2), w(r)=0 at r = a. Hence, unless w is identically equal to zero, it will take on an extremal value in the interior of flow field where f= &.@r will become equal to zero. Nearby, we must, therefore, havef
531
On certain steady flows of anisotropic fluids
In short, this rather sketchy comparison shows that the material function introduced by us have, more or less, similar properties as those introduceds in [12]. It is, therefore, obvious that the analyses presented in [ 12, 131where more stress has been laid to determine these functions from an experimental point of view, will hold true in our case also. We
feel it unnecessary to repeat those arguments here, once again. The material functions we have used in the present paper bear the following relationship with others introduced in [12, 111 and in a slightly different form in [351. This is recorded below where the functions are written in the same order as the references cited above : Pl(f)=tll-t33=Zl(k)=V1-V2=Viii_Vi P2(f)=t22-tJJ=Z2(k)=V1-V3=Vii*-Vii a>
= t, 2 = G)
= 3V4(3)
(7.3) = $Jtl(j)
f=k=j.
Acknowledgements-1 especially wish to thank Prof. J. L. Ericksen for sending me all his reprints and clarifying certain aspects of his theories. I also wish to thank the referee for certain comments and helpful suggestions. Finally I express my gratitude to Dr. G. Bandyopadhyay aad Prof. B. R. Seth for their advice and encouragement.
REFERENCES [l] J. L. ERICKSEN,Koll. Z. 173, (117-122) (1960). [2] J. L. ERICKSEN,Arch. Rut. Mech. Anal. 4,231-237 (1960). [3] [4] [5] [6]
J. L. J. L. J. L. J. L.
ERICKSEN, Trans. Sot. Rheol. 4,29-39 (1960). ERICKSEN,Arch. Rut. Mech. Anal. 8, l-8 (1961). ERICKSEN, J. Polyrn. Sci. 47, 327-333 (1960). ERICKSEN, Proc. 7th Congress of Theoretical and Applied Mechanics,
Bombay, India 7,21 l-218
(1961). [7j J. L. ERICKSEN,Trans. Sot. Rheol. 6,275-292 (1962). [8] P. D. S. VERMA,Arch. Rut. Mech. Anal. 10, 101-107 (1962). [9] G. L. HAM), Arch. Rut. Mech. Anal. 7, 81-86 (1961). [lo] G. B. JEPPERY, Proc. roy. Sot. A102, 161-179 (1922). [ll] H. MARKOVITZ,7hzs. Sot. Rheo. 1, 37-52 (1957). [12] B. D. COLEMANand W. NOLL, Arch. Rut. Mech. Anal. 3,289-303 (1959). [13] B. D. CoLEMANand W. NOLL, J. appl. Phys. 30,1508-1512 (1959). [14] W. NOLL, Arch. Rut. Mech. Anal. 2, 197-226 (1958).
[15] R. [16] R. [17] A. [18] A.
S. RIVLIN,J. Rat. Mech. Anal. 5, 179-189 (1956). S. RrvLIN and J. L. ERICKSEN, J. Rat. Mech. Anal. 4, 323-425 (1955). C. ERINGEN,Non-linear Theory of Continuous Media, McGraw-Hill, New York (1962). E. GREEN,Proc. Cur&. Phil. Sot. 60, 123-128 (1964).
[19] A. C. MERRINGTON, Nuture 152,663 (1943). [20] S. MIDDLEMAN and J. GAVIS,Phys. Fluids 4, 355 (1961). [21] S. MIDDLEMANand J. GAVIS,Phys. Fluids 4, 963 (1961). [22] J. GAMYand S. MIDDLEMAN, J. Appl. Polymer. Sci. 7,493 (1963). [23] J. GAV~S,Phys. Fluids (1964). [24] K. W~~ENBERG, Nature 159, 310 (1945). [25] J. L. ERICKS~, Quart. appl. Maths. 14,319-321 (1956). [26] M. REINER,Amer. J. Math. 67, 350-362 (1945). [27] R. S. RIVLIN,Nature 160,611-613 (1945). [28] W. LANGLOIS and R. S. RIVLIN,Tech. Report No. 3, Div. appl. Math.. Brown University (1959). [291 A. E. GREENand R. S. RIVLIN, Quart. uppI. Moth. 14,299-308 (1956). [30] W. LANGLOISand R. S. RIVLIN,Rendiconti di Matematica 22, 169-185 (1963). [31] W. 0. CRIMINALE JR., J. L. ERICK~EN and G. L. FILBEYJR., Arch. Rat. Mech. Anal. 1,410-417 (1958).
532
P. N. KALONI
[32] D. E. STONE,Quart. appl. Mark 15, 257-262 (1957). [33] C. TRUESDELL,Principles of Continuum Mechanics, Socony Mobil Oil Co. Inc., Dallas (1960). [34] K. WEISSENBERG, Proc. 1st ht. Rheo. Congress, Amsterdam 1, 29-45 (1949). [35] H. MARKOVITZand D. R. BROWN,To appear in Proc. Z.U.T.A.M. Symp., Haifa, Israel (1962). [36] J. L. ERICKSEN,Znt. J. Engng. Sci. 1, 157-161 (1963). (Received 8 June 1964)
R&r&--L’article present& est une etude de l’ecoulement stable classique de certains fluides orientables qui tendent a se desorienter au repos. L’analyse de l’auteur est basee sur une theorie des fluides anisotropes, developpee par Ericksen et qui semble Ctre susceptible dune interpretation assez g&r&ale. L’auteur montre que les problemes presentes sont bien exposes mathtmatiquement et que, pour des gradients
de
vitesse
mod&es,
le
fluide
se
comporte
comme
un
fluide
visco-elastique,
laissant
prevoir
un
En faissant usage de la notion de fonctions materielles, l’auteur fait aussi remarquer, lorsque c’est necessaire, que les differents resultats obtenus sont en relation etroite avec ceux de Coleman et No11 concernant les “fluides simples”. comportement
similaire
a ceux
qui
sont
observes
dans
les
solutions
de
polymeres.
Zusammenfassung-Die vorliegende Arbeit untersucht einige klassische stationare Stromungen gewisser orientierbarer Fliissigkeiten, die im Ruhezustand dazu neigen, unorientiert zu sein. Die Analyse grtlndet sich auf eine von Ericksen entwickelte Theorie der anisotropen Fltissigkeiten, die in ziemlich allgemeiner Weise ausgelegt werden kann. Es wird gezeigt, dass die gestellten Probleme mathematisch gut formuliert sind, und dass bei mlssigem Geschwindigkeitszuwachs die Fltlssigkeit sich wie eine viskoelastische Fhlssigkeit verhllt. Dadurch llsst sich auf ein den Polymerlosungen lhnliches Verhalten schliessen. Der Begriff der Stoffunktionen wird dazu herangezogen, urn-wo dieses notwendig erscheint-auf die enge Verwandtschaft der erhaltenen Resultate mit den von Coleman und No11 fur ‘einfache Fhissigkeiten’ gefundenen Ergebnissen hinzuweisen. Sommario-La presente memoria verte sullo studio di qualche flusso costante classic0 di certi fluidi orientabili the tendon0 al non orientamento in fase di riposo. L’analisi B basata su una teoria di fluidi anisotropici sviluppata da Ericksen, the sembra essere d’interpretazione piuttosto generale. E’ mostrato the i problemi formulati sono matematicamente ben impostati e the a gradiente di velocita moderata il fluid0 si comporta come un fluid0 viscoelastico, predicendo il comportamento simile a quell0 osservato nelle soluzioni polimeriche. Impiegando la nozione delle funzioni materiali t: anche fatto presente, quando necessario, the vari risultati ottenuti sono in stretto rapport0 con i corrispondenti risultati di Coleman e No11 per “fluidi semplici”. ABcrparcr-Hacroamaa HeKOTOpbIX A~arm3
OCHOBbIBaeTCa
pIIpeTauIlA YeCKII ynpyras
pa6ora
OpHeHTUpyeMbIX KaxeTCa
XOpOmO
Ha
TeOpIIH
nOBOJIbH0
nOCTaBJIeHa
mnnKOCTb,
KaCaeTCa
XWAKOCTei,
II,
aHn30TpOWbIX
HeKoTopbIX
B COCTORHIIU
IIpeACKa3bIBaff
YYIITIIBaa
nOHRTIIe
nOJIyYeHHbIe
pa3Hbie
pe3yJIbTaTbI
“npOCTbIX
mAKOCTefI”.
nOBeneHKe
nOXOXCOe
MaTepIEUIbHbIX TeCHO
CBa3aHbI
&HKI@ C
KrIaccIIYecKHX
IIOKOX
WIAKOCTei,
cTaI.IIIoHapHbIx
CTpeMRTCa
pa3BnTyIO
o61ueZi. ,!lOKa3bIBaeTCa, YTO YTO Aria He6onbmux rpanIIeHTOB
rIOJIIiMepOB.
Henna nJrs
nccJIenOaaHnff KOTOpbIe
3pIIKCZHOM,
C@OpMyJInpOBaHHaa
TOMy,
CKOpOCTR KOTOpOe
yKa3bIBaeTCII,
COOTBeTCTByKWIMII
XCHnKOCTb
KOTOpaX
rAe
B WHTe-
npo6neMa MaTeMaTIIBeAeT ce6a KaK Ba3KO
Ha6rIIOnaeM TaM
TeYeHKk
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pp. 515-532. Pergamon Press 1965. Printed in Great Britain
ON CERTAIN STEADY FLOWS OF ANISOTROPIC FLUIDS P. N. KALONI Indian Institute of Technoiogy,
Kharagpur, India
Abstract-The present paper is concerned with the study of some classical steady flows of certain orientable fluids which tend to be unoriented at rest. The analysis is based on a theory of anisotropic fluids developed by Ericksen which seems to be rather general in interpretation. It is shown that the problems formulated are mathematically well set and that at moderate velocity gradients the fluid behaves like a viscoelastic fluid, predicting the behaviour similar to that observed in polymer solutions. Using the notion of material functions it is also pointed out, wherever necessary, that various results obtained are closely related with the corresponding results of Coleman and No11 for ‘simple fluids’.
1. INTRODUCTION
IN A SERIESof papers [l-7], Ericksen has formulated and developed a class of properly invariant theories of anisotropic fluids. Some of the results of these papers suggest that these theories cover a wide class of other theories of isotropic fluids, and describe some of them comparatively in a simpler and better way. Ericksen’s analysis [3,7] of simple shearing flow and homogeneous flows predicts that, at moderate velocity gradients, a class of these theories describes the behaviour similar to that observed in non-Newtonian fluids. For a different sub-class of these theories, which may be called the simplest theory of anisotropic iluids, Ericksen’s [1, 41 and Verma’s [8] analyses illustrate that the behaviour shown resembles that of Bingham materials and that, without using any yield criterion, one set of constitutive equations are sufficient to describe such materials. Further, Hand [9] has shown that his theory of dilute suspensions, which is obtained from Jeffery’s [lo] calculations for the motion of an ellipsoid suspended in a Newtonian Auid, falls as a special case of Ericksen’s theory [I] of anisotropic fluids. The object of the present work is to investigate the behaviour of such fluids in certain classical steady flows, using the general theory of anisotropic fluids [2]. The flow problems considered are: Poiseuille flow, Couette flow, helical flow and flow through pipes of arbitrary cross section.* Our analysis (including that of ffow between rotating cones, which will appear separately) reveals that the predicted behaviour is similar to that observed in viscoelastic fluids. In fact, it is observed that, if we follow the notion of material functions as has been introduced by Markovitz [11] or by Coleman and No11 112, 131,our calculations are simplified and yield results closely similar to those obtained by Coleman and No11 [12, 131for ‘simple fluids’ [14] and by Rivlin [I 51for Rivlin-Ericksen fluids [16]. Because of the vast literature on this subject, the references in this paper are limited to a few which are most closely related to our work. For the general background and an organized account of various isotropic theories we refer the reader to Eringen [17] with its extensive bibliography.
* Simple shearing flow and homogen#us flows with constant velocity gradients have already been analysed by Ericksen 13, 71.
516
P.N.
KALONI
Solutions for some of the flows that we consider here have already been analysed by Ericksen [I, 41 and Verma [8], but their derivations are based upon the simplest theory of anisotropic fluids [l], Recently, Green [18] has pointed out some difficulties with this type of theory on the grounds that it does not give a satisfactory definition of a ‘fluid’. We have therefore reworked these flows, along with others, using a more general theory [2] restricting our attention to fluids for which n+O when they are motionless. This restriction not only clarifies the doubts raised by Green [18] but has some physical precedent for it. In dilute solutions of high polymers at rest, the long-chain flexible molecules are supposed to be randomly coiled, and therefore define no preferred direction. In shear it is believed that they stretch out and align themselves at some definite angle relative to the streamlines. Tn the theory proposed this tendency towards alignment is represented macroscopically by associating with each particle of the fluid, a vector II of variable magnitude, which represents the direction of orientation and whose magnitude measures the amount of stretch. In equilibrium, i.e. at rest, we should therefore have n=O. 2. BASIC
EQUATIONS
The theory considered involves a vector II of variable magnitude which represents a preferred direction in the fluid and is related to the fluid velocity Ri by equations of the form Gi=wiknk+(P1
(2.1)
+Clzd,,n,n,)ni+llJdikn~.
Here dik and Oik are, respectively, the symmetric and anti-symmetric gradient tensor given by 2dij=ii,
j+ij,
i
2Oij=~_i,
j-~j,
i
parts of the velocity
(2.2)
while p’s are functions of n2=nknk to be determined by experiments, and dots denote material derivative. The constitutive equations for such fluids have the form tij = -p6ij + (1, + I,dr,n,n,)ninj + 2Aadrj+ 2J.,(dikn,nj + dikn,ni)l
(2.3)
where tii is the stress tensor, p an arbitrary pressure and A’sare functions of n2. We assume the fluid to be incompressible which means that velocity ii must satisfy the equation . xk,
(2.4)
k- -0.
Finally, we have the equations of motion which in the absence of body forces are
(2.5)
tij, j=pf,
where p is the mass density. In view of what equation (2.3) implies, we have seven equations (2.1), (2.4) and (2.5) to determine seven unknowns p, 1, and n,. The simplest theory of anisotropic fluids has been obtained by imposing the constraint n2 = 1, which implies that 2’s and p’s should then be constant. Also the consistency condition then requires that (2.6) Pl =O, p3=-p2* 3. POISEUILLE
FLOW
We consider the steady flow of the anisotropic fluid through an infinite circular pipe of radius a. We use cylindrical polar coordinates (I, 8, z) and assume the velocity field as
517
On certain steady flows of anisotropic fluids i=o, Assuming
O
i = w(r),
tL0,
(3.1)
that the fluid adheres to the walls of the pipe, we have the boundary condition w(a) = 0.
(3.2)
The physical components of the tensors dii and wii are 0
0
f/2-
0
0
0
f/2 -
0
0
I [dij]=
COijl = -
0
0
-fi
0
0
0
f/2
0
0
(3.3)
-
where f stdnds for doldr. On using (3.3) equation (2.3) becomes t11= -p+ab:+2&n,n,f t2*=
-p+aw:
133 = -p+ml:+21,n,n,f tl,=~flln,+[~3+~4(n:.+n:)]f
(3.4)
t,2=@)111112+L&n3f t,,=@n*n,+A,n,n,f
where @=(1,+Qz,n,)J
(3.5)
On assuming n=n(r, t) equation (2.1) gives
(3.6)
Equations (3.6) can be satisfied by taking n i=O and in that case the theory reduces to that of Newtonian fluid with viscosity A,(O). However, as is discussed by Ericksen [3, 71, such solutions are of less significance because of their unstable nature, in comparison to those with ni*O. In the analysis that follows we restrict our attention to fluids for which 1140 when there is no motion. Following Ericksen [7] we find that all components of n, tend to zero when the fluid is at rest and take steady values for Iarge times when the fluid is in motion, only if Pl(O)
&+
P3’ 19
llftan
$>O
(3.7)
where tan24=k.
-1
/13+1
(3.8)
518
P. N. KALONI
Assuming that the above conditions hold, we then have a critical rate of shear given by
(3.9) where ~~(0) and ~~(0) are the values of pi and jt3 evaluated at a2 =O. It then follows that (cf. Ericksen [7]) n+O when,f’
(3.11)
where N denotes that value of n which corresponds to stable steady-state orientation. In short, all the above indications are that at low rates of shear the fluid will behave like a Newtonian fluid, but will behave differentIy above the critical shear rate. We now solve equations (3.6) for the components nits for steady-state flow by putting &z,/&=O and assume that aforesaid conditions hold true. It is then a simple matter to verify that these equations have the solutions ltl = N sin 4,
112=o,
n,=Ncos
(3.12)
n,‘S
in the first equation of (3.6)
1) cot 4+pUziV2 sin 2tp]f+2p1=0
(3.13)
where 4 is determined by (3.8). On setting above values of we arrive at the equation [(yj-
cf,
which along with (3.8) determines the value of N in terms of motion. equation is solvable. Using (3.12) we then obtain t11=
-p+a,N’
(3.14)
sin2 ++Af
(3.15)
f22 = -p tj3=
-p+a&v cos2l#+Bf
(3.16) (3.17)
t13=C+Df t,,=t,,-
We assume that this
-0
(3.18)
where A, B, C and D are given by 2A=N2
sin 2c$[A,??2 sin’c#+&]
(3.19)
2B= N2 sin 2$[A,N2 COS~~+~~~]
(3.20)
2C= A,N 2 sin 24
(3.21)
4D=4(~~+~~~2)~~2N4sin2 and where N 2 is given by (3.13).
24.
(3.22)
On certain steady flows of anisotropicfluids
519
We now introduce the material functions defined as (C+Df)=cU) (&IV sin24 + Af) = PI(f)
(3.23)
(&IV2 cos2~+Bf)=p2(f) where LYand /3’s are functions of ,f’ only, as is evident from equations (3.19) to (3.22) and (3.13). Following the time honoured terminology we call, a(f )/f‘shear dependent viscosity’ and PI(f), P2(f) ‘normal stress functions’. We shall, however, for the present, postpone the discussion of these functions to a subsequent section, where we shall try to establish the relationship between these and others introduced in [II] and [12]. The equations of motion (2.5) in the absence of body forces reduce to
(3.24)
Assuming that a and /I’s are independent of temperature of (3.24) : p= -a’z+h(r),
&f)=
with
we have the following solutions
a’r b - yf +r
s
“(f)&.+L
W=Bl(f)+
r
(3.25)
(3.26)
where a’, b and L are constants, of which the middle one, i.e. b should be taken equal to zero, since at r=O, f must be bounded. The second equation of (3.25) is a differential equation for the velocity w(r) to be solved subject to (3.2). However, if we assume, which we shall justify later, that an inverse of the function cl(f) exists, then (3.25) can be written as (3.27) so that with the help of (3.2), o(r) turns out to be (I o(r)=
cl-l $
s ,
dr.
(3.28)
0
The total mass flux across the tube is given by Q=27rp “m(r)dr s
0
(3.29)
which on using (3.2) and (3.27) gives (3.30)
520
P.N.
KALONI
Thus from equations (3.28) and (3.30) it is clear to observe that the determination of both velocity and mass flux simply requires the knowledge of material function g(f). In other words, if the mass flux Q can be determined as a function of pressure head a’, we can calculate the material function ix(S) very easily. To complete the solution we now calculate the normal stresses. From equations (3.14) to (3.16), (3.25) and (3.26), we have (3.31)
(3.32)
(3.33) wherefis
given by (3.27)
It is interesting to observe from the above equations that both radial and downstream normal stresses are always in excesss over the classical values a’z of these quantities. It therefore follows that the well-known swelling phenomenon at the exit section of the tube, often referred to as the Merrington effect [ 191in the literature, will be accounted for qualitatively by this theory. At this stage, we now recall some interesting allied experimental results that have been reported in a series of papers 120-231by Middleman and Gavis in capillary jets of Newtonian and viscoelastic fluids. They have found that not only viscoelastic fluids, but also the laminar jets of Newtonian liquids, expand or contract from a capillary nozzle depending upon the fluid properties and jet velocity. These authors suggest that, besides the effect of surface tension, there are three main sources of normal stresses which can contribute to the observed changes in the jet diameter; viz. (i) a non linear normal stress developed in the tube (ii) a viscous normal stress due to the velocity profile relaxation outside the capillary tube (this exists in Newtonian fluids also) and (iii) a normal stress caused by elastic response to the velocity profile changes. Merrington 1191,on the other hand, attributes this phenomenon to the elastic recovery of the fluid stretched in the capillary tube, meaning thereby that only those class of fluids which possess elastic properties would be capable of predicting such changes. In the view of Middleman and Gavis, this conclusion is, therefore, not perfectly satisfactory as there are other equally important sources which are responsible for such changes. In fact, it has been pointed out by Middleman and Gavis [22] that at high ejection velocities and for highly viscous fluids, the internal normal stress is mainly responsible in comparison to these of the profile relaxation and external normal stresses. The effect of surface tension being, comparatively, negligible [23]. However, since the quantitative estimation of various normal stresses, which have been quoted above, is still incomplete, it is therefore difficult to assess the relative importance of these factors, till the question is resolved completely. To fix the ideas further, we assume, following Middleman and Gavis, that internal nonlinear stress is chiefly responsible for the swelling of the fluid. We suppose that the fluid is issued from the pipe into an atmosphere at pressure pO. The force exerted on the output cross section of the pipe, neglecting the effect of surface tension, is, therefore, equal to nazpo. At the exit section we must. therefore, have the following force balance system
521
On certain steadyflows of anisotropicfluids a
J
7ra2po=-27t
t,;rdr
0
and, which on employing the value of t,, from (3.33) gives xa'pO=-7z s
(3.34)
"12p,(f.)-Bl(f)frd~+na2~. 0
Eliminating the constant L. between (3.31) and (3.34) we get (3.35) Let again P = (-t,,)y =a, i.e. the force per unit area that the fluid exerts on the pipe. Then from (3.35) we get
J
P-P,=$
(3.36)
a @P2V)--1303>rdr. 0
Equation (3.36) demands that for a stream of fluid to swell at the exit section, we must have 2Mf+
(3.37)
~~(~j
where f is given by (3.27). Hence, we conclude that for a fluid under consideration to show the swelling effect the material functions (3.23) must satisfy the condition (3.37). 4. COUETTE
FLOW
We consider the steady flow of the fluid contained between two infinite coaxial cylinders of radii rl and r2(r2>rl) which are rotating about the common axis with constant angular velocities R, and C& respectively. We use cylindrical polar coordinates (r, 8, z) taking zaxis to be the common axis of the cylinders and assume the velocity field as
$=O,
~=~~(r),
i=o
(4.1)
which satisfy (2.4). Assuming that the fluid adheres to the walls we have the boundary conditions as w(r*) = R,. 0@1)=Q*, (4.2) The physical components of the tensors d, and wU are 0 Cdij]=
A2
0 .-
.f/2
0
0-
O
O
0
0
,
-i_ (f-t-20)
[Oij]= i(f+2W)
0 -
0
0
0
0
0
(4.3)
!
where f now stands for r &D/C%. On using (4.3) the stress components given by (2.3) reduce to the same expressions as given by equations (3.4) with suffixes 1, 2, 3 now replaced by r, z, 8 in that order. Also on assuming n=n(r, t), the equations determining the physical components of n,‘s reduce to the same equations as given by (3.6) with the suffixes 1, 2, 3, now replaced by r, z, 8, in that order. For the stable steady-state values of ni)s, corresponding to N+ 0, we therefore have the same restrictions as given by equations (3.7) to (3.11), where now _f denotes r do/dr. The values of Ri’S then turn out to be
522
P.N.
n,=N
KALONI
n,=Ncos
sin t$
q5
n,=O
(4.4)
where 4 and N2 are given by equations (3.8) and(3.13) respectively. The stress components then reduce to the same expressions as given by (3.14) to (3.18) with the suffixes replaced in the order stated above. The dynamical equations (2.5) now take the form (4.5)
(4.6)
(4.7) where p is the density of the fluid and g is the acceleration due to gravity. An elementary analysis shows that the symmetrical solutions of the above equations are - t,, = p(r,z) = pgz + h(r)
(4.8)
u,=$ W=Pdf)+
s
(4.9)
'Bl(fw32(f)dr+
r
pro2
f
(4.10)
where K is the torque per unit height required to maintain the relative motion of the cylinders. For a known value of K, equation (4.9) along with (4.2) is a differential equation to determine the velocity profile o(r). However, as before, if we assume the existence of the inverse of a(f), then we get (4.11) which on integration and using (4.2) gives (4.12) where K is determined from the equation (4.13) Thus, if the material function LX(~)is known, equation (4.12) determines the velocity profile, and (4.13) determines the differences of angular velocities of the cylinders as a function of torque per unit height K. Amongst other expressions, which also occur in the case of Newtonian fluid, equation (4.8) shows the additional variation of the vertical normal pressure which is given by
523
On certain steady flows of anisotropic fluids
P’w=Bl(f>+
r~1(f)-~2(f)dr+COnStallt .r
(4.14)
I
wherefis given by (4.11). This expression is therefore responsible for the so-called Weissenberg climbing effects [24]. It follows, therefore, that pressures must be applied perpendicular to the axis of rotation in order to keep the upper surface horizontal. If such pressures are not present the fluid will tend to rise or sink, the exact shape of which will depend upon the function p’(r). 5. HELICAL
FLOW
The problem of helical flow for simple fluids [14] and for Rivlin-Ericksen fluids [16] has been studied, respectively, by Coleman and No11 [ 131and Rivlin [ 151. Our aim here is to analyse this problem for anisotropic fluids and to show that similar results can be obtained by much simpler calculations. We consider the steady motion of the fluid contained between two coaxial cylinders of radii rl and rz(rz > I~) which are rotating about their common axis with angular velocities R, and Sz,, respectively, and a pressure gradient and/or a gravitational force is acting on the fluid parallel to the axis of rotation. We again use cylindrical polar coordinates (r, 8, z) taking the z-axis to be the common axis of the cylinders and assume the velocity field as i=o, i=u(r). 4=rw(r), (5.1) which satisfy (2.4). Assuming that the fluid does not slip at the solid boundaries, the boundary conditions take the form u(rr) = u(rJ = 0.
W(Tz)= %,
c-01)=%
(5.2)
The physical components of the tensors d, and oy are 0
_.
--I
Yl
Yz
2
2
0
Yl
Cdijl = 2
0
0
0
0
9
Ewijl =
(?+CO> -72 (1)
Y2 2
(5.3) 0
0
_
_I
where du do . y1 = r dr and y2 = do’
(5.4)
On using (5.3) the stress components reduce to t,, = -p+cDn;+2L&y, tzz= -p+@r$+2L,n,n,y, t,,=
++yz)n,
-p+@r;+2a,n,n,y,
tl,=~,y,+~nln,+~4(n2-n32)yl+;l,n2n3y, t13=a3~2+cDnln3+~,~n2-n~~~2+~,n2n3~l t23=~'n2n3+~4(yln3+Y2n2)nl
(5.5)
524
P.N.
KALONI
where @=
(4 +J,(n,y, +n,y,)i.
(5.6)
Assuming n = n(r,t), equations (2.1) give
an1 dt=
dn, -= at
( 1 cL3-1
-
c
2
P3+1 -
2
(Yrn,+y,n,)+[C1l+~zfYtn,+y,n,)n,ln,
(5.7)
n~y~+[~~+~,(y,n,+y~n3)n,ln,
(5.8)
>
an, Cr,+l
( >
at= - 2
nly2 +
[k
+~2(hn2fy2n3hln3.
(5.9)
We again restrict to fluids for which n-0 at rest and assume N to denote that value of n which corresponds to stable steady-state orientation. To determine the stabte values of N, we write (cf. Ericksen 171) ni=Ni+mi
(5.10)
and linearize equations (5.7) to (5.9) with respect to mi and then determine the stability conditions. We thus observe that all solutions of the above equations will approach N=O in time, only if /p3(0)l > I and y <
-2P1(o) MO)- 1>*
(5.11)
where (5.12)
We notice that conditions (5.11) are satisfied for small values of y. If its value exceeds the critical value yCgiven by 4&(O) (5.13) y,z= @L:(O)- 1) the associated configuration will be unstable. It therefore follows that if conditions (5.11) hold true, then the ffuid will behave like a Newtonian fluid and when (5.1 I) fails then N=O will represent unstable con~guration. We are, however, interested, in those solutions which correspond to stable steady state N+O when (5.11) is violated. For this we linearize equations (5.8) and (5.9) with respect to mi, which reduce to am,
-_=
(5.14)
at
am3 P3fl -= --7j-~l~2+[IJil~1?2~~I~2+~2~3)~llm3 at
( 1
(5.15)
where bars over p’s denote their values at PZ’= N2. On multiplying (5.14) by yZ and (5.15) by y1 and then subtracting we get 84 at=C~,+~2(~1N2+~2N3)N115
(5.16)
525
On certain steady flowsof anisotropicfluids
where <=(m,y, -m,y,)+O. This equation gives one of the necessary conditions, showing that for a stable value of N we must have (5.17) Combining (5.17) with (5.8) (after setting l&/at (5.17) as
=0
in the latter) we get an alternative to (5.18)
So long as above conditions hold, we can solve equations (5.7) to (5.9) for stable values of ili’S by putting h,/lJt=O. We thus get n, = N sin 4,
n,=Nsin
n,=Ncosecos&
0 cos 4
(5.19)
where tan e=e
(5.20) Yl
and I#Jis determined by (3.8). Using (5.19) equation (5.8) then gives [(pa+ 1) tan 4+p2NZ sin 24]y+2p, =O.
(5.21)
which along with (3.8) and (5.12) determines N2 in terms of motion. To summarize what has been said above, we notice from equation (5.21) that if the conditions (5.11) hold and when y < y. then N does not take the real values until y reaches its critical value yc given by (5.13) and that in the latter case it tends to zero. However,if y 2 yc (y not exceeding too much from yc cf. Ericksen [7]) then N turns out to be a real vector having the components given by (5.19). Assuming that the above conditions for real values of N hold true and that yr and yZ are well-behaved inside the annulus, the stress components on using (5.19) become t11=
-P+P,(Y)
(5.22)
*,,=
_p+fi’oyf
(5.23)
_,+!&,:
t2,=
Y
f12=-Y,
t23
44 Y
7
_b2(y) -2YlY2 Y
U(Y)
t13 =-Y2
(5.25)
Y
(5.26)
where a and p’s are the material functions introduced earlier and y’s are given by equations (5.4) and (5.12). It is clear from the above equations that stresses are functions of r only, except for a dependence on z occurring in p. Hence the dynamical equations (2.5) when transferred to cylindrical polar coordinates take the form
526
P. N.KALONI
t,r-
--
be
= _pro2
I‘
dl,e 24, - +-=o dr
(5.27)
r
dr,,
t,, dr,__ -& + ;+z f&7,=0
where g, is the z-component of the gravitational field. An elementary inspection of the above equations suggests that these are satisfied only if p=p(r,
(5.28)
z)=Pz+h(r)
4Y)
trs=~ y1=
K
(5.29)
g2
(5.30)
and (5.31) where K is the torque per unit height required to maintain the motion of the Q and K, are constant while P= (+/a~ - pgZ)is a constant which may be called the pressure gradient’. Equations (5.29) and (5.30) are a pair of simultaneous non-linear differential to determine w(r) and u(r) subject to the boundary conditions (5.2). From these we observe that
cylinders, ‘frictional equations equations
(5.32) so that a(y)=&r)
(5.33)
or y =a-'[G(r)]
where 6(r) is purely a function of r only. It is thus possible to write (5.29) and (5.30) in the form do K cc-‘[G(r)] -=(5.34) dr 2nr3 a(r) du
(5.35)
-= dr
These equations along with the boundary conditions (5.2) give
s
w(r)=$rI, ~-llfi(r)ldr+R r36(r)
(5.36)
u(r)=
(5.37)
j:,
{(~+~)‘-~~l)dr
1
527
On certain steady flows of anisotropic fluids
where the constants involved in these equations are to be evaluated with the help of following equations rz ~-‘C4’>1,, &-R,=K 27rs I, r36(r) (5.38)
The volume discharge rate per unit time is given by (5.39)
Q=27rnp ‘*m(r)dr s r, which on using (5.35) and (5.38) turns out to be
(5.40) From the above equations it therefore follows that velocity profile and volume discharge rate, once again, can be determined by the simple knowledge of the material function c+J). The pressure distribution at any point is given by
Ypr~2+Kl
p(r, 2)= Pz + PI(Y)+
(5.41)
where w(r,), yl, y2 and y have the values already obtained earlier. Introducing this value of p in equations (5.22) to (5.24), it is straight forward to determine the normal stresses and their differences without any difficulty. 6. FLOW
THROUGH
A NON-CIRCULAR
PIPE
We now consider the flow of anisotropic fluids through tubes of arbitrarycross section under uniform pressure gradient. Several years ago Ericksen [25] considered this problem for Reiner-Rivlin fluids [26, 271 and showed that rectilinear flow of such fluids is possible only for special shapes of the tube and that in general (except when the coefficients that occur in the constitutive equation satisfy certain definite conditions), it is accompanied with a secondary flow. The significance of this conclusion has been examined by a number of authors [28-311 for various other theories and have obtained the results in agreement to the findings of Ericksen [25]. Our interest here is to examine the possibility of rectilinear flow in the case of anisotropic fluids.
We use Cartesian coordinates (x, y, z) and assume the velocity field as k=O,
I;=o,
i =f(x, y).
(6.1)
which satisfy (2.4). The physical components of the symmetric and antisymmetric velocity gradient tensors are 0 0 0 -Y2/2 [dij]
=
0 Y2P
0
hiI =
0 Y2P
0
-Y1/2
Y1/2
0
(6.2)
P. N. KALONI
528
where now (6.3)
On using (6.2) the stress components given by (2.3) reduce to the same expressions as given by equations (5.5) with suffixes 1,2, 3, now replaced by z, y, x in that order, and where now yt and yz are given by (6.3) Assuming n=n(x, y, t) equation (2.1) gives (6.4) dn y=
( ) c1&
at
2
an,_ dt
-
!
‘+
Yllrz+ [cl, +C(2(Y2n,$yln,)n,ln,,
(6.5)
(~2n,+yl~~,)+[~l
(6.6)
+~2(Y2~x+Ylny)n,]nZ.
>
Stability conditions for stable steady state N=i=Oin the present case can also be worked out in the previous section. We find that the conditions (5.11) to (5.13), (5.18) and the discussion presented these still hold true except that now y stands for (6.7) It is thus possible to obtain the steady-state solution of equations (6.4) to (6.6) as n,=Nsinflsin4,
n,=NcosUsincp,
nz=
Ncos&
(6.8)
where C#Iis given by (3.8) and 0 is determined by (5.20) with y1 and y2 now given by (6.3). With the above values of ni’s, the equation determining N in terms of motion again reduces to (5.21) where now y has the value given by (6.7). On substituting the values of n,‘s from equation (6.8) in the equation determining the stress components we obtain the same expressions as given by (5.22) to (5.26) with the suffixes 1,2,3, now replaced by z, y, x in that order, where y’s have the new values defined in this section. The equations of motion in the absence of body forces are
“,‘; atxY-() 1
aY
3&c, i)x+-=O
ai,,
(6.9)
JY
%z at,, z+-=O ay where it is evident from (6.1) that such motions are accelerationless. expressions for stress components in the above equations we get
Employing the
On certain steady flows of anisotropic fluids
529
(6.10)
where ,=!z!$) Y (6.11)
and @=a(y) -. Y
Equations precisely of the same form, as above, have been analysed by Ericksen [25]. From his analysis it can therefore be concluded that for a solution of equations (6.10) to exist, we must have a constant q and a function R(f) such that
a a 4=&w,)+-m)
(6.12)
aY
(6.13) and that, then P = N_fl +.kW2)
(6.14)
+qz.
Equations (6.12) and (6.13) are two partial differential equations to determine a single unknown functionf(x, y). Ericksen [25] has shown that when x and 0 are independent functions, the rectilinear motion is not physically possible except for rigid motions and for motions in which the curves of constant speed are circles or straight lines. For a special value of 0, @=m/y where m is a constant, Stone [32] has shown that curve~=constant are straight lines if q=O and circles of radius l//mq] if q=l=O. However if x and 0 are not independent but are linearly related such that x=MO
(6.15)
where M is a constant, then all solutions of (6.12) satisfy equation (6.13) with R(f)=Mq..f, so that the ffow in such cases is rectilinear. In our present analysis under general conditions, neither (6.15) holds nor 0 has the form ~~~ unless or(y)is constant. It therefore follows that for each choice of q, the solutions of (6.12) will not satisfy (6.13) for any choice of R(f) and that the flow cannot be rectilinear. Although it is rather difficult to conclude at this stage, that secondary flows will always be observed in the flow of anisotropic fluids through the tubes of arbitrary cross section, but the results discussed here and those of isotropic fluids suggest that such inferences are likely to be correct. 7. SOME REMARKS
ON THE MATERIAL
FUNCTIONS
In the course of our present study we have seen that the three material functions introduced by us through equations (3.23), determine velocity profile, stress and pressure distribution completely. In the present section we shall compare these, making some observations, with those introduced by Coleman and No11 [12] in the theory of ‘simple
530
P. N. KALONI
fluids’ [14], which is thought to be the most general fluids reported
hitherto
and appropriate
theory
of isotropic
[33].
We begin our discussion with the remark that the existence of our material functions, which are merely the differences of the normal stress functions, ascertains that, in general, neither the famous conjecture due to the Weissenberg [34] nor the symmetry relation predicted by the Reiner-Rivlin theory, regarding the normal stress function, hold true. This is in agreement with the findings of Markovitz and Brown [35]. We also observe that even for complex flows, such as those which have been discussed in sections 5 and 6, our three material functions are sufficient to determine the stresses and velocity profiles completely. It follows, therefore, that if experimental measurements are performed with any one of the simpler solutions to determine these functions, then complete velocity profiles and stresses can be predicted for other complicated flows. From equations (3.13) to (3.23), and remembering the fact that n-+0 when the fluid is at rest, we find that all material functions vanish iff=O, i.e. a(0) = PI(O) =&(O) = 0.
(7.1)
Also from the same equations it is noticed that if we substitute f= -f in the equations determining these material functions, then LX(~)changes its sign whereas pi(f) and /IX(f) remain unaffected. The second law of thermodynamics requires the energy dissipation to be positive. This means that f&(f)>0 for f* 0. (7.2) Since c$f) is an odd function 0f.h (7.2) implies that f and g(f) must have the same sign. All the above results are in agreement with [ 121. In the foregoing analyses we have not emphasized the fact that derivatives of a(f) and c1-‘(f) will be discontinuous of f=f,,the discontinuity generally being finite. This rather weak discontinuity does not invalidate the solutions. Whether it occurs naturally depends on, whether singular value f,is attained in the flow regime. That such a singular value does occur in some of the flows discussed can be accounted for in the following manner. In the case of analysis of section 3, for example, we have from equation (3.2), w(r)=0 at r = a. Hence, unless w is identically equal to zero, it will take on an extremal value in the interior of flow field where f= &.@r will become equal to zero. Nearby, we must, therefore, havef
531
On certain steady flows of anisotropic fluids
In short, this rather sketchy comparison shows that the material function introduced by us have, more or less, similar properties as those introduceds in [12]. It is, therefore, obvious that the analyses presented in [ 12, 131where more stress has been laid to determine these functions from an experimental point of view, will hold true in our case also. We
feel it unnecessary to repeat those arguments here, once again. The material functions we have used in the present paper bear the following relationship with others introduced in [12, 111 and in a slightly different form in [351. This is recorded below where the functions are written in the same order as the references cited above : Pl(f)=tll-t33=Zl(k)=V1-V2=Viii_Vi P2(f)=t22-tJJ=Z2(k)=V1-V3=Vii*-Vii a>
= t, 2 = G)
= 3V4(3)
(7.3) = $Jtl(j)
f=k=j.
Acknowledgements-1 especially wish to thank Prof. J. L. Ericksen for sending me all his reprints and clarifying certain aspects of his theories. I also wish to thank the referee for certain comments and helpful suggestions. Finally I express my gratitude to Dr. G. Bandyopadhyay aad Prof. B. R. Seth for their advice and encouragement.
REFERENCES [l] J. L. ERICKSEN,Koll. Z. 173, (117-122) (1960). [2] J. L. ERICKSEN,Arch. Rut. Mech. Anal. 4,231-237 (1960). [3] [4] [5] [6]
J. L. J. L. J. L. J. L.
ERICKSEN, Trans. Sot. Rheol. 4,29-39 (1960). ERICKSEN,Arch. Rut. Mech. Anal. 8, l-8 (1961). ERICKSEN, J. Polyrn. Sci. 47, 327-333 (1960). ERICKSEN, Proc. 7th Congress of Theoretical and Applied Mechanics,
Bombay, India 7,21 l-218
(1961). [7j J. L. ERICKSEN,Trans. Sot. Rheol. 6,275-292 (1962). [8] P. D. S. VERMA,Arch. Rut. Mech. Anal. 10, 101-107 (1962). [9] G. L. HAM), Arch. Rut. Mech. Anal. 7, 81-86 (1961). [lo] G. B. JEPPERY, Proc. roy. Sot. A102, 161-179 (1922). [ll] H. MARKOVITZ,7hzs. Sot. Rheo. 1, 37-52 (1957). [12] B. D. COLEMANand W. NOLL, Arch. Rut. Mech. Anal. 3,289-303 (1959). [13] B. D. CoLEMANand W. NOLL, J. appl. Phys. 30,1508-1512 (1959). [14] W. NOLL, Arch. Rut. Mech. Anal. 2, 197-226 (1958).
[15] R. [16] R. [17] A. [18] A.
S. RIVLIN,J. Rat. Mech. Anal. 5, 179-189 (1956). S. RrvLIN and J. L. ERICKSEN, J. Rat. Mech. Anal. 4, 323-425 (1955). C. ERINGEN,Non-linear Theory of Continuous Media, McGraw-Hill, New York (1962). E. GREEN,Proc. Cur&. Phil. Sot. 60, 123-128 (1964).
[19] A. C. MERRINGTON, Nuture 152,663 (1943). [20] S. MIDDLEMAN and J. GAVIS,Phys. Fluids 4, 355 (1961). [21] S. MIDDLEMANand J. GAVIS,Phys. Fluids 4, 963 (1961). [22] J. GAMYand S. MIDDLEMAN, J. Appl. Polymer. Sci. 7,493 (1963). [23] J. GAV~S,Phys. Fluids (1964). [24] K. W~~ENBERG, Nature 159, 310 (1945). [25] J. L. ERICKS~, Quart. appl. Maths. 14,319-321 (1956). [26] M. REINER,Amer. J. Math. 67, 350-362 (1945). [27] R. S. RIVLIN,Nature 160,611-613 (1945). [28] W. LANGLOIS and R. S. RIVLIN,Tech. Report No. 3, Div. appl. Math.. Brown University (1959). [291 A. E. GREENand R. S. RIVLIN, Quart. uppI. Moth. 14,299-308 (1956). [30] W. LANGLOISand R. S. RIVLIN,Rendiconti di Matematica 22, 169-185 (1963). [31] W. 0. CRIMINALE JR., J. L. ERICK~EN and G. L. FILBEYJR., Arch. Rat. Mech. Anal. 1,410-417 (1958).
532
P. N. KALONI
[32] D. E. STONE,Quart. appl. Mark 15, 257-262 (1957). [33] C. TRUESDELL,Principles of Continuum Mechanics, Socony Mobil Oil Co. Inc., Dallas (1960). [34] K. WEISSENBERG, Proc. 1st ht. Rheo. Congress, Amsterdam 1, 29-45 (1949). [35] H. MARKOVITZand D. R. BROWN,To appear in Proc. Z.U.T.A.M. Symp., Haifa, Israel (1962). [36] J. L. ERICKSEN,Znt. J. Engng. Sci. 1, 157-161 (1963). (Received 8 June 1964)
R&r&--L’article present& est une etude de l’ecoulement stable classique de certains fluides orientables qui tendent a se desorienter au repos. L’analyse de l’auteur est basee sur une theorie des fluides anisotropes, developpee par Ericksen et qui semble Ctre susceptible dune interpretation assez g&r&ale. L’auteur montre que les problemes presentes sont bien exposes mathtmatiquement et que, pour des gradients
de
vitesse
mod&es,
le
fluide
se
comporte
comme
un
fluide
visco-elastique,
laissant
prevoir
un
En faissant usage de la notion de fonctions materielles, l’auteur fait aussi remarquer, lorsque c’est necessaire, que les differents resultats obtenus sont en relation etroite avec ceux de Coleman et No11 concernant les “fluides simples”. comportement
similaire
a ceux
qui
sont
observes
dans
les
solutions
de
polymeres.
Zusammenfassung-Die vorliegende Arbeit untersucht einige klassische stationare Stromungen gewisser orientierbarer Fliissigkeiten, die im Ruhezustand dazu neigen, unorientiert zu sein. Die Analyse grtlndet sich auf eine von Ericksen entwickelte Theorie der anisotropen Fltissigkeiten, die in ziemlich allgemeiner Weise ausgelegt werden kann. Es wird gezeigt, dass die gestellten Probleme mathematisch gut formuliert sind, und dass bei mlssigem Geschwindigkeitszuwachs die Fltlssigkeit sich wie eine viskoelastische Fhlssigkeit verhllt. Dadurch llsst sich auf ein den Polymerlosungen lhnliches Verhalten schliessen. Der Begriff der Stoffunktionen wird dazu herangezogen, urn-wo dieses notwendig erscheint-auf die enge Verwandtschaft der erhaltenen Resultate mit den von Coleman und No11 fur ‘einfache Fhissigkeiten’ gefundenen Ergebnissen hinzuweisen. Sommario-La presente memoria verte sullo studio di qualche flusso costante classic0 di certi fluidi orientabili the tendon0 al non orientamento in fase di riposo. L’analisi B basata su una teoria di fluidi anisotropici sviluppata da Ericksen, the sembra essere d’interpretazione piuttosto generale. E’ mostrato the i problemi formulati sono matematicamente ben impostati e the a gradiente di velocita moderata il fluid0 si comporta come un fluid0 viscoelastico, predicendo il comportamento simile a quell0 osservato nelle soluzioni polimeriche. Impiegando la nozione delle funzioni materiali t: anche fatto presente, quando necessario, the vari risultati ottenuti sono in stretto rapport0 con i corrispondenti risultati di Coleman e No11 per “fluidi semplici”. ABcrparcr-Hacroamaa HeKOTOpbIX A~arm3
OCHOBbIBaeTCa
pIIpeTauIlA YeCKII ynpyras
pa6ora
OpHeHTUpyeMbIX KaxeTCa
XOpOmO
Ha
TeOpIIH
nOBOJIbH0
nOCTaBJIeHa
mnnKOCTb,
KaCaeTCa
XWAKOCTei,
II,
aHn30TpOWbIX
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