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An investigation of non-newtonian flow past a sphere
107
Journal of Non-Newtonian Fluid Mechanics, 3 (19?7/19?8) 107-125 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
AN INVESTIGATION OF NON-NEWTONIAN FLOW PAST A SPHERE
K. ADACHI, Department (Received
N. YOSHIOKA
and K. SAKAI
of Chemical Engineering,
March 12, 1977; in revised
Kyoto University, Kyoto, 606 {Japan) form May 23, 1977)
Summary Any experimental work on the flow of a polymer solution or any theoretical analysis on the basis of a visoelastic constitutive equation does not always bring out viscoelastic effects but may be showing a non-Newtonian viscosity effect. Therefore, in order to obtain a clear understanding about viscoelastic effects, it is desirable to have a sufficient knowledge of the non-Newtonian viscosity effect. To facilitate this, finis-difference numerical solutions of non-Newtonian flow were carried out using a non-Newtonian viscous model for the Reynolds numbers of 0.1, 1.0, 20 and 60. Drag force measurements and flow visualization experiments were also performed over a wide range of experimental conditions using polymer solutions. The present work appears to support the following idea: When compared with the Newtonian case on the basis of ~~~I~~, where q. is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines. 1. Introduction The slow uniform flow of viscoelastic liquids past submerged objects is an idealization of flow situations commonly encountered in industrial processing. It is, furthermore, a flow configuration which has been studied in great detail for Newtonian liquids, both theoretically and experimentally. Hence, it seems to be useful to investigate the effect of the rheological fluid properties on the flow.
108
Using perturbation methods, Leslie [ 11, Caswell and Schwarz [ 21 and Giesekus [ 31 predicted a small downstream shift in the streamlines at 0( We) and a decrease in the drag at 0( We2), where We is the Weissenberg number. More recently, Ultman and Denn [ 41 solved the Oseen equations for a simplified Maxwell liquid and predicted an upstream shift which was contradictory to the previous prediction but agreed with their experimental data of flow visualization. However, Mena and Caswell [ 51 obtained again the same theoretical results as the previous investigators with the exception of Ultman and Denn. Zana et al. [ 61 investigated the Oseen-linearization for viscoelastic flow past a submerged body and showed that the prediction due to Ultman and Denn [ 41 is incorrect because the Oseen-linearization does not even provide a uniformly valid first approximation to small elastic corrections. Broadbent and Mena [ 71 found from their experiments that any shifting of the streamlines is too small to be discerned clearly and the drag force decreases, and Zana et al. [6] reported another experimental result that the streamline shift is very smaller than predicted by the theoretical analysis of Ultman and Denn [ 41 although some upstream shifting of the streamlines was observed for Re . We = O(1). It must be noted that all the experimental and theoretical results may contain not only viscoelastic effects but also non-Newtonian viscosity effects. Previous investigators may have a situation in which non-Newtonian viscous effects are dominant enough to veil viscoelastic effects (such as the normal force and high extension viscosity), even if the analyses of non-Newtonian flow were based on viscoelastic models. Here, the viscoelastic effect is regarded as one which cannot be taken into account by means of any non-Newtonian viscous model. Therefore, in order to obtain a clear understanding of viscoelastic effects, one must first have a sufficient knowledge of the non-Newtonian viscosity effect. This will be done in Section 2: Non-Newtonian effects will be studied in detail by means of a numerical finite-difference analysis for the flow of a non-Newtonian viscous fluid past a sphere. The existing experimental data were taken in a range of low Reynolds numbers and low strain rates, so the effects of viscoelasticity on the drag and flow patterns is expected to have been small even if it arose. Present drag force measurements and flow visualization measurements were performed over a wide range of experimental conditions so as to produce clearer evidences of changes in flow due to viscoelasticity. The relevant experimental results will be described in Sections 3 and 4. As is well known, polymer fluids are viscoelastic and at the same time have a non-Newtonian viscosity. All the experimental data, which are based on polymer fluids, are influenced by both viscoelasticity and non-Newtonian viscosity. Net-effects of viscoelasticity become clear when the contributions due to the non-Newtonian viscosity are subtracted from experimental data for polymer fluids. The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV,p/qo, where
109
no is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it seems that it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines. 2. Flow of a non-Newtonian
viscous fluid past a sphere
The steady flow of an incompressible less form (see [ 81) as: [$-(&)
!$-
4 +Re
$(*)
y]sin
8 = +$E’(wrsin
-% wrsinf3)++$-$ I[ ar
-- cot 0 orsine 2
+ L9.. t r iir
oi__- sine
a%
L
ar2
r2 s
2
I(
aG
r ae
)
r2
ar
+cot~C!!!+i!C!i! E!L &)=-
3w
r
&-2
in dimension-
0)
$]$+$[$(wrsint9)
+2a2J/_cote r arae
*
ar
fluid can be expressed
arae I 7
E2G
a?,
a6 1 ae
a%
+L J
&2?& r* (
(1) (2)
r sin e ’
with (3) where (r, 8, 4) are spherical polar coordinates, $, o and 7) are the stream, vorticity and viscosity functions, respectively, and Re (= DV,p/qo) is the Reynolds number. The boundary conditions for flow past a sphere are given as:
Q?$=() J/ + -$r’
onr=l, sin28
asr+m.
(5)
For the solution of the above boundary value problem, the viscosity func: tion 7) should be given. In the present analysis, the extended Williamson model is used which has the following dimensionless form: q =7), + 1-z
rlo
7)o
[
I
/[l + (2oIZZ#7)
(6)
110
where r), is the limiting viscosity
zze= ___-l
( [( 1
r2 sin20 r2
+ late ( rae
a’+ _--arae
cotead2 ar
2aG2 r ae )
11+2 (la*
and + -_ a’$ +cot&+X2 ar ( arae
r ar
av ar2
r
time2 2
11 ’
ae ) (7)
This model was recommended by Cramer and Marchello [ 91 for the following reason: Reasonable estimates of the extended Williamson model parameters can readily be made upon inspection of experimental, rheological data because the material constants vary monotonically with the concentrations of the solutions, while this is not true for other good models such as the five parameter Powell-Eyring model. It must be noted here that the results obtained in the present analysis are valid not only for the extended Williamson model but also for various non-Newtonian viscous models whose viscosity curve is shown in Fig. 1. Equations (l), (2) and (6) were solved simultaneously using standard numerical finite-difference techniques. Convergence of the computation was too difficult to achieve in the case of q-/~Ia = 0.1 because of too large variations of the vorticity function very near the sphere surface. Therefore, the computation was restricted to the case where +/qO = 0.5 and a = jI = 1.0. In what follows, comparison of the Newtonian case with the non-Newtonian case will be made at the same value of the Reynolds number, which is based on the zero shear viscosity, unless any other indication is given. The fundamental computational results are presented in Table 1, where the values of the drag coefficients for Newtonian fluids are in good agreement with those due to Jenson [lo] and Hamielec et al. [ll]. The pressure distribution over the surface of the sphere is shown in Fig. 2. For non-Newtonian flow, the higher pressure over the rear surface of the sphere gives rise to a smaller form drag C, than for Newtonian flow, as presented in Table 1. The figure also indicates that the difference between the pressure distributions for Newtonian and non-Newtonian fluids becomes smaller as the Reynolds number increases. The pressure distribution is influenced not by the local flow near the sphere but by the whole flow, and
1
ObM-~-e--l-j J2II, L-1 Fig. 1. Viscosity
curve.
10
20
111 TABLE
I
between Newtonian and Non-Newtonian ~___l_._--__ Re(DVdlrlo) 0.1 1.0 ____...-._-_.--~-__...__ cc@.) 259.1 262 + 27.65 27.5 + Cd(N.N.) 197.5 21.27 Cd(N.N.),c,(N.) 0.763 0.769
flows on the basis of DV,pl~o _.._____l_-_.-.__-~-20 60 2.871 2.307 0.804
2.946 ++
1.472 1.118 0.760
C,,(N.) Cp( N.N.) C,W.N.YC,(N.)
83.08 52.25 0.629
8.886 5,941 0.669
1.051 0.977 0.930
1.089 ++
0.620 0.529 0.853
C&N.) CffN.N.) CPN.N.)/Cf(N.)
176 145.3 0.826
18.76 15.33 0.817
1.820 1.330 0.731
1.857 +’
0.852 0.589 0.691
Comparison
SID(N.) S/D(N.N.)
-
-
-
0,tN.I 8,tN.N)
-
-
-
0.458 0.586 43.4 47.4
0.57 * 45 *
%tN.) 0.699 ** 0.687 ** 0.630 ** 0.639 ** ?jlN.N.) 0.649 0.640 0.617 0.641 ______.. ._ ___. .___~~~ -...___.. - ..__-.-~~~~-..--.N. Newtonian flow; N.N. Non-Newtonian flow. .t I-Iamielec [ 111, ++ Jenson [lo], * Taneda 1121; ** Calculated from II, for Newtonian flow by using the viscosity function eqn. (6).
Fig. 2. Pressure Fig. 3. Vorticity
49 Cdegrrel distribution over the sphere. distribution
over the sphere.
e
Cdegrccl
112
the viscous effect extends farther from the sphere at lower Reynolds numbers. This is the reason why the non-Newtonian viscous effect on the pressure distribution is more striking at the low Reynolds numbers. On the other hand, the vorticity distribution over the sphere is influenced not by the whole flow but by the local flow in the neighbourhood of the sphere surface, and the viscous effect is limited to the flow closer to the sphere at higher Reynolds numbers. As is seen in Fig. 3, this is the reason why the nonNewtonian viscous effect on the vorticity distribution is more striking at the higher Reynolds numbers. The higher vorticity for non-Newtonian flow also means that the non-Newtonian fluid flows closer to the sphere except in a part of the wake. This phenomenon may result from the fact that the fluid has a tendency to flow through a region of less mechanical energy loss. Figures 4-7 show the streamlines in their upper halves and the equivorticity lines in their lower halves, where the fluid flows right to left. As has been described above, the non-Newtonian fluid flows closer to the sphere except in a part of the wake. In the case of the Reynolds number of 0.1, the flow pattern is found from the vorticity distribution to be symmetric before and after the sphere for the non-Newtonian fluid as well as for the Newtonian fluid. On the other hand, in a range of Reynolds numbers higher than 0.1, the non-Newtonian fluid flows closer to the front part of the sphere surface and, because of the inertial effects, this causes the fluid to flow
Fig. 4. Streamlines
and equivorticity
lines (Re
= 0.1).
113
/
Fig. 5. Streamlines and equivorticity lines
.
_ .........
(Re =
1.0),
(Re =
20).
U , !
Fig. 6. Streamlines and equivorticity fines
Fig. 7. Streamlines
and equivorticity
lines (Re = 60).
farther away than the Newtonian fluid from the rear part of the sphere surface. The separation angle 0, and the vortex wake length S at Re = 60 are presented in Table 1 and compared with experimental results due to Taneda [12] for a Newtonian fluid. In the case of the separating flow, the separation point moves upstream and the steady twin vortices are elongated when compared with the Newtonian flow. This is clearIy shown in Fig. 7. In the case of the non-Newtonian flow, a higher vorticity is created over the sphere surface, as is shown in Fig. 3, and, moreover, the dissipation energy of recirculation is smaller in a non-Newtonian vortex wake than in a Newtonian wake of similar size because the non-Newtonian fluid is less viscous in the fringe of the steady vortex wake, as compared with the inner region. This is the reason for the larger vortices for the non-Newtonian fIuid. The drag coefficients are presented in Table 1. At the same value of the Reynolds numbers, these drag coefficents are lower for the non-Newtonian flow than for the Newtonian flow. As the Reynolds number increases, C~(N,N.)/C~(N.) increases and C~(N,N.)/C~(N.) decreases, but G(N.N.)/G(N.), however, remains almost constant, where N. and N.N. indicate the Newtonian and non-Newtonian cases, respectively. Moreover it must be noted that for creeping flow, the ratio C#Z’,, should be exactly 2.0 for Newtonian flow but it is expected to be much larger than 2.0 for nonNewtonian flow.
115
30
0
60 8
Fig. 8. Viscosity
SO 120 [degree]
distribution
150
180
over the sphere.
The comparison between the Newtonian and non-Newtonian cases presented above has been made at the same value of the Reynolds number, which is based on the zero shear viscosity qo. Roughly speaking, it has also indicated that the non-Newtonian flow patterns are more washed away than the corresponding Newtonian flow patterns. This is caused by the fact that the viscosities over the sphere are lower for the non-Newtonian flows than for the corresponding Newtonian flows. A more meaningful comparison may be obtained by using the surface-average viscosity in the definition of the Reynolds number. The averaged non-Newtonian viscosity q(N.N.) is given in Table 1 together with q(N.), where q(N.N.) and T(N.) are, respectively, the averaged values of r)(N.N.) and q(N.) which are shown in Fig. 8. The quanti-
TABLE
2
Comparison
between
Newtonian
and non-Newtonian
flows on the basis of DV,plr)
Re(DV.&77)
0.154
1.563
32.41
Cd(N.) C,(N.N.) C,(N.N.)/C,(N.)
172 * 197.6 1.15
19.0 * 21.27 1.12
2.10 * 2.307 1.10
1.11 * 1.118 1.01
C,(N.) C,(N.N.) C,(N.N.)/C,(N.)
55 * 52.25 0.95
6.2 * 5.941 0.96
0.83 * 0.977 1.18
0.50 0.529 1.06
Cr(N.) Cf(N.N.) Cf(N.N.)/Cr(N.)
117 * 145.3 1.24
12.8 * 15.33 1.20
1.27 * 1.330 1.05
0.61 * 0.589 0.97
S/Q
N. )
S/D(N.N.) @AN.) 0JN.N.) ~__~~~~~..~. ...__~~~~~~~ * Interpolated or presumed values.
-
-
93.6
0.72
*
0.586 .-------
53.0 * 47.4 ~~~~~~ ~~~--. -
116
ties r)(N.N.) and n(N.) are calculated respectively by substituting the nonNewtonian and Newtonian strain-rates into the non-Newtonian viscosity function, eqn. (6). When the averaged non-Newtonian viscosity T(N.N.) is not known, @N.) may be a good substitute for it. Table 2 gives a comparison between the Newtonian and non-Newtonian cases at the same value of the Reynolds number which is based on the averaged viscosity ?j. In this comparison, the drag coefficients are higher for the non-Newtonian flows than for the corresponding Newtonian flows, and it is remarkable that this result is apparently opposite to the previous one. It may be reasonable that the present results for DVp/Fj of 93.6 are compared with predictions due to Adachi et al. [S] for non-Newtonian flow of a power-law model at Re . o = 60 because D2-“v”,p m
Re.o=
DV,p
(8)
-7’
Then, the following conclusions are in agreement between these two predictions: When the fluid changes from Newtonian to non-Newtonian, the friction drag coefficients Cf decreases, and the pressure and total drag coefficients C,, C, increase. However, the results concerning the twin vortices are in disagreement between them. This disagreement appears to result from the defect of infinite zero-shear viscosity for the power-law model because this defect occurs at the separation point. Therefore, the present predictions, that the separation point moves downstream and the twin vortices shorten when the fluid changes from Newtonian to non-Newtonian, seem probable. The present drag predictions are compared with the existing ones for the power-law model due to Adachi et al. [ 81 and Wasserman and Slattery [ 131 in Figs. 11 and 12, where the value of 0.639 is used as the power index n for the present model, which is evaluated from Fig. 1. Leslie, like other investigators, predicted a small downstream shift in the streamlines at O((X,--X,)V,/R) and a decrease in the drag at O((hl--X&e K?/R2) for the slow flow of Oldroyd’s viscoelastic model past a sphere when comparing with the corresponding Newtonian flow: 7ik + A,
s7ik +po7;eik 6t
=
eik +
Q,,
6 ~~-~z/z
,
6t
where F/6 t is the Oldroyd convected flow, this viscoelastic model gives
time derivative.
(9) In a steady simple shear
(10) where X1 > X2 since the limiting viscosity cosity qo. Taking account of @I
-
h2)
/Jot*
-
(A,
-
A*)
PO
(
%
1
*
n_ is lower than the zero shear vis-
(11)
117
for flow past a sphere, it seems obvious that the non-Newtonian viscosity is the most dominant factor for the drag decrease, but that it does not cause the downstream shift at O((h,-h,)V,/R). This downstream shift is the net viscoelastic effect which was predicted by means of the Oldroyd model. Moreover, Broadbent and Mena [7] reported a decrease in the drag which is quadratic in velocity. However, the fact seems to be that the drag decreases as (V,/D)” (2 > Q > 0) since the non-Newtonian viscosity is represented better by eqn. (6) than by eqn. (10). 3. Drag force measurements A decrease in the drag for flow of a viscoelastic fluid past an object, which was predicted theoretically by several investigators [l-3], was found in the last section to be due to the non-Newtonian viscosity. Available experimental data show the drag decrease [ 141. On the other hand, a large pressure drop is well known as an experimental fact for converging flow of a polymeric fluid and this effect has been considered to be due to viscoelasticity [15]. An increase in the drag due to viscoelasticity, which is superposed on a decrease in the drag due to the non-Newtonian viscosity, is expected to be observed experimentally for the flow past a sphere as well. However, as the former increase is usually much smaller than the latter decrease, it is difficult to observe the viscoelastic effect, when compared with the Newtonian case on the basis of DV,p/r),. The viscoelastic effect may be discerned more clearly when the difference in the drag between the Newtonian and non-Newtonian cases is investigated on the basis of DV,p/q, because the drag for Newtonian and non-Newtonian viscous fluids are then almost the same. An average viscosity over the sphere was calculated by Q = m( V, /D)“-1
(12)
for each falling-sphere test, where the material constants m and n were determined so that the power-law model might represent the shear-stress-shearrate relationship over the sphere. First, the value of n was assumed for a particular falling-sphere test, and then the maximum shear rate over the sphere was estimated using ?nl,X = C(n) V-/D,
(13)
and C(n) = {A(n)‘.’
+ [B(~)
. Re . 0’/(l+n)]0.6}1:0.6,
(14)
where A(n) was derived from Wasserman and Slattery’s work [ 131 and B(n) from both the work of Adachi et al. [8] and a boundary layer analysis for flow of the power-law fluid past a sphere. The functions A(n) and B(n) are represented in Fig. 9, where broken lines are only extrapolated ones. The power index n was taken so as to represent the rheological curve in a range
118
..
I
I
I
1
7yo40
‘t, ‘I.,
50
-Y, -
20 1
r
7
,. -
a
I
y .. ~... ._.\ .Y. ~_.. '... ... '. ..
4 ,035
~~~-__c---_.~~..:_ \
-1 0.30 ~.
_ 5- t
1
o.25 0.20
2-
__j\jj 1
I 02
03
I
I
0.4
05
I
I
0.6
0.7
I
I
08
09
IO.15 I.0
nr--1
Fig. 9. Coefficients
for calculating
the characteristic
strain rate.
of strain rate from ymax to ymax /lo. This trial-and-error procedure was repeated until the convergence of n was achieved. For the experimental falling-sphere investigation reported here, aqueous glycerol solutions were used as Newtonian fluids, aqueous solutions of sodium carboximethyl cellulose (CMC) as weakly elastic fluids and aqueous solutions of sodium polyacrylate (SPA) and Separan-AP30 as strongly elastic
t-
3x16’~ 1’ I 2x1o-2
I
I
I
III
16' 3
Fig. 10. Viscosity
I
loo c1/scc1
curves for CMC solutions.
III
I 10'
I
I II lo2
119
Present
Re.ozGBO)
(10s
work
c [ WI
%I
1.04
flylOll
1.1 1
A
tungsten
.
others
A I
I
0 I
I
1.2 Adachl
1.0 -
/_g Presen(
ii
0.5
0.6
0.7
0.8
0.9
1.0
II
x
0.6 0.5
n C-l
l
theory
I
I
I
I
0.6
0.7
0.6
0.9
n
Fig. 11. Drag ratio for weakly
elastic
fluids (Re Q 0.1).
Fig. 12. Drag ratio for weakly
elastic
fluids (Re 2 10).
et al. 0
1.0
C-l
fluids. The falling-sphere apparatus used consisted of a circular cylinder of 20 cm in diameter and 200 cm in height, and of nylon, aluminum, glass, steel, lead and tungsten speres. The sphere diameters ranged from 0.1 to 1.5 cm and the terminal falling velocities from 0.01 to 10 cm/s. The measured terminal velocities were corrected for wall effects using Faxen’s formula [ 141. The concentrations for the CMC solutions were 1.11, 1.84, 3.15 and 4.99% by weight and their shear viscosity curves are presented in Fig. 10. The experimental data for these weakly elastic fluids are shown in Fig. 11 for Re < 0.1 and in Fig. 12 for 10 < Re < 80. They are compared with the available theoretical works [ 8,131. For weakly elastic fluids, the experimental data agree approximately with those theoretical predictions which are based on an inelastic fluid model. In the case of slow flow, X k 1.1 _+0.1 and in the case of flow with inertial effects and large strain rates, X = 1.05 f 0.05, where the drag ratio X is defined as X=
Cd(Non-Newtonian) -= Cd(Newtonian)
C,Re( Non-Newtonian) C,Re( Newtonian)
(15)
at the same value of the Reynolds number
D%-“P _ m
Re
_
o
(16)
As Re increases, the drag ratio X decreases as was shown theoretically in the last section.
1101 s q
scpamn
0.70
100 2x16' 16' 5X1$ 13
2xd
Fig. 13. Viscosity
10" p lllucl
10'
curves for SPA and Separan-AP30
solutions.
Present
work
(&/DLlO.O)
Rc.0 i
present work v-/o
1.G
0.1)
( Re.o L D -0.1
0.7 v/i%
Cl
10
S.P
6
LO-
01-1.0
Sepamn
05wtV.
A
A
A
S.P. A.
1 0 WI%
0
a
l
I
I
A.
I
F
.
I
z % ”
0
O.Swt%
A
S. P A.
1
0
WV.
.
A
0 _
o
A ..___h._
.___--
Exp. data
Present
1.0 a2
0.7wt+
S. F! A.
1.0 - 10.0
..,..$._..__:__,.45
1.2
II
Sepamn
1.6
*
1.4
x
0.1 - 1.0
0.3
0.4 "C-l
0.5
03
0.7
0.2
elastic fluids (Re < 0.1).
Fig. 15. Drag ratio for strongly
elastic fluids (Re > 0.1).
++
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
IlC-1
Fig. 14. Drag ratio for strongly
theory
I” Flg.12
121
The concentrations for the SPA solutions were 0.5 and 1.0 wt% and the concentration for the Separan-AP30 solution was 0.7 wt%, whose viscosity curves are presented in Fig. 13. The experimental data for these strongly elastic fluids are shown in Fig. 14 for Re < 0.1 and in Fig. 15 for Re > 0.1. In the case of slow flow, X z 1.25 + 0.25 and in the case of flow with large strain rates, X = 1.45 + 0.25. As Re and L/D increases, the drag ratio X increases. This phenomenon is opposite to the weakly elastic case and seems to be the effect of viscoelasticity. Moreover, the drag ratio for strongly elastic fluids are larger than that for weakly elastic fluids. The mechanism for the drag increase due to the viscoelasticity will be discussed in detail in the following section.
4. Flow patterns and a flow mechanism In order to investigate a characteristic feature of viscoelastic flow, flow visualization was accomplished using fine aluminum dust. In the fahingsphere test, which has been described in the last section, attention was also paid to the global flow in a small cylindrical container. Photographs of the motion of fluids in the whole flow field were taken with long exposure times by a camera which was fixed in space. In the case of a glycerol-water solution, the fluid was made to descend by the falling sphere in a central vertical region of the container and to ascend near the container wall. There was a large circulating flow as is illustrated in Fig. 16(a). The right half of this figure shows schematically a long-time trace of a fluid particle near the path of the falling sphere. The particle is first pushed forward and then dragged by the sphere. On the other hand, in the case of a 1% sodium polyacrylatewater solution, no global circulating flow was observed and a long-time trace of a fluid particle is shown in Fig. 16(b). It indicates that at the end of the motion the particle goes back considerably towards its original location. This reverse flow was observed in a different aspect by Sigli and Coutanceau [ 161 also. They made similar experiments of flow visualization using larger spheres,
T I !
!
7
! I t a.
Viscous
I flow
Fig. 16. Displacement
b. Viscoelastic
of a fluid particle.
flow
122
in which their exposure time was short enough to give clear photographs for instantaneous absolute flow around a sphere moving with constant speed. They found that in a region of the flow field near the sphere, a polymer solution moved in the same direction as the sphere, as in the Newtonian case, but farther down this region the fluid moved in the opposite direction. This implies that, for uniform flow past a sphere which is fixed in space, a part of the viscoelastic fluid is accelerated, to a little more than the uniform speed, far downstream from the sphere. For investigation of viscoelastic effects on the streamlines, other flow visualization experiments were conducted in a horizontal tank with dimensions 50 X 50 X 200 cm. The experimental apparatus and procedure were similar to those used by Taneda [ 121. Photographs were taken with a camera which was connected with a sphere and moved at the same speed as the sphere. Aqueous solutions of sodium polyacrylate were used of concentrations 0.5, 1.0 and 2.0 wt%. The ranges of sphere diameters and speeds were, respectively, 1.11 < D G 2.23 cm and 0.03 < V, < 15 cm/s. In spite of such a wide range of experimental conditions any shifting in the streamlines was too small to be discerned clearly. Some thought should be given to the fact that we could not observe any viscoelastic effect by the present technique, while Sigli and Coutanceau [ 161 could. The changes in the flow patterns due to viscoelasticity might be too small to be observed when referred to a fluid moving at a constant speed, but large enough when referred to a stationary fluid. Another possible reason is that the present container walls might be too far from the sphere to contribute to an increase in the difference between the Newtonian and non-Newtonian flows [ 161. Laminar separating flows with steady twin vortices behind a sphere were also observed. Very dilute aqueous solutions of sodium polyacrylate (0.030, 0.031 wt%) were used to make the twin vortices. Their viscosity curves are shown in Fig. 13. The vortex wake length, the separation angle and the position of the vortex center are presented, respectively, in Figs. 17, 18 and 19. In the case of Newtonian fluids, the present experimental data agree well wih Taneda’s results [ 121. Therefore, the present experiment may be considered to be satisfactorily performed. The vortices for the polymer solutions become smaller and the separation point moves downstream when compared with those for Newtonian fluids on the basis of DV,~/Q. This phenomenon is not considered to be a non-Newtonian viscous effect but to be a net viscoelastic effect because the non-Newtonian viscosity makes the vortices elongate as was shown in Section 2. This shortening of the vortices for separating flow seems to correspond to the upstream shift in the streamlines for slow flow. It is interesting to consider a probable mechanism of the upstream shift in the streamlines. When a fluid comes closer to an object, a part of the fluid is prevented from flowing uniformly together with its remaining part by the presence of the object, so the fluid is deformed near the body. The elongation of an element of a viscoelastic fluid gives rise to a characteristic feature
123
A
4.70 0,094
0
4.70
0.015-
0.15 0.053 I I I lIiIl_
102
1
1111111
1
I1111111 lo3
Ret-l
Ret-1
Fig. 17. Vortex
I
lo2
103
wake length.
Fig. 18. Separation
angle.
of viscoelastic flow as Dealy [ 171 has reviewed. When an element of a viscoelastic fluid is subject to an elongation, a tensile stress responds to it as a result of the memory which the element has. This tensile normal stress is very much higher than the frictional shear stress at the same magnitude of shear rates. Therefore, the viscoelastic fluid is expected to flow in a region of convergent flow so as to suppress a rapid development of such an extensional flow for the purpose of maintaining a force balance: Then, the stream-
; 0.8t
c 4hl-
22
A L70
0.094
0 015
0
0 15
0 053
170
Tonedo
Re 1-l Fig. 19. Position
of the vortex
center.
124
lines may respond to a change of the flow situation (such as the presence of an object) earlier in a viscoelastic fluid than in an inelastic fluid. On the other hand, in a region of divergent flow, the deformed element tends to recover, by the action of the tensile force, the original configuration and position which the element had, and shows a recoil which is illustrated in Fig. 16(b). Behind an object, a part of the fluid, which was retarded by the presence of the object, is considerably accelerated by the tensile force and the streamlines return sooner to uniform streaming following the object than in an inelastic fluid. Thus, due to the characteristic tensile force and recoil, the viscoelastic fluid appears to show a flow, similar to the wine-glass flow in a convergent flow region, before the body, and a flow, similar to the swelling flow in a divergent flow region, after the object. This seems to be the mechanism for the upstream shift. The flow, which is produced by the large tensile stress, is expected to show a large pressure drop in the downstream direction and a large drop of a total stress normal to the stream, although the latter drop is considered to be smaller than the former one. This gives rise to an increase in the normal force drag for the flow of a viscoelastic fluid past an object. However, as polymer fluids are non-Newtonian viscous as well as viscoelastic, the friction drag decreases but the normal force drag may increase or decrease, when compared with the Newtonian case on the basis of DV,p/qo. Whether the normal force drag increases or decreases, depends upon which property is stronger, non-Newtonian viscosity or viscoelasticity, in the flow under consideration. As the streaming velocity increases, the departure from the Newtonian value of the total drag decreases due to the non-Newtonian viscosity but the departure increases due to the viscoelasticity because stretch rates become larger. When comparison is made on the basis of DK.,p/T’f, for increasing Reynolds numbers, the drag ratio X cannot be expected to become larger for weakly elastic fluids, but it may be expected to become larger for strongly elastic fluids. This prediction seems to be confirmed by the present drag measurements. The observations for flow of a viscoelastic fluid past a sphere have been given a consistent explanation. The upstream shift in the streamlines has become a more unquestionable fact although the shift is small. The experimental fact of the upstream shift is opposite to the existing theoretical predictions which are based on constitutive equations of a different type. 4. Conclusions The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV,p/q,, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines in low Reynolds numbers. On the other hand, it is due to the viscoelasticity that the normal force drag increases, that the
125
separating vortices behind the sphere become shift occurs in the streamlines.
smaller, and that an upstream
Acknowledgements The authors wish to express their gratitude to Professor S. Taneda for helpful comments on the experimental apparatus for flow visualization. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
F.M. Leslie, Q. J. Mech. Appl. Math., 14 (1961) 96. B. Caswell and W.H. Scharz, J. Fluid Mech., 13 (1962) 417 H. Giesekus, Rheol. Acta, 3 (1963) 59. J.S. Ultman and M.M. Denn, Chem. Eng. J., 2 (1971) 81. B. Men and B. Caswell, Chem. Eng. J., 8 (1974) 125. E. Zana, G. Tiefenbruck and L.G. Leal, Rheol. Acta, 14 (1975) 891. J.M. Broadbent and B. Mena, Chem. Eng. J., 8 (1974) 11. K. Adachi, N. Yoshioka and K. Yamamoto, Chem. Eng. Sci., 28 (1973) 2033. S.D. Cramer and J.M. Marchello, AIChE J., 14 (1968) 980. V.G. Jenson, Proc. R. Sot. London, 249 (1959) 346. A.E. Hamielec, T.W. Hoffman and L.L. Ross, AIChE J., 13 (1967) 212. S. Taneda, Rep. Res. Inst. Appl. Mech. Kyushu Univ. (Japan), 4 (1956) 99. M.L. Wasserman and J.C. Slattery, AIChE J., 10 (1961) 383. R.M. Turian, AIChE J., 13 (1967) 999. C.D. Han and M. Charles, Trans. Sot. Rheol., 15 (1971) 371. D. Sigli and M. Coutanceau, J. Non-Newtonian Fluid Mech., 2 (1977) 1. J.M. Dealy, Polym. Eng. Sci., 11 (1971) 433. H. Kato, H. Tachibana and K. Oikawa, Bull. JSME, 38 (1972) 809.
Journal of Non-Newtonian Fluid Mechanics, 3 (19?7/19?8) 107-125 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
AN INVESTIGATION OF NON-NEWTONIAN FLOW PAST A SPHERE
K. ADACHI, Department (Received
N. YOSHIOKA
and K. SAKAI
of Chemical Engineering,
March 12, 1977; in revised
Kyoto University, Kyoto, 606 {Japan) form May 23, 1977)
Summary Any experimental work on the flow of a polymer solution or any theoretical analysis on the basis of a visoelastic constitutive equation does not always bring out viscoelastic effects but may be showing a non-Newtonian viscosity effect. Therefore, in order to obtain a clear understanding about viscoelastic effects, it is desirable to have a sufficient knowledge of the non-Newtonian viscosity effect. To facilitate this, finis-difference numerical solutions of non-Newtonian flow were carried out using a non-Newtonian viscous model for the Reynolds numbers of 0.1, 1.0, 20 and 60. Drag force measurements and flow visualization experiments were also performed over a wide range of experimental conditions using polymer solutions. The present work appears to support the following idea: When compared with the Newtonian case on the basis of ~~~I~~, where q. is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines. 1. Introduction The slow uniform flow of viscoelastic liquids past submerged objects is an idealization of flow situations commonly encountered in industrial processing. It is, furthermore, a flow configuration which has been studied in great detail for Newtonian liquids, both theoretically and experimentally. Hence, it seems to be useful to investigate the effect of the rheological fluid properties on the flow.
108
Using perturbation methods, Leslie [ 11, Caswell and Schwarz [ 21 and Giesekus [ 31 predicted a small downstream shift in the streamlines at 0( We) and a decrease in the drag at 0( We2), where We is the Weissenberg number. More recently, Ultman and Denn [ 41 solved the Oseen equations for a simplified Maxwell liquid and predicted an upstream shift which was contradictory to the previous prediction but agreed with their experimental data of flow visualization. However, Mena and Caswell [ 51 obtained again the same theoretical results as the previous investigators with the exception of Ultman and Denn. Zana et al. [ 61 investigated the Oseen-linearization for viscoelastic flow past a submerged body and showed that the prediction due to Ultman and Denn [ 41 is incorrect because the Oseen-linearization does not even provide a uniformly valid first approximation to small elastic corrections. Broadbent and Mena [ 71 found from their experiments that any shifting of the streamlines is too small to be discerned clearly and the drag force decreases, and Zana et al. [6] reported another experimental result that the streamline shift is very smaller than predicted by the theoretical analysis of Ultman and Denn [ 41 although some upstream shifting of the streamlines was observed for Re . We = O(1). It must be noted that all the experimental and theoretical results may contain not only viscoelastic effects but also non-Newtonian viscosity effects. Previous investigators may have a situation in which non-Newtonian viscous effects are dominant enough to veil viscoelastic effects (such as the normal force and high extension viscosity), even if the analyses of non-Newtonian flow were based on viscoelastic models. Here, the viscoelastic effect is regarded as one which cannot be taken into account by means of any non-Newtonian viscous model. Therefore, in order to obtain a clear understanding of viscoelastic effects, one must first have a sufficient knowledge of the non-Newtonian viscosity effect. This will be done in Section 2: Non-Newtonian effects will be studied in detail by means of a numerical finite-difference analysis for the flow of a non-Newtonian viscous fluid past a sphere. The existing experimental data were taken in a range of low Reynolds numbers and low strain rates, so the effects of viscoelasticity on the drag and flow patterns is expected to have been small even if it arose. Present drag force measurements and flow visualization measurements were performed over a wide range of experimental conditions so as to produce clearer evidences of changes in flow due to viscoelasticity. The relevant experimental results will be described in Sections 3 and 4. As is well known, polymer fluids are viscoelastic and at the same time have a non-Newtonian viscosity. All the experimental data, which are based on polymer fluids, are influenced by both viscoelasticity and non-Newtonian viscosity. Net-effects of viscoelasticity become clear when the contributions due to the non-Newtonian viscosity are subtracted from experimental data for polymer fluids. The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV,p/qo, where
109
no is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it seems that it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines. 2. Flow of a non-Newtonian
viscous fluid past a sphere
The steady flow of an incompressible less form (see [ 81) as: [$-(&)
!$-
4 +Re
$(*)
y]sin
8 = +$E’(wrsin
-% wrsinf3)++$-$ I[ ar
-- cot 0 orsine 2
+ L9.. t r iir
oi__- sine
a%
L
ar2
r2 s
2
I(
aG
r ae
)
r2
ar
+cot~C!!!+i!C!i! E!L &)=-
3w
r
&-2
in dimension-
0)
$]$+$[$(wrsint9)
+2a2J/_cote r arae
*
ar
fluid can be expressed
arae I 7
E2G
a?,
a6 1 ae
a%
+L J
&2?& r* (
(1) (2)
r sin e ’
with (3) where (r, 8, 4) are spherical polar coordinates, $, o and 7) are the stream, vorticity and viscosity functions, respectively, and Re (= DV,p/qo) is the Reynolds number. The boundary conditions for flow past a sphere are given as:
Q?$=() J/ + -$r’
onr=l, sin28
asr+m.
(5)
For the solution of the above boundary value problem, the viscosity func: tion 7) should be given. In the present analysis, the extended Williamson model is used which has the following dimensionless form: q =7), + 1-z
rlo
7)o
[
I
/[l + (2oIZZ#7)
(6)
110
where r), is the limiting viscosity
zze= ___-l
( [( 1
r2 sin20 r2
+ late ( rae
a’+ _--arae
cotead2 ar
2aG2 r ae )
11+2 (la*
and + -_ a’$ +cot&+X2 ar ( arae
r ar
av ar2
r
time2 2
11 ’
ae ) (7)
This model was recommended by Cramer and Marchello [ 91 for the following reason: Reasonable estimates of the extended Williamson model parameters can readily be made upon inspection of experimental, rheological data because the material constants vary monotonically with the concentrations of the solutions, while this is not true for other good models such as the five parameter Powell-Eyring model. It must be noted here that the results obtained in the present analysis are valid not only for the extended Williamson model but also for various non-Newtonian viscous models whose viscosity curve is shown in Fig. 1. Equations (l), (2) and (6) were solved simultaneously using standard numerical finite-difference techniques. Convergence of the computation was too difficult to achieve in the case of q-/~Ia = 0.1 because of too large variations of the vorticity function very near the sphere surface. Therefore, the computation was restricted to the case where +/qO = 0.5 and a = jI = 1.0. In what follows, comparison of the Newtonian case with the non-Newtonian case will be made at the same value of the Reynolds number, which is based on the zero shear viscosity, unless any other indication is given. The fundamental computational results are presented in Table 1, where the values of the drag coefficients for Newtonian fluids are in good agreement with those due to Jenson [lo] and Hamielec et al. [ll]. The pressure distribution over the surface of the sphere is shown in Fig. 2. For non-Newtonian flow, the higher pressure over the rear surface of the sphere gives rise to a smaller form drag C, than for Newtonian flow, as presented in Table 1. The figure also indicates that the difference between the pressure distributions for Newtonian and non-Newtonian fluids becomes smaller as the Reynolds number increases. The pressure distribution is influenced not by the local flow near the sphere but by the whole flow, and
1
ObM-~-e--l-j J2II, L-1 Fig. 1. Viscosity
curve.
10
20
111 TABLE
I
between Newtonian and Non-Newtonian ~___l_._--__ Re(DVdlrlo) 0.1 1.0 ____...-._-_.--~-__...__ cc@.) 259.1 262 + 27.65 27.5 + Cd(N.N.) 197.5 21.27 Cd(N.N.),c,(N.) 0.763 0.769
flows on the basis of DV,pl~o _.._____l_-_.-.__-~-20 60 2.871 2.307 0.804
2.946 ++
1.472 1.118 0.760
C,,(N.) Cp( N.N.) C,W.N.YC,(N.)
83.08 52.25 0.629
8.886 5,941 0.669
1.051 0.977 0.930
1.089 ++
0.620 0.529 0.853
C&N.) CffN.N.) CPN.N.)/Cf(N.)
176 145.3 0.826
18.76 15.33 0.817
1.820 1.330 0.731
1.857 +’
0.852 0.589 0.691
Comparison
SID(N.) S/D(N.N.)
-
-
-
0,tN.I 8,tN.N)
-
-
-
0.458 0.586 43.4 47.4
0.57 * 45 *
%tN.) 0.699 ** 0.687 ** 0.630 ** 0.639 ** ?jlN.N.) 0.649 0.640 0.617 0.641 ______.. ._ ___. .___~~~ -...___.. - ..__-.-~~~~-..--.N. Newtonian flow; N.N. Non-Newtonian flow. .t I-Iamielec [ 111, ++ Jenson [lo], * Taneda 1121; ** Calculated from II, for Newtonian flow by using the viscosity function eqn. (6).
Fig. 2. Pressure Fig. 3. Vorticity
49 Cdegrrel distribution over the sphere. distribution
over the sphere.
e
Cdegrccl
112
the viscous effect extends farther from the sphere at lower Reynolds numbers. This is the reason why the non-Newtonian viscous effect on the pressure distribution is more striking at the low Reynolds numbers. On the other hand, the vorticity distribution over the sphere is influenced not by the whole flow but by the local flow in the neighbourhood of the sphere surface, and the viscous effect is limited to the flow closer to the sphere at higher Reynolds numbers. As is seen in Fig. 3, this is the reason why the nonNewtonian viscous effect on the vorticity distribution is more striking at the higher Reynolds numbers. The higher vorticity for non-Newtonian flow also means that the non-Newtonian fluid flows closer to the sphere except in a part of the wake. This phenomenon may result from the fact that the fluid has a tendency to flow through a region of less mechanical energy loss. Figures 4-7 show the streamlines in their upper halves and the equivorticity lines in their lower halves, where the fluid flows right to left. As has been described above, the non-Newtonian fluid flows closer to the sphere except in a part of the wake. In the case of the Reynolds number of 0.1, the flow pattern is found from the vorticity distribution to be symmetric before and after the sphere for the non-Newtonian fluid as well as for the Newtonian fluid. On the other hand, in a range of Reynolds numbers higher than 0.1, the non-Newtonian fluid flows closer to the front part of the sphere surface and, because of the inertial effects, this causes the fluid to flow
Fig. 4. Streamlines
and equivorticity
lines (Re
= 0.1).
113
/
Fig. 5. Streamlines and equivorticity lines
.
_ .........
(Re =
1.0),
(Re =
20).
U , !
Fig. 6. Streamlines and equivorticity fines
Fig. 7. Streamlines
and equivorticity
lines (Re = 60).
farther away than the Newtonian fluid from the rear part of the sphere surface. The separation angle 0, and the vortex wake length S at Re = 60 are presented in Table 1 and compared with experimental results due to Taneda [12] for a Newtonian fluid. In the case of the separating flow, the separation point moves upstream and the steady twin vortices are elongated when compared with the Newtonian flow. This is clearIy shown in Fig. 7. In the case of the non-Newtonian flow, a higher vorticity is created over the sphere surface, as is shown in Fig. 3, and, moreover, the dissipation energy of recirculation is smaller in a non-Newtonian vortex wake than in a Newtonian wake of similar size because the non-Newtonian fluid is less viscous in the fringe of the steady vortex wake, as compared with the inner region. This is the reason for the larger vortices for the non-Newtonian fIuid. The drag coefficients are presented in Table 1. At the same value of the Reynolds numbers, these drag coefficents are lower for the non-Newtonian flow than for the Newtonian flow. As the Reynolds number increases, C~(N,N.)/C~(N.) increases and C~(N,N.)/C~(N.) decreases, but G(N.N.)/G(N.), however, remains almost constant, where N. and N.N. indicate the Newtonian and non-Newtonian cases, respectively. Moreover it must be noted that for creeping flow, the ratio C#Z’,, should be exactly 2.0 for Newtonian flow but it is expected to be much larger than 2.0 for nonNewtonian flow.
115
30
0
60 8
Fig. 8. Viscosity
SO 120 [degree]
distribution
150
180
over the sphere.
The comparison between the Newtonian and non-Newtonian cases presented above has been made at the same value of the Reynolds number, which is based on the zero shear viscosity qo. Roughly speaking, it has also indicated that the non-Newtonian flow patterns are more washed away than the corresponding Newtonian flow patterns. This is caused by the fact that the viscosities over the sphere are lower for the non-Newtonian flows than for the corresponding Newtonian flows. A more meaningful comparison may be obtained by using the surface-average viscosity in the definition of the Reynolds number. The averaged non-Newtonian viscosity q(N.N.) is given in Table 1 together with q(N.), where q(N.N.) and T(N.) are, respectively, the averaged values of r)(N.N.) and q(N.) which are shown in Fig. 8. The quanti-
TABLE
2
Comparison
between
Newtonian
and non-Newtonian
flows on the basis of DV,plr)
Re(DV.&77)
0.154
1.563
32.41
Cd(N.) C,(N.N.) C,(N.N.)/C,(N.)
172 * 197.6 1.15
19.0 * 21.27 1.12
2.10 * 2.307 1.10
1.11 * 1.118 1.01
C,(N.) C,(N.N.) C,(N.N.)/C,(N.)
55 * 52.25 0.95
6.2 * 5.941 0.96
0.83 * 0.977 1.18
0.50 0.529 1.06
Cr(N.) Cf(N.N.) Cf(N.N.)/Cr(N.)
117 * 145.3 1.24
12.8 * 15.33 1.20
1.27 * 1.330 1.05
0.61 * 0.589 0.97
S/Q
N. )
S/D(N.N.) @AN.) 0JN.N.) ~__~~~~~..~. ...__~~~~~~~ * Interpolated or presumed values.
-
-
93.6
0.72
*
0.586 .-------
53.0 * 47.4 ~~~~~~ ~~~--. -
116
ties r)(N.N.) and n(N.) are calculated respectively by substituting the nonNewtonian and Newtonian strain-rates into the non-Newtonian viscosity function, eqn. (6). When the averaged non-Newtonian viscosity T(N.N.) is not known, @N.) may be a good substitute for it. Table 2 gives a comparison between the Newtonian and non-Newtonian cases at the same value of the Reynolds number which is based on the averaged viscosity ?j. In this comparison, the drag coefficients are higher for the non-Newtonian flows than for the corresponding Newtonian flows, and it is remarkable that this result is apparently opposite to the previous one. It may be reasonable that the present results for DVp/Fj of 93.6 are compared with predictions due to Adachi et al. [S] for non-Newtonian flow of a power-law model at Re . o = 60 because D2-“v”,p m
Re.o=
DV,p
(8)
-7’
Then, the following conclusions are in agreement between these two predictions: When the fluid changes from Newtonian to non-Newtonian, the friction drag coefficients Cf decreases, and the pressure and total drag coefficients C,, C, increase. However, the results concerning the twin vortices are in disagreement between them. This disagreement appears to result from the defect of infinite zero-shear viscosity for the power-law model because this defect occurs at the separation point. Therefore, the present predictions, that the separation point moves downstream and the twin vortices shorten when the fluid changes from Newtonian to non-Newtonian, seem probable. The present drag predictions are compared with the existing ones for the power-law model due to Adachi et al. [ 81 and Wasserman and Slattery [ 131 in Figs. 11 and 12, where the value of 0.639 is used as the power index n for the present model, which is evaluated from Fig. 1. Leslie, like other investigators, predicted a small downstream shift in the streamlines at O((X,--X,)V,/R) and a decrease in the drag at O((hl--X&e K?/R2) for the slow flow of Oldroyd’s viscoelastic model past a sphere when comparing with the corresponding Newtonian flow: 7ik + A,
s7ik +po7;eik 6t
=
eik +
Q,,
6 ~~-~z/z
,
6t
where F/6 t is the Oldroyd convected flow, this viscoelastic model gives
time derivative.
(9) In a steady simple shear
(10) where X1 > X2 since the limiting viscosity cosity qo. Taking account of @I
-
h2)
/Jot*
-
(A,
-
A*)
PO
(
%
1
*
n_ is lower than the zero shear vis-
(11)
117
for flow past a sphere, it seems obvious that the non-Newtonian viscosity is the most dominant factor for the drag decrease, but that it does not cause the downstream shift at O((h,-h,)V,/R). This downstream shift is the net viscoelastic effect which was predicted by means of the Oldroyd model. Moreover, Broadbent and Mena [7] reported a decrease in the drag which is quadratic in velocity. However, the fact seems to be that the drag decreases as (V,/D)” (2 > Q > 0) since the non-Newtonian viscosity is represented better by eqn. (6) than by eqn. (10). 3. Drag force measurements A decrease in the drag for flow of a viscoelastic fluid past an object, which was predicted theoretically by several investigators [l-3], was found in the last section to be due to the non-Newtonian viscosity. Available experimental data show the drag decrease [ 141. On the other hand, a large pressure drop is well known as an experimental fact for converging flow of a polymeric fluid and this effect has been considered to be due to viscoelasticity [15]. An increase in the drag due to viscoelasticity, which is superposed on a decrease in the drag due to the non-Newtonian viscosity, is expected to be observed experimentally for the flow past a sphere as well. However, as the former increase is usually much smaller than the latter decrease, it is difficult to observe the viscoelastic effect, when compared with the Newtonian case on the basis of DV,p/r),. The viscoelastic effect may be discerned more clearly when the difference in the drag between the Newtonian and non-Newtonian cases is investigated on the basis of DV,p/q, because the drag for Newtonian and non-Newtonian viscous fluids are then almost the same. An average viscosity over the sphere was calculated by Q = m( V, /D)“-1
(12)
for each falling-sphere test, where the material constants m and n were determined so that the power-law model might represent the shear-stress-shearrate relationship over the sphere. First, the value of n was assumed for a particular falling-sphere test, and then the maximum shear rate over the sphere was estimated using ?nl,X = C(n) V-/D,
(13)
and C(n) = {A(n)‘.’
+ [B(~)
. Re . 0’/(l+n)]0.6}1:0.6,
(14)
where A(n) was derived from Wasserman and Slattery’s work [ 131 and B(n) from both the work of Adachi et al. [8] and a boundary layer analysis for flow of the power-law fluid past a sphere. The functions A(n) and B(n) are represented in Fig. 9, where broken lines are only extrapolated ones. The power index n was taken so as to represent the rheological curve in a range
118
..
I
I
I
1
7yo40
‘t, ‘I.,
50
-Y, -
20 1
r
7
,. -
a
I
y .. ~... ._.\ .Y. ~_.. '... ... '. ..
4 ,035
~~~-__c---_.~~..:_ \
-1 0.30 ~.
_ 5- t
1
o.25 0.20
2-
__j\jj 1
I 02
03
I
I
0.4
05
I
I
0.6
0.7
I
I
08
09
IO.15 I.0
nr--1
Fig. 9. Coefficients
for calculating
the characteristic
strain rate.
of strain rate from ymax to ymax /lo. This trial-and-error procedure was repeated until the convergence of n was achieved. For the experimental falling-sphere investigation reported here, aqueous glycerol solutions were used as Newtonian fluids, aqueous solutions of sodium carboximethyl cellulose (CMC) as weakly elastic fluids and aqueous solutions of sodium polyacrylate (SPA) and Separan-AP30 as strongly elastic
t-
3x16’~ 1’ I 2x1o-2
I
I
I
III
16' 3
Fig. 10. Viscosity
I
loo c1/scc1
curves for CMC solutions.
III
I 10'
I
I II lo2
119
Present
Re.ozGBO)
(10s
work
c [ WI
%I
1.04
flylOll
1.1 1
A
tungsten
.
others
A I
I
0 I
I
1.2 Adachl
1.0 -
/_g Presen(
ii
0.5
0.6
0.7
0.8
0.9
1.0
II
x
0.6 0.5
n C-l
l
theory
I
I
I
I
0.6
0.7
0.6
0.9
n
Fig. 11. Drag ratio for weakly
elastic
fluids (Re Q 0.1).
Fig. 12. Drag ratio for weakly
elastic
fluids (Re 2 10).
et al. 0
1.0
C-l
fluids. The falling-sphere apparatus used consisted of a circular cylinder of 20 cm in diameter and 200 cm in height, and of nylon, aluminum, glass, steel, lead and tungsten speres. The sphere diameters ranged from 0.1 to 1.5 cm and the terminal falling velocities from 0.01 to 10 cm/s. The measured terminal velocities were corrected for wall effects using Faxen’s formula [ 141. The concentrations for the CMC solutions were 1.11, 1.84, 3.15 and 4.99% by weight and their shear viscosity curves are presented in Fig. 10. The experimental data for these weakly elastic fluids are shown in Fig. 11 for Re < 0.1 and in Fig. 12 for 10 < Re < 80. They are compared with the available theoretical works [ 8,131. For weakly elastic fluids, the experimental data agree approximately with those theoretical predictions which are based on an inelastic fluid model. In the case of slow flow, X k 1.1 _+0.1 and in the case of flow with inertial effects and large strain rates, X = 1.05 f 0.05, where the drag ratio X is defined as X=
Cd(Non-Newtonian) -= Cd(Newtonian)
C,Re( Non-Newtonian) C,Re( Newtonian)
(15)
at the same value of the Reynolds number
D%-“P _ m
Re
_
o
(16)
As Re increases, the drag ratio X decreases as was shown theoretically in the last section.
1101 s q
scpamn
0.70
100 2x16' 16' 5X1$ 13
2xd
Fig. 13. Viscosity
10" p lllucl
10'
curves for SPA and Separan-AP30
solutions.
Present
work
(&/DLlO.O)
Rc.0 i
present work v-/o
1.G
0.1)
( Re.o L D -0.1
0.7 v/i%
Cl
10
S.P
6
LO-
01-1.0
Sepamn
05wtV.
A
A
A
S.P. A.
1 0 WI%
0
a
l
I
I
A.
I
F
.
I
z % ”
0
O.Swt%
A
S. P A.
1
0
WV.
.
A
0 _
o
A ..___h._
.___--
Exp. data
Present
1.0 a2
0.7wt+
S. F! A.
1.0 - 10.0
..,..$._..__:__,.45
1.2
II
Sepamn
1.6
*
1.4
x
0.1 - 1.0
0.3
0.4 "C-l
0.5
03
0.7
0.2
elastic fluids (Re < 0.1).
Fig. 15. Drag ratio for strongly
elastic fluids (Re > 0.1).
++
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
IlC-1
Fig. 14. Drag ratio for strongly
theory
I” Flg.12
121
The concentrations for the SPA solutions were 0.5 and 1.0 wt% and the concentration for the Separan-AP30 solution was 0.7 wt%, whose viscosity curves are presented in Fig. 13. The experimental data for these strongly elastic fluids are shown in Fig. 14 for Re < 0.1 and in Fig. 15 for Re > 0.1. In the case of slow flow, X z 1.25 + 0.25 and in the case of flow with large strain rates, X = 1.45 + 0.25. As Re and L/D increases, the drag ratio X increases. This phenomenon is opposite to the weakly elastic case and seems to be the effect of viscoelasticity. Moreover, the drag ratio for strongly elastic fluids are larger than that for weakly elastic fluids. The mechanism for the drag increase due to the viscoelasticity will be discussed in detail in the following section.
4. Flow patterns and a flow mechanism In order to investigate a characteristic feature of viscoelastic flow, flow visualization was accomplished using fine aluminum dust. In the fahingsphere test, which has been described in the last section, attention was also paid to the global flow in a small cylindrical container. Photographs of the motion of fluids in the whole flow field were taken with long exposure times by a camera which was fixed in space. In the case of a glycerol-water solution, the fluid was made to descend by the falling sphere in a central vertical region of the container and to ascend near the container wall. There was a large circulating flow as is illustrated in Fig. 16(a). The right half of this figure shows schematically a long-time trace of a fluid particle near the path of the falling sphere. The particle is first pushed forward and then dragged by the sphere. On the other hand, in the case of a 1% sodium polyacrylatewater solution, no global circulating flow was observed and a long-time trace of a fluid particle is shown in Fig. 16(b). It indicates that at the end of the motion the particle goes back considerably towards its original location. This reverse flow was observed in a different aspect by Sigli and Coutanceau [ 161 also. They made similar experiments of flow visualization using larger spheres,
T I !
!
7
! I t a.
Viscous
I flow
Fig. 16. Displacement
b. Viscoelastic
of a fluid particle.
flow
122
in which their exposure time was short enough to give clear photographs for instantaneous absolute flow around a sphere moving with constant speed. They found that in a region of the flow field near the sphere, a polymer solution moved in the same direction as the sphere, as in the Newtonian case, but farther down this region the fluid moved in the opposite direction. This implies that, for uniform flow past a sphere which is fixed in space, a part of the viscoelastic fluid is accelerated, to a little more than the uniform speed, far downstream from the sphere. For investigation of viscoelastic effects on the streamlines, other flow visualization experiments were conducted in a horizontal tank with dimensions 50 X 50 X 200 cm. The experimental apparatus and procedure were similar to those used by Taneda [ 121. Photographs were taken with a camera which was connected with a sphere and moved at the same speed as the sphere. Aqueous solutions of sodium polyacrylate were used of concentrations 0.5, 1.0 and 2.0 wt%. The ranges of sphere diameters and speeds were, respectively, 1.11 < D G 2.23 cm and 0.03 < V, < 15 cm/s. In spite of such a wide range of experimental conditions any shifting in the streamlines was too small to be discerned clearly. Some thought should be given to the fact that we could not observe any viscoelastic effect by the present technique, while Sigli and Coutanceau [ 161 could. The changes in the flow patterns due to viscoelasticity might be too small to be observed when referred to a fluid moving at a constant speed, but large enough when referred to a stationary fluid. Another possible reason is that the present container walls might be too far from the sphere to contribute to an increase in the difference between the Newtonian and non-Newtonian flows [ 161. Laminar separating flows with steady twin vortices behind a sphere were also observed. Very dilute aqueous solutions of sodium polyacrylate (0.030, 0.031 wt%) were used to make the twin vortices. Their viscosity curves are shown in Fig. 13. The vortex wake length, the separation angle and the position of the vortex center are presented, respectively, in Figs. 17, 18 and 19. In the case of Newtonian fluids, the present experimental data agree well wih Taneda’s results [ 121. Therefore, the present experiment may be considered to be satisfactorily performed. The vortices for the polymer solutions become smaller and the separation point moves downstream when compared with those for Newtonian fluids on the basis of DV,~/Q. This phenomenon is not considered to be a non-Newtonian viscous effect but to be a net viscoelastic effect because the non-Newtonian viscosity makes the vortices elongate as was shown in Section 2. This shortening of the vortices for separating flow seems to correspond to the upstream shift in the streamlines for slow flow. It is interesting to consider a probable mechanism of the upstream shift in the streamlines. When a fluid comes closer to an object, a part of the fluid is prevented from flowing uniformly together with its remaining part by the presence of the object, so the fluid is deformed near the body. The elongation of an element of a viscoelastic fluid gives rise to a characteristic feature
123
A
4.70 0,094
0
4.70
0.015-
0.15 0.053 I I I lIiIl_
102
1
1111111
1
I1111111 lo3
Ret-l
Ret-1
Fig. 17. Vortex
I
lo2
103
wake length.
Fig. 18. Separation
angle.
of viscoelastic flow as Dealy [ 171 has reviewed. When an element of a viscoelastic fluid is subject to an elongation, a tensile stress responds to it as a result of the memory which the element has. This tensile normal stress is very much higher than the frictional shear stress at the same magnitude of shear rates. Therefore, the viscoelastic fluid is expected to flow in a region of convergent flow so as to suppress a rapid development of such an extensional flow for the purpose of maintaining a force balance: Then, the stream-
; 0.8t
c 4hl-
22
A L70
0.094
0 015
0
0 15
0 053
170
Tonedo
Re 1-l Fig. 19. Position
of the vortex
center.
124
lines may respond to a change of the flow situation (such as the presence of an object) earlier in a viscoelastic fluid than in an inelastic fluid. On the other hand, in a region of divergent flow, the deformed element tends to recover, by the action of the tensile force, the original configuration and position which the element had, and shows a recoil which is illustrated in Fig. 16(b). Behind an object, a part of the fluid, which was retarded by the presence of the object, is considerably accelerated by the tensile force and the streamlines return sooner to uniform streaming following the object than in an inelastic fluid. Thus, due to the characteristic tensile force and recoil, the viscoelastic fluid appears to show a flow, similar to the wine-glass flow in a convergent flow region, before the body, and a flow, similar to the swelling flow in a divergent flow region, after the object. This seems to be the mechanism for the upstream shift. The flow, which is produced by the large tensile stress, is expected to show a large pressure drop in the downstream direction and a large drop of a total stress normal to the stream, although the latter drop is considered to be smaller than the former one. This gives rise to an increase in the normal force drag for the flow of a viscoelastic fluid past an object. However, as polymer fluids are non-Newtonian viscous as well as viscoelastic, the friction drag decreases but the normal force drag may increase or decrease, when compared with the Newtonian case on the basis of DV,p/qo. Whether the normal force drag increases or decreases, depends upon which property is stronger, non-Newtonian viscosity or viscoelasticity, in the flow under consideration. As the streaming velocity increases, the departure from the Newtonian value of the total drag decreases due to the non-Newtonian viscosity but the departure increases due to the viscoelasticity because stretch rates become larger. When comparison is made on the basis of DK.,p/T’f, for increasing Reynolds numbers, the drag ratio X cannot be expected to become larger for weakly elastic fluids, but it may be expected to become larger for strongly elastic fluids. This prediction seems to be confirmed by the present drag measurements. The observations for flow of a viscoelastic fluid past a sphere have been given a consistent explanation. The upstream shift in the streamlines has become a more unquestionable fact although the shift is small. The experimental fact of the upstream shift is opposite to the existing theoretical predictions which are based on constitutive equations of a different type. 4. Conclusions The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV,p/q,, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines in low Reynolds numbers. On the other hand, it is due to the viscoelasticity that the normal force drag increases, that the
125
separating vortices behind the sphere become shift occurs in the streamlines.
smaller, and that an upstream
Acknowledgements The authors wish to express their gratitude to Professor S. Taneda for helpful comments on the experimental apparatus for flow visualization. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
F.M. Leslie, Q. J. Mech. Appl. Math., 14 (1961) 96. B. Caswell and W.H. Scharz, J. Fluid Mech., 13 (1962) 417 H. Giesekus, Rheol. Acta, 3 (1963) 59. J.S. Ultman and M.M. Denn, Chem. Eng. J., 2 (1971) 81. B. Men and B. Caswell, Chem. Eng. J., 8 (1974) 125. E. Zana, G. Tiefenbruck and L.G. Leal, Rheol. Acta, 14 (1975) 891. J.M. Broadbent and B. Mena, Chem. Eng. J., 8 (1974) 11. K. Adachi, N. Yoshioka and K. Yamamoto, Chem. Eng. Sci., 28 (1973) 2033. S.D. Cramer and J.M. Marchello, AIChE J., 14 (1968) 980. V.G. Jenson, Proc. R. Sot. London, 249 (1959) 346. A.E. Hamielec, T.W. Hoffman and L.L. Ross, AIChE J., 13 (1967) 212. S. Taneda, Rep. Res. Inst. Appl. Mech. Kyushu Univ. (Japan), 4 (1956) 99. M.L. Wasserman and J.C. Slattery, AIChE J., 10 (1961) 383. R.M. Turian, AIChE J., 13 (1967) 999. C.D. Han and M. Charles, Trans. Sot. Rheol., 15 (1971) 371. D. Sigli and M. Coutanceau, J. Non-Newtonian Fluid Mech., 2 (1977) 1. J.M. Dealy, Polym. Eng. Sci., 11 (1971) 433. H. Kato, H. Tachibana and K. Oikawa, Bull. JSME, 38 (1972) 809.