Rheologica Acta Rheol Acta 32:490-498 (1993)
Flow of a non-Newtonian fluid between intersecting planes of which one is moving
Y.-N. Huang 1, R.K. Bhatnagar 2, and K.R. Rajagopal 1
1 Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA 2 Department of Mathematics, University of Pittsburgh, Greensburg, Pennsylvania, USA
Abstract: We study the flow of an Oldroyd-B fluid between two intersecting plates, one of which is fixed and the other moving along its plane. This problem was first considered by Strauss (1975) for the Maxwell fluid using a similarity transformation. We find that even in the case of a Maxwell fluid, which can be obtained by setting a specific parameter, say a, in the Oldroyd-B model to zero, our results disagree with those of Strauss (1975). We find that circulating cells are present, adjacent to the stationary plate while Strauss (1975) finds them adja- cent to the moving plate. We also delineate the effect of the coefficient a, which is a measure of the elasticity of the flow, on the flow pattern. We find that an increase in the elastic parameter reduces the cellular structure.
Key words." Oldroyd-B fluid - viscoelasticity - relaxation time - retardation time
Introduction
Fluids of the differential and rate type (cf. Truesdell and Noll, 1965) have been used quite suc- cessfully for modeling the behavior of dilute polymeric fluids and biological fluids. Amongst the many rate type models that are used, one that has gained considerable acceptance is the Oldroyd-B fluid (cf. Oldroyd, 1950). Several boundary value problems have been studied within the context of this model, and some of these results compare favorably in a qualitative sense with the experimental d a t a , when available. One such example is the interesting theoretical and experimental study of Strauss (1974) wherein he studies the converging flow of a Maxwell fluid between a pair of fixed intersecting planes; here, by converging, we mean flow towards the apex of the intersecting planes. Strauss (1974) found good qualitative agreement between theory and experi- ment. He allowed for the flow to be non-radial, and studied the problem by means of series expansion which is singular at r = 0, the point of intersection of the two planes. The study of Strauss (1974) has been recently extended by Bhatnagar, Rajagopal and Gup- ta (1993) for the exact same geometry and boundary conditions as the study of Strauss (1974), to the case of an Oldroyd-B fluid. They find that the pattern of the flow undergoes marked changes as the Reynolds
number is increased beyond the range of values con- sidered by Strauss (1974). The elastic properties of the fluid hasten the change in the flow structure in that these changes occur at much lower Reynolds numbers, in the presence of increasing elastic effects. While the flow at very low Reynolds numbers (Re<19.67) has a two-cell structure, at higher Reynolds numbers they have a four-cell structure.
In a later study, Strauss (1975) analyzed the flow of a Maxwell fluid between two intersecting planes, one of the planes being allowed to move (cf. Fig. 1). Here, we extend his work in two ways. Firstly, we study the effect of the elastic parameter a = A2/A l, where A 1 and A2 are the relaxation and retardation time, respectively, by considering the flow of an Oldroyd-B fluid, and secondly we carry out our calculations at much higher values of the Reynolds number to see if there are any dramatic changes in the structure of the flow, as was found in the previous study (cf. Bhat- nagar, Rajagopal and Gupta, 1993). Our results show in the case of a Maxwell fluid (a = 0), though our equations agree with those of Strauss' (1975), our results are completely different. While Strauss' (1975) finds circulating cells adjacent to the moving plate, we find circulating cells adjacent to the stationary plate. At this juncture, it would be appropriate to point out that Strauss (1975) solved his exceedingly complicated equations analytically while we solved the same equa-
Huang et al., Flow of a non-Newtonian fluid between intersecting planes of which one is moving 491
tions numerically. When our results proved to be dif- ferent from Strauss (1975) we realized we had to check our numerical calculations very carefully. We set the moving plate velocity to zero and allowed for the flow to be that between two fixed plates due to a prescribed flow rate, namely the problem studied pre- viously by Strauss (1974) for the Maxwell fluid, and Bhatnagar, Rajagopal and Gupta (1993) for an Oldroyd-B fluid. The results agreed perfectly and this gave us some confidence regarding the numerical work.
2. Equations of motion
The Cauchy stress T in an Oldroyd-B fluid is given by (cf. Oldroyd, 1950)
T = - p l + S , (2.1)
S + A 1 ( S - L S - S L T) =/2 [A1 +A2 (AI - L A I
- A l L r)] , (2.2)
where - p l is the spherical part of the stress due to the constraint of incompressibility, L is the velocity gradient, the dot denotes material time derivative,/2 is the viscosity, and A 1 and A 2 material time con- stants referred to respectively as relaxation time and retardation time. The kinematical tensor A1 is defin- ed through
A I = L + L T , (2.3)
where
L = grad v . (2.4)
We notice that when A1 = A2 = 0, the model (2.2) reduces to the classical linearly viscous fluid model. When A2 = 0, the model reduces to a Maxwell fluid.
The fluid is incompressible and hence it can only undergo isochoric motion, and thus
div v = 0 . (2.5)
For the flow between two intersecting plates, one of which is moving (cf. Fig. 1), we shall assume the following forms for the velocity field and the com- ponents of the extra stress:
v = u(r,O)e~+v(r,O)eo , (2.6)
Srr = Srr(r, O) , Sro = Sro(r, O) , Soo = Soo(r , O) , (2.7)
in a cylindrical-polar coordinate system (r,O,z), er and e0 being the unit vectors in the r and 0 directions.
It follows from (2.5) and (2.6) that the conservation of mass reduces to
~r 8v ( r u ) + - - = div v = 0 . (2.8) 80
The balance of linear momentum
a/v div T + ~ b = ~ - - , (2.9)
dt
reduces to (neglecting the body force ~9 b)
_ _ 1 8Sro 81) + OSrr q- Srr - Soo + 8r 8r r r 80
= 6 8u v 8u ;2.]
U--+
8r r 80 (2.10)
_! op +as o +2 Sro+ Z OSo__ o r 80 Or r r 80
[uSU+VSU uv] = e L 8r rO0 '
(2.11)
8p 0 . 82
(2.12)
Substituting (2.5) into (2.2) yields
[ ( u 8 + V 8"] 8u 2 8 u S Srr+A1 \ ~ 7-~// Srr--Z~fSrr------ r 80 rO
=2/28u+2Az/2[(uS+V-~)8r \ 8rr r 8('8r)
- 2 -- _----7_ + r - - - v , (2.13) 8r
soo+Al[( +vo ov Or r 8 0 / / S ° ° - 2 -~ Soo
2U 80 S°° + 2 ( v - r Svx~ Or./
492 Rheologica Acta, Vol. 32, No. 5 (1993)
\ 00/j
+ 2A2/x U - - + +__1 0V Or
r 0 0 \ r rO--0/ r \ r r
+r '5 v-r -~r \--Or-r O r - / J ' (2.14)
°u
\ Or rO0,/ w S°°
+ -- S r r r Or/
=1 (8,,+rSV - v) \oo Or
Or r O0/ C.' - / )
r Or \Or Or /
_l_(3 0U+rOV v) (U+l Ov']] r \ - ~ -~-r- \ r r~ - -0 , / J " (2.15)
We shall find it convenient to non-dimensionalize the above equations by introducing
r u P r - , a = - - , p = - - , (2.16)
UA1 U (I-I/A1)
~ j_ Sq (2.17) Cu/A1)
We also introduce the flux
Q= j urdO, (2.18) - -a
and
Q = Q (2.19) U2AI
For convenience, we shall drop the bars that appear over the various quantities*).
Finally, we introduce the stream function qs(r,O) through
1 0 ~ O~ u = - , v = - - - (2.20)
r O0 Or
Since
~u(r,a ) - q/(r, - a) = - Q , (2.21)
we use
q/(r, a ) = ----Q , ~u(r, - a ) = ---Q . (2.22) 2 2
We shall seek solutions for q /and Si} of the form
q / ( r ,0 )= ~ q/(n)(O) , (2.23) n= -1 r n
n = 1 r n f n n = l
Soo= ~ c,,(O) n = l r e
(2.24)
The series solution is not a perturbation in which successive terms in the series are multiplied by a higher order term in the perturbation parameter, say e. When e is sufficiently small, then it might be suffi- cient to carry out the perturbation up to linear or quadratic terms in the perturbation parameter, and yet capture the property of the function. In fact, by the very nature of the series, it is highly unlikely that the series will converge when r ~ 1, and we do expect the solution to be singular at r = O. However, when r~> 1, the solution is probably quite reliable as n in- creases, as the effect of successive terms are less sig-
*) However, when we find it necessary to discuss the properties of the streamlines and distinguish between f and r, we shall do so by retaining the bar and explicitly stating this. When it is apparent from the discussion that there is no need for the bar by virtue of the context, we shall drop it.
H u a n g et al., F low o f a n o n - N e w t o n i a n f lu id be tween intersect ing planes o f which one is mov ing 493
nificant. In fact, we find that max {i I/J(1) (0)11 < 0.053. Moreover , if the streamline patterns at r_> 1 are valid, we can indeed get a picture of flow in the whole do- main f rom the tendencies suggested by the results at r _ l . Of course, the streamlines por t rayed in the figures for f_< 1 may not have the precise structure that has been depicted. Also, since we have non- dimensionalized the radial coordinate r by dividing by
q ] /~ l , i.e., f = r/] /qA1, it is possible that the solu-
t ion holds for values o f r < 1, provided q ] / ~ l is suf- ficiently small, and thus the result will be useful in a much larger domain. The smaller the value of
q ] /~ l , the larger the domain of possible applicability for our solution.
It follows f rom (2.20) and (2.23) that
u(r,O) = q/~_l)(0) + ~'~°)(0) + q/ii)(0) + W~z)(0) + F #.2 F3 " " "
q/(n)(O) = ~ (2.25)
n = -1 Fn+l
and
w(1) (0) v(r,O) = - w ( _ l ) ( 0 ) + r2
+ 2 qJ(2) (0) + 3 gJ(3) (0) + .... r 3 r 4
n g](n) (O) = S (2.26)
n= -1 r n+I
The adherence boundary condit ions imply that
u ( r , a ) = 0 , (2.27)
u ( r , - a ) = - I , (2.28)
v ( r , a ) = 0 , (2.29)
v(r, - a ) = O , (2.30)
q/(o) (a ) - Q ~(o) ( - a ) Q 2 2
(2.31)
~ ' (n ) ( -+a)=O , n = - 1 , 1 , 2 . . . . (2.32)
w(n)(a) = 0 , n = - 1 , 0 , 1 . . . . (2.33)
(2.34) = - 1
W ~ , ) ( - a ) = 0 , n = 0 , 1 . . . . (2.35)
Next, substituting (2.23) and (2.24) into (2.13), (2.14) and (2.15) helps us determine the funct ions an(O), b,(O) and c,(O). We find that by equating powers of r n, n = - 1,0, 1 , . . . we obtain (up to n = 3):
al = 0 , (2.36)
a 2 = - 2 ~o) + 2 (1 - a ) ty~'_ 1)(gt~' 1) + ~ ( - 1)) , (2.37)
a 3 = - 4 q/~l) + (1 - (7) [4 ~@) ty~" i) - 4 q/~o) ~'~- 1)
+ 10 ~u(_ 0 ~u~_ 1) W(_ 1) + 6 ~u(_ 1) W(_ 1) q/i" 1)
+ 2 q/~_ i) W~"- i)~91)1 , (2.38)
bl = g/i" 1) + W(- 1) , (2.39)
b2 = (1 - (7) [ w ( _ 1) w ( - 1 )
+ 2 q/(_ 1) ~'~-1) + ~u(1) ~t ~"_ 1)] , (2.40)
b 3 = ty~i ) - 3 t//(1 ) + (1 - (7) [q/(_ l) W ~)') + 3 q/(o) q / i " i)
+ 2 . . . . . ± 4 . . . . ~(o)~(-1)- ~)!Y(-1)T I/s(-l)l/s(-O~s(_ 0
+ 5 I//(_ 1) q/(@1) + 6 v/(_ 1) ~u~ 2- i) + 2 ~,~21) ~'{" a)
+ 6 ~J~- i) ~'('-- 1) + W~_ i) WI [ i)1 , (2.4•)
ci = 0 , (2.42)
C 2-- 20 '~o )+2( I - (7 )~ ' (_1 ) [~ ' ( '_1 )+ ~t(_O] , (2.43)
( 7 # # I t c3 41/s~O+(1- )[41//(o)!l/(_l)+4~(_l)!~'(o)
+ 4 q/2- OtY("-l) + 8 q/(- l) q/~_ l) ~('_ ,)
+ 12 ~ _ 1) gt~_ 1)1 • (2.44)
The pressure p can be eliminated f rom (2 .10 ) - (2 .12 ) and we find that this leads to
Q U 2 A I [ 0 ~ v 0 " 0 ~ ' 0 ]
°[° + - - Srr "4- ! Sr t
O0 ar r
494 Rheologica Acta, Vol. 32, No. 5 (1993)
- r I~r2(Sro)+~-~r (Sro) - -~o2(Sro) ]
i
00 Lor r J (2.45)
where A denotes the Laplacian in cylindrical polar coordinates. Now, it follows from (2.23) with the ex- pressions (2.24) for S~, Sro and Soo, and (2 .36)- (2.44) for the a'~s, bns and c'~s (up to n = 3) that
i~ ¢~,_ = ~u(_~)+2 ~)+q/(-1) 0 ,
i v t t q/(o) + 4 q/(o) = 0 ,
(2.46)
(2.47)
iv ,, = - - ( i -- O ' ) { 8 q / ( _ l ) ~ _ l ) q / ( _ l ) ~u(1) + 10 ~u(1) + 9 ~u(l )
~) + 12 1) ~ ( - 1)
+ 2 + 8 . . . . 3 - 1) 7J (_ 1)-- I//it-31)
+ 28 q/(_ 1) ~u~_ 1) ~u~" 1) + 5 ~u(_ ~) ~ , ( 2 )
+32q/~21)~u~,_l)+,, 2 iv .+ 2 . vi / q/(- ~) q/(- 1) q/(- 1) ~'(- 1)
3 ~ ,2 , + U q/(_ 1) q/(_ 1) + 6 q/~_ 1) q/(_ 1)} . (2.48)
We notice that there is no effect due to the elasticity on the solutions to ~u(_ 1) or q/(0), and the first time a makes an appearance is in the equation governing ~t(1 ). It is possible to establish an analytical solution for q/(~), however, as the equation is quite cumber- some, we choose to solve it numerically.
We have thus found the stream function q/(r, 0) to within terms of order n = 2 as
~(r,O)=r~/(_l)+~(o)(O) + ~/(1)(0)+ O (-~'~ , r \ r - /
(2.49)
Srr=a2(O)+a3(O)+o(-~)r2 r3 , (2.50)
Sr°=b~(O)+b2(O)+b3(O)+o(-~)r r 2 r 3 , (2.51)
r ~ c2(0) + c3(0) + 0 ( 7 ) . _ _ 7 Soo=---r -- . (2.52)
Of course, the series expansions that have been used blow up as r ~ 0. In fact as we mentioned previ- ously, the series solution can only be considered
reliable if f > 1. If q]/-~l is small enough, then the results would be valid for values of r appropriately small, possibly much smaller than unity. Of course, the solutions would be reasonable approximations valid for appropriately large L However, as we shall see, the flow pattern (solutions) for large r gives us adequate information as to the nature of the solution for values of g< 1, in view of the fact that the solu- tions are expected to have a smooth enough structure.
Equations (2 .46)- (2.48) have to be solved subject to boundary conditions (2.31)-(2.35).
It follows from (2.46), (2.48) and (2.31)-(2.35) that
~u(_l) = A sin O+Bcos O+COsinO+DOcos 0 , (2.53)
wiih
- a cos a - a sin a A - , B - , (2.54)
2a - sin (2a) 2a + sin 2a
cos a sin a C - , D - , (2.55)
2a + sin 2a 2a - sin 2a
and
q/(0) (0) = A 0 + B sin 2 0
with
(2.56)
A = Q cos 2a
sin 2 a - 2 a cos 2 a
B _ _.~ - O 2[sin 2a - 2a cos 2a]
(2.57)
Equation (2.40) for qJO) has been solved numerically. Strauss (1975) obtained an exact solution for q/(1). However, the governing equation for ~(1) has so many highly non-linear non-hom*ogeneous terms that it is easy to make an error in the calculations. In fact, we feel that this is indeed the case, as our equations agree with those of Strauss (1975) except for some ob- vious printing errors while our calculations yield results that are totally at odds.
Results and discussions
rN
We first discuss the case when the elastic parameter a = 0, which was the case considered by Strauss (1975). It is appropriate to point out at this juncture that we cannot obtain the results for a :~ 0, f rom those obtained by Strauss (1975) for the case a = 0, though it appears at first sight there might be a way to do so (i.e., by replacing A s by A 1 - A 2 in the equa- tions). However, this is not possible as we have to change the non-dimensionalization, and this will make the whole analysis inapplicable.
I f we notice the streamline patterns portrayed in the figures, a port ion of the streamlines go towards the origin while others bend away f rom the origin and proceed towards r = oo, but at the other (fixed) plate. Continuity of the streamline structure predicts a dividing streamline, on the one side of which streamlines proceed towards the origin and the others tending towards r = oo. In the case of a Newtonian fluid the dividing streamline would hit the fixed wall at 0 = a and the radial coordinate is given by
p(r,O)= ~ pn(o) , n = l F n
Q sin 2 a (4a 2 - sin 2 2 a )
[sin 2 a - 2 a cos 2 a ] 2
I f the case of an Oldroyd-B fluid, it can be shown that the dividing streamline hits the plane 0 = a, to within terms of n = 2, by
1 [ / ~ / 2 - ~ (]) (4-------a- 2 -- sin2 2 a) ] 1/2 r n - N = 2 r u + + 2 sin 2 a - 4 - - ~ c o s ~ a J
We note that Strauss's calculation for the dividing streamline has a negative sign within the parentheses which is probably a printing error.
I f we assume that the pressure p ( r , O ) can be ex- pressed as
then it can be shown that
Pn(O) = _ 1 [(1 - n )an(O) + b '~(O)- c,(O)] . n
Once the solution has been obtained up to ~(~), we can obtain the tractions acting on the plates as the tractions t (0 = _+a) through Fig. 2. Streamlines for a = 0.9, a = 30 °, Q = 0.3
t = T T n ,
when n is the unit outward normal to the plates. Figure 1 shows the flow field when there is no
elasticity, i.e., a = 0, when the angle between the plates is 60 °. We see that the streamlines below the dividing streamlines all proceed towards the origin so
2-
Fig. 1. Streamlines for a = 0, a = 30, Q = 0.3
Huang et al., Flow of a non-Newtonian fluid between intersecting planes of which one is moving 495
496 Rheologica Acta, Vol. 32, No. 5 (1993)
.
Fig. 3. Streamlines for a = 0, a = 45 °, Q = 0.3
, \ (.,
Fig. 5. Streamlines for a = 0, a = 60 °, Q = 0.3
Fig. 4. Streamlines for a = 0.9, a = 45 °, Q = 0.3 Fig. 6. Streamlines for cr = 0.9, a --= 30 °, Q -~ 0.3
that the streamlines enter the origin tangential to the moving plate. The small circulating cell adjacent to the origin cannot be trusted as being accurate for the reasons mentioned earlier regarding the reliability of the solution close to the origin. We find f rom Fig. 2 that increasing the elasticity parameter moves the dividing streamline towards the origin, that is rn_ N
decreases with increasing a , thereby squeezing the
streamlines below the dividing streamlines. Moreover, we see that the circulating cell structure near the origin has disappeared thereby implying a reduction in secondary flow as the elasticity increases. We find f rom Figs. 3 and 4, corresponding to the angle be- tween the plates being 90 °, that qualitatively we once again see a similarity in the structure of the streamline patterns with Figs. 1 and 2, respectively. Increasing
Huang et al., Flow of a non-Newtonian fluid between intersecting planes of which one is moving 497
i
Fig. 7. Streamlines for a = 0.9, a = 30 °, Q = 1 Fig. 8. Streamlines for a = 0.99, a = 30 °, Q = 1
the angle between the plates still further, we see for the first time that the cellular pattern is large enough to exist beyond r = 1, as evidenced in Fig. 5. Increas- ing the elasticity once again inhibits the cellular struc- ture (cf. Fig. 6). We also find that as the angle be- tween the plates increases, the point where the dividing streamline meets the moving plate moves fur- ther away f rom the origin. The same effect can be ob- served when the angle is fixed and the flow rate is in- creased in that the cellular structure is diminished and the dividing streamline moves away f rom the origin. Figure 8 shows when that the elasticity parameter is increased significantly to a = 0.99, then the structure of the solution changes qualitatively in that not all the streamlines approach the origin tangential to the fixed
plate. In fact, some of the streamlines are tangential to the moving plate while others approach radially. The point where the dividing streamline meets the fixed plate has moved away f rom the origin. Finally, we discuss Fig. 9 which has been constructed by reflecting Fig. 3, for the flow between two plates in- clined at 90 ° to one another with the plate along the y-axis moving along its plane, about the y-axis. This would correspond to a flow due to the y-axis in the figure being drawn downwards, the plates along x- axis lying on ( - ~ , 0 ) and (0, ~ ) being fixed.
Acknowledgement
K.R. Rajagopal thanks the Air Force Office of Scientific Research for its support.
Fig. 9. Streamlines for cr = 0.9, Q = 0.3; case where the two plates are inclined at 90 ° to one another by reflecting Fig. 3
498 Rheologica Acta, Vol. 32, No. 5 (1993)
References
Bhatnagar RK, Rajagopal KR, Gupta, G (1993) Flow of Oldroyd-B fluid between intersecting planes. J Non- Newtonian Fluid Mechanics 46:49- 67
Oldroyd JG (1950) Proceedings of the Royal Society, Ser A 200:523 - 541
Strauss K (1974) The flow of a viscoelastic fluid in a con- tracting channel. Part I: steady flow. Acta Mechanica 20:233 - 246
Strauss K (1975) Modell der StrOmung, die sich am Rakelmesser einer Beschichtungsanlage einstellt. Rheol Acta 14:1058- 1065
Truesdell C, NoU W (1965) In: Flugge (ed) Non-linear field theories of Mechanics. Handbuch der Physik, Berlin- Heidelberg-New York
Correspondence to:
Prof. R.K. Bhatnagar Dept. of Mathematics University of Pittsburgh Greensburg, PA 15601 USA
(Received December 14, 1992; in revised form July 28, 1993)
