- Email: [email protected]
Complex flow of viscoelastic fluids through oscillating pipes. Interesting effects and applications
427
Journal of Non-Newtonian Fluid Mechanics, 6 (1979) 427-448 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
COMPLEX FLOW OF VISCOELASTIC FLUIDS THROUGH OSCILLATING PIPES. INTERESTING EFFECTS AND APPLICATIONS *
B. MENA, 0. MANERO and D.M. BINDING Centro de Znvestigacidn de Materiales and Faculty of Engineering, Mexico. Apdo. Postal 70-360, Mexico City 20. D.F. (Mexico) (Received
National
University
of
August 10.1978)
summary In the present communication, the flow of liquids through a straight pipe which is oscillating longitudinally about a mean position is examined. The basic flow is generated by a constant pressure gradient and the effect of the superimposed oscillations upon the flow is analyzed with particular attention to the flow rate. In the viscous case as well as for an elastic fluid with constant viscosity, no variation in the flow rate is present. This is in agreement with the theoretical analysis. Nevertheless, for viscoelastic fluids, increases in the flow rate of up to twenty times are possible when compared to purely rectilinear flow. This effect is examined for various viscoelastic fluids and relations are found with the basic properties of such fluids. The complex flow situation is analyzed using flow visualization techniques. As a result, the flow appears to be dominated by a shear-thinning effect. A numerical solution using a power-law fluid predicts increases in flow rate which agree qualitatively with the experimental data but are quantitatively different. It is therefore concluded that a more general model must be used for agreement between experiments and theory. In the light of the experimental results, applications are being presently undertaken for the flow of polymer melts in situations of industrial interest. __Introduction and previous work During the last decade attention has been given to nearly viscometric flows of viscoelastic liquids in pipes. In many cases the viscoelastic effects give rise * hesented at the IUTAM Symposium on Non-Newtonian la-Neuve, Belgium, 28 Auguekl September, 1978.
Fluid Mechanics,
Louvain-
428
substantial differences with respect to purely viscous liquids flowing under similar circumstances. One such situation is the flow of non-Newtonian liquids through a pipe which oscillates longitudinally about a mean position. This problem has been recently examined [l-3], and presents very interesting features, which are not feasible in any other pipe flow situation, such as increases of an order of magnitude in the flow rate when compared to purely rectilinear flow. These differences do not make it possible to analyze the problem by using small perturbation schemes as for other pipe flow problems [ 4,5]. In this paper, the experimental situation will be described in detail and a plausible theoretical consideration will be given to the problem. to
I. Experimental considerations Experimental arrangement The experimental arrangement is depicted schematically in Fig. 1. Fluid is pumped by a peristaltic pump (P) through a smoothing bottle (B) into a test section consisting of an interchangeable pipe of circular cross-section. Pipe diameters of 0.26 cm and 0.52 cm were examined. The pipe is supported by low friction bearings and is free to slide in the horizontal direction. A variable speed motor (M) is used to oscillate the pipe at frequencies ranging from 0 to 75 rad/sec. An eccentric shaft provides three different amplitudes of oscillation e.g. 0.65,l and 1.3 cm. The frequencies are monitored by a stroboscopic lamp (S). Two pressure holes connected to a differential pressure transducer (T) provide measurements of the pressure gradient along the pipe. The fluid leaves the test section through a regulated tap and falls into a reservoir (R) where it is recirculated by the pump. A conventional beakerstopwatch (W) method is used to measure the flow rate. TEST SSTZON
Fig. 1. Schematic description of experimental apparatus.
429
Test fluids (a) Newtonian. Water, glycerol and water-glycerol solutions of various viscosities. (b) Viscoehstic. Solutions of polyacrylamide Separan AP-30 (Dow Chemical of Mexico) in distilled water at 0.6 and 1.5% in weight. A solution of 1% Separan AP-30 in 50-50s water-glycerol mixture was also examined. The viscometric characteristics of the above fluids are summarized in Fig. 2. In addition, a solution of 200 p.p.m.w. of Separan AP-30 in corn syrup was used. This solution shows properties of a second-order fluid, i.e. a constant viscosity through a wide range of shear rates, while exhibiting substantial normal-stress differences. For completeness, a 0.05% toluene--sulphonic acid and Cetil-trimethylammonium solution in water was tested. This dilute “soap” solution has been shown to possess high elastic properties in other non-viscometric flow situations [4].
0
0.6%
A 1.5 Y.
AP-30
IN
DISTILLED
AP-30
IN DISTILLED
SHEAR
Fig. 2. Viicometric
WATER WATER
RATE
o’hd
properties of the experimental
fluids.
430
Results Following the notation used in previous articles [l-3], the increase in flow rate Z from the stationary to the oscillating case is defined as: z (%) =
Q
“““e- Qstatx 100%) stat
(1)
where Q,, and Qstitatare the flow rates corresponding to the oscillatory and stationary (Poiseuille) conditions respectively. Fig. 3a shows the percentage increase in flow rate (Z) plotted as a function of the Reynolds number (Re) for the 0.6% aqueous Separan AP-30 solution. The Reynolds number is based on the viscosity value calculated from an experimental curve of flow rate vs. pressure gradient under Poiseuille flow condi-
005
b
Re
0.6 % Separan AP-30 in Hz0 d = 0.26 cm w = 75 Rad/Sec a-A=l.3cm o-A= km o -A = 065cm
431
I
D
>
0
DOL
I
.Ol
I
I
I
.02
.03
.04
RI
Fig. 3. Increase in flow rate (I) vs. Reynolds number (Re)for a 0.6% Separan AP-30 solution in distilled water for various values of amplitude, frequency and pipe diameter,
432
WI
cl I.6 % SEPARAN d 80.26cm A -1.3 cm A-W=7l O-W860 o-w 8 45 +-w= 30
IO0
W-30
in
Il.0
100 -
200 -
100 -
o-
(2 +
#CC ,005
1 +-+-
!
-
.olo
o 1;; -
--,015
0 km~/mc)
b-A 81.3 em 0-Aal.Ocm
I
I
.ca
.OlO
I ,015
o*mV=l
433
1.5 X SEPAltAN d = 0.52 cm A=L3cm
&-0 g --
-+-
e
I
.Ol
0
+ t+ .02
l-3
+
:
AP-30
in Ii.<
n +
.03
0
+
.04
Olctnpc)
d w* 75
M
Rod /Set
loo0
sot
lol (
Fig. 4. Increase in flow rate (I) vs. stationary flow rate (Q& for a 1.5% Separan AP-30 solution in distilled water at various values of amplitude, frequency and pipe diameter.
434
a (WI
b W = 75 Rod/me A-A=L3cm Q-A*l.Ocm 0-A*O.63cm
a 0
a
0
a a
Q
0
”
Ei
0 I 0
. 0
I
I
I
1
I
.02
.04
.o(I
.a
h
J
435
C WI
I 0
0 + +
I .ool
+
I .003
I .Oi
I .OIS
RI
d ox
w=?G Rod /See A-A=I.3cm 0-A=1.0cm 0-A=O.SG cm
Fig. 5. Increase in flow rate (I) vs. Re for a 1% Separan AP-30 solution in 50%~50% glycerol-water for various values of amplitude, frequency and pipe diameter.
436
tions for the pipe considered. The results are for a fixed amplitude and several oscillating frequencies. It may be observed that extremely high increases in flow rate are obtained for low values of Re. The increase becomes moderately high as Re increases and finally approaches asymptotically the Newtonian zero value at higher flow rates. Fig. 3b shows a similar plot of I vs. Re for the same solution as in Fig. 3a but at fixed frequency and various amplitudes. It is clear that in the range of experimental values the maximum increase in flow rate is obtained at the maximum frequency and amplitude. This behaviour should reach a value where it ceases to be valid, but the experimental limitations did not allow us to attain such a point. The effect of increasing the pipe diameter by a factor of two is shown in Figs. 3c and 3d. This effect appears to decrease the values OfI. The behaviour of a different solution, 1.5% Separan AP-30 in distilled water under the same experimental conditions is shown in Figs. 4a-d. Although the same general results are obtained it should be noted that in this case the larger pipe diameter yields a higher increase in flow rate. It may also be observed that at sufficiently low values of the Reynolds number, the increase in flow rates becomes smaller, returning to the origin for zero Re. Results for another solution, 1% AP-30 in 50-50% glycerol-water are shown in Figs. 5a-d. These results follow the same general lines as the ones before but lower values of I are attained when compared to the above solutions. At this point it was thought that the striking increases in flow rate should be due to a combination of elastic and viscous effects in the highly non-viscometric flow situation under consideration; therefore, two more fluids were examined: a dilute soap solution and a second-order type fluid (200 p.p.m.w. Separan AP-30 in corn syrup). The first solution showed no increase in flow rate when compared to the stationary case making it obviously clear that dilute solutions do not present the effect, which is not surprising. Subsequently, the effect of elasticity upon the flow was examined for the 200 p.p.m.w. Separan AP-30 in corn syrup solution; again, no increase in flow rate was presented under the experimental range considered. It is of interest to examine the dependence of the increase in flow rate upon the product amplitude X frequency of the oscillations. This is shown in Fig. 6 for the 0.6% solution. Initially the increase in flow rate I varies as a quadratic function of the product oA and for higher increases, say beyond 400% the dependance is a logarithmic function of wA. Similar results are applicable for the other solutions examined as may be seen in Fig. 7 for the 1% AP-30 in 50-50% glycerol-water solution. Flow visualization
experiments
In order to examine closely the nature of the flow field inside the oscillating
437
I’ h.7 cwO-4 w!imolo’
Ro*3.3xIO“
IO’
_/--_-47 */-+-
l/f
R.. I.Rx lo-8 IO’
a’
.’
/’
I
/3
./ ./’
Fig. 6. Dependence of the increase in flow rate (I) on the parameters OA and (uA)~ several values of the Reynolds number (0.6% aqueous Separan AP-30 solution).
for
pipe, a series of flow visualization experiments were performed. A dye of the same density as the test fluid was injected upstream and photographed,first under unsteady conditions and then once the oscillatory steady state had been reached. The velocity profiles inside the pipe for a purely viscous glycerol solution are shown in Fig. 8 for the stationary case and for an oscillatory flow of 75 rad/sec with an amplitude of 1.3 cm. It may be seen that both profiles are the same. This is not surprising since for a purely viscous fluid there is no increase in flow rate from the stationary to the oscillatory case. Figure 9 shows velocity profiles for the viscoelastic 0.6% Separan AP-30 in distilled water solution flowing under identical conditions as for Fig. 8. It may be seen that at 30 rad/sec the velocity profile has changed drastically suffering an elongation of the centre part. At 45 rad/sec, the change is even
438 I
I
400
200
IO0
looo 3ooo 5ooo Fig. 7. Same as Figure 6 for 1% AP-30 solution in 50%-50%
glycerol-water.
F&:9. Velocity profiles for a solution diameter d = 0.62cm, Re = 0.003.(a) 1.3 cm; (c) oscillatory, o = 46 radlsec, 1.3 cm; (e) oscillatory, o = 75 rad/sec,
of 0.6% Separan AP-30 in distilled water. Pipe Stationary; (b) oscillatory, o = 30 rad/sec, A = A = 1.3 cm; (d) oscillatory, w = 60 rad/sec, A = A = 1.3 cm.
440
more obvious and at 75 rad/sec, it bears no resemblance to the original prome. It should be noti4zed that at this point the increase in flow rate is the largest ce to the experimental results of Figs. 3c and 3d. In order to separate elastic from viscous effects, the 200 p.p.m.w. Separan AP-30 in corn syrup solution was examined under the same conditions as in Figs. 8 and 9. The results are shown in Fig. 10 where enlarged superposition of the negatives corresponding to stationary and oscillatory conditions showed no measurable effect upon the flow pattern. This is in agreement with the theoretical analysis for a linear viscoelastic fluid as will be shown later in this artiole. It would appear f&m the observation of Fig. 9 that perturbations both in the axial and radial directions are present when the oscillations are imposed; to examine this point, a series of time sequence exposures were taken at 45 rad/sec. Each photograph (Fig. 11) has a time lapse of 1 sec. It may be seen that although a radial perturbation appears to be present, it is the axial perturbation that dominates the effect. The centre part is accelerated and the flow velocity is drastically increased. A similar result is&own in Fig. 12 for a frequency of 60 rad/sec. The first three photographs (a, b, c) follow the same line as for Fig. 11 while the last exposures (d and e) were taken after the os&ations had ceased. Photograph d was taken 20 set after the oscillations had been interrupted and it took approximately 2 min for the flow to return to its initial condition (Fig. 12e).
Fig. 10. Velocity profiles for a 200 p.p.m.w. Separan A.P-30 in corn mp. Pipe diameter d = 0.52 cm, Re = 0.003. (a) Stationary; (b) oscillatory, o = 75 rad/sec, A = 1.3 cm.
441
J?ig_ll. Time sequence velocity development, 0.6% AP-80 in water, Re = 0.003, o = 45 radlsec, time inter& 1 aec, A = 1.3 cm.
Experimental
conclusions
(1) The experimental results confirm the drag reduction effects exhibited in other non-viscometric pipe flows showing at the same time very large viscoelastic effects upon the flow due to the superimposed oscillations on the pipe. Increases in flow rate of 20 times are possible for the fluids examined. The increases appear at moderate wlue~ of amplitude and frequency of the oscillations and take place while the pressure gradient along the pipe remains virtually constant. (2) For small increases in flow rate, a quadratic dependence on the product amplitude X frequency (oA) is observed, but as the increases become larger, the dependence becomes a logarithmic function of oA. (3) From the experimental data taken for a second-order type fluid together with a visual examination of the flow pattern inside the pipe, it becomes appar-
442
Fig. 12. Time sequence velocity development, 0.6% AP-30 in water, Re = 0.003, w = 60 rad/sec, A = 1.3 cm (the last two photographs are for 20 set (d) and 2 min (e) after cessation of flow).
443
ent that the dominating factor in the oscillating flow of viscoelastic liquids in pipes is the shear dependent viscosity. The elastic properties of the liquid are of secondary importance in the phenomenon. II. Theoretical considerations Linear viscoelastic
theory
Although a linear viscoelastic model is not capable of predicting increases in flow rate, it does provide information regarding the influence of elasticity upon the flow field under consideration; therefore, a simple analysis will be presented which includes the Newtonian solution as a limiting case [ 61. All physical quantities are referred to cylindrical polar coordinates (I; 8, z), where the z direction is along the axis of the pipe. The pipe has a circular cross section of radius a. The flow is generated by a constant pressure gradient G in the z direction; in addition an oscillatory motion is imposed on the wall r = a. This motion is of the form u = a! exp(int), where the real part is implied and a! represents the product of the amplitude and frequency of the oscillations. It may be shown that the velocity gradually becomes a periodic function of time with the same frequency as the velocity of the boundary. Only this steady periodic state will be considered and any transient phenomena disregarded. The physical components of the velocity vector under such a situation are U, = 0, u = 0, U, = u(r, t) which satisfy the equation of continuity identically. The relevant stress equation of motion is in the z direction and for an incompressible, homogeneous liquid is given by
where p is the density of the liquid, G is the generating constant pressure gradient and p’,, is the relevant component of the extra stress tensor. Equation (2) is to be solved subject to the boundary conditions: 24= (Yexp(int)
at r = a,
au -= ar
at r = 0 for all t .
(3) 0
The viscoelastic liquid is characterized by equations of state of the form: Pik
=
-Pgik
+e’k
I
9
(4)
t lJ& = 2
J -00
$(t - t’) e$)(zc, t’) dt’,=
(5)
where Jl(p--t’)=
s Ep 0
exp(-(t
- t')/T}
dr .
(6)
444
Following the usual notation Pik is the stress tensor, p an arbitrary isotropic pressure, gik is the metric tensor of a fixed coordinate system xi, e#) is the rate of strain tensor and N(T) is the relaxation spectrum. The liquid represented by eqns. (4)-(6) contains as a special case Oldroyd’s B liquid by setting x1---2
N(r) = rlo 2
S(r) + 7)o 7
(7)
Q~-h), 1
1
where q, is the zero shear rate viscosity, X1 and X2 are the relaxation and retardation times respectively and 6 represents a Dirac delta function. It is easily shown that the relation between P,, and the velocity component in the direction of flow u,, reduces to:
au, p:, = ar
m J $(x)
exp(inx) d.r .
(8)
0
Equation (2) can be made homogeneous using as dependent variable the departure of the velocity from the steady state value G(a2 - r2)/4 rlo, i.e. o(r, t) = u(r, t) - G(a2 - r2)/4 q.
(9)
Equation (2) then becomes
aa a2w +;a? -=- q. at p ar2 r ar 1
(
(10)
with appropriate boundary conditions. The solution to eqn. (10) is then,
JoW
w(r, t) = QIJoW)
exp(int)
,
(11)
where, using Equation (7), k is given by
(12) It is easily seen that the purely viscous Newtonian solution is obtained by setting Xi = X2 and the Maxwell fluid solution for A2 = 0. The effect of elasticity upon the velocity profile is shown in Fig. 13 at different times for several values of (n, Xl) and (n, A,). The difference in profile from the purely viscous case is seen to depend on the values of the elastic parameters X, and X2 as expected, but in any case the profiles are not very different. (The velocity profile in this analysis corresponds to Fig. 10 of the flow visualization experiments). Inelastic theory The previous analysis and the experimental evidence seem to indicate that the presence of elasticity in the fluid has a minor effect upon the flow rate.
445
t = - 0.4
t=o
t = 0.4
Fig. 13. Velocity profiles at various times for different values of the elastic parameters nhI and n&. The dashed line is the Newtonian solution.
It is therefore of interest to analyze the oscillating pipe problem for an inelastic model. For the purpose of this analysis the fluid may be characterized by the following equation of state: Pik =-pSik
+ 2 Tj(II1, II,, IIS) et:),
(13)
where U1, II, and 11s are the three principal invariants of the first rate of strain tensor elk) and q is the apparent viscosity of the fluid. It is easily seen that II, = II, = 0 and II, = : (a u/a r)2. Hence the viscosity may be regarded as a function of the rate of shear only. That is, 77= V(1), where + = au/ar.
(14)
The relevant equation of motion is identical to eqn. (2) of the previous section, i.e. (15) with p’,, = ~TJ(+) and the same boundary conditions as in the previous analysis, i.e. u=Awcoswt
atr=a,
au/at-
atr=Oforallt.
= 0
(16)
The flow rate Q at any instant will be given by Q = 2 r p r(u -A
cosot)dr.
(17)
0 The
time average flow rate Q,,,
once the flow has reached a steady oscil-
446
latory flow is then, 2njw =Q osc
"J 2lr
Qdt.
(18)
0
For simplicity a power-law model, i.e. Pi, = KT” was used. The constants K = 18.2 and n = 0.36 were determined from experimental stress-shear rate data. The solution of eqn. (15) was effected using a standard finite difference method.
w=4G
w=30 IO
/ w=o (EXPERIMENTAL)
IO
IO’
/
Y’
I
W-O
IO< PRESSURE
GRADIENT
POWER LAW SOLUTION
IO' G(dynnlcmz~
Fig. 14. Numerical solution of flow rate (Q) vs. pressure gradient (G) for a 0.6% Separan AP-30 solution in distilled water for various frequencies of oscillation at an amplitude of 1.3 cm and a pipe diameter of 0.26 cm. The solid line represents the theoretical zero-frequency curve and the dashed line is the experimental curve.
447
The results are shown in Fig. 14 as a plot of flow rate Q vs. pressure gradient G along the pipe. The solid line represents the stationary numerical solution while the points represent the experimental data for zero frequency. It is seen that the agreement is good until the power-law approximation ceases to be valid. This is shown by the dashed line. The lines corresponding to 30 rad/ set and 45 rad/sec are also shown as obtained from the numerical method for the 0.6% solution at an amplitude of 1.3 cm and a pipe diameter of 0.26 cm. The results for the 30 and 45 rad/sec show very large increases in flow rate when compared to the zero-frequency curve (more than an order of magnitude). This increase exceeds the experimental results of previous sections although the qualitative behaviour is preserved. The region of the maximum increase in flow rate may be predicted by the inelastic approximation, since the rates of shear involved are sufficiently high, but not so for the eero-frequency curve in that region where the shear rates are too small for the approximation to be valid. Therefore it becomes apparent that a more general model is necessary to improve the theoretical results. That is, a model which will include a shear rate dependent viscosity as well as elastic effects. This will involve more sophisticated numerical schemes for the solution of the equations. Theoretical conclusions (1) A linear viscoelastic model predicts small differences in the flow pattern when compared to a purely viscous model. This is in agreement with experimental results. Nevertheless due to the obvious limitations of the model, no increase in flow rate can be obtained. (2) An inelastic “power-law type” model predicts large increases in flow rate for the regions of small pressure gradient in qualitative agreement with the experimental data. Nevertheless quantitative agreement is not possible and is different in some cases by an order of magnitude. (3) It becomes apparent that although the elastic properties of the fluid might be of secondary importance in the flow through an oscillating pipe, it is not possible to neglect them completely for an accurate theoretical description of the flow problem. Some interesting applications The interesting effects exhibited by the flow of viscoelastic solutions through oscillating pipes suggest numerous possibilities for industrial applications. At present, a project is being undertaken to examine the problem for polymer melts, with the purpose of modifying and improving the extrusion process. Although the programme is still at an early stage it is expected to obtain useful results in the near future.
443
Acknowledgements The authors wish to acknowledge the invaluable photographic aid and comments of Dr. Rati Valenzuela of the Centro de InvestigacGn de Materiales, Mexico. Helpful discussions with Prof. A.B. Metzner of the University of Delaware are also acknowledged. The interest in this work was pioneered and encouraged by Prof. K. Walters of the University College of Wales, to whom we are deeply grateful.
1 B. Mena and 0. Manero, in C. Klaxon and J. Kubat (Edr.), Proc. VIIth. Int. Congr. on Rheology, Gothenburg, Sweden, 1976, p. 400. 2 0. Manero and B. Mena, Rheol. Acta, 16(6) (1977) 673. 3 0. Manero, B. Mena and R. Valenzuela, Rheol. Acta, 17 (in press). 4 K. Walters, Rheometry, Chapman and Hall, London, 1975. 5 R. Gunn, B. Mena and K. Walters, Z. Angew. Math. Phys., 25 (1974) 591. 6 A. Kaye and B. Mena, Rev. Mex. Fis., (in prers).
Journal of Non-Newtonian Fluid Mechanics, 6 (1979) 427-448 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
COMPLEX FLOW OF VISCOELASTIC FLUIDS THROUGH OSCILLATING PIPES. INTERESTING EFFECTS AND APPLICATIONS *
B. MENA, 0. MANERO and D.M. BINDING Centro de Znvestigacidn de Materiales and Faculty of Engineering, Mexico. Apdo. Postal 70-360, Mexico City 20. D.F. (Mexico) (Received
National
University
of
August 10.1978)
summary In the present communication, the flow of liquids through a straight pipe which is oscillating longitudinally about a mean position is examined. The basic flow is generated by a constant pressure gradient and the effect of the superimposed oscillations upon the flow is analyzed with particular attention to the flow rate. In the viscous case as well as for an elastic fluid with constant viscosity, no variation in the flow rate is present. This is in agreement with the theoretical analysis. Nevertheless, for viscoelastic fluids, increases in the flow rate of up to twenty times are possible when compared to purely rectilinear flow. This effect is examined for various viscoelastic fluids and relations are found with the basic properties of such fluids. The complex flow situation is analyzed using flow visualization techniques. As a result, the flow appears to be dominated by a shear-thinning effect. A numerical solution using a power-law fluid predicts increases in flow rate which agree qualitatively with the experimental data but are quantitatively different. It is therefore concluded that a more general model must be used for agreement between experiments and theory. In the light of the experimental results, applications are being presently undertaken for the flow of polymer melts in situations of industrial interest. __Introduction and previous work During the last decade attention has been given to nearly viscometric flows of viscoelastic liquids in pipes. In many cases the viscoelastic effects give rise * hesented at the IUTAM Symposium on Non-Newtonian la-Neuve, Belgium, 28 Auguekl September, 1978.
Fluid Mechanics,
Louvain-
428
substantial differences with respect to purely viscous liquids flowing under similar circumstances. One such situation is the flow of non-Newtonian liquids through a pipe which oscillates longitudinally about a mean position. This problem has been recently examined [l-3], and presents very interesting features, which are not feasible in any other pipe flow situation, such as increases of an order of magnitude in the flow rate when compared to purely rectilinear flow. These differences do not make it possible to analyze the problem by using small perturbation schemes as for other pipe flow problems [ 4,5]. In this paper, the experimental situation will be described in detail and a plausible theoretical consideration will be given to the problem. to
I. Experimental considerations Experimental arrangement The experimental arrangement is depicted schematically in Fig. 1. Fluid is pumped by a peristaltic pump (P) through a smoothing bottle (B) into a test section consisting of an interchangeable pipe of circular cross-section. Pipe diameters of 0.26 cm and 0.52 cm were examined. The pipe is supported by low friction bearings and is free to slide in the horizontal direction. A variable speed motor (M) is used to oscillate the pipe at frequencies ranging from 0 to 75 rad/sec. An eccentric shaft provides three different amplitudes of oscillation e.g. 0.65,l and 1.3 cm. The frequencies are monitored by a stroboscopic lamp (S). Two pressure holes connected to a differential pressure transducer (T) provide measurements of the pressure gradient along the pipe. The fluid leaves the test section through a regulated tap and falls into a reservoir (R) where it is recirculated by the pump. A conventional beakerstopwatch (W) method is used to measure the flow rate. TEST SSTZON
Fig. 1. Schematic description of experimental apparatus.
429
Test fluids (a) Newtonian. Water, glycerol and water-glycerol solutions of various viscosities. (b) Viscoehstic. Solutions of polyacrylamide Separan AP-30 (Dow Chemical of Mexico) in distilled water at 0.6 and 1.5% in weight. A solution of 1% Separan AP-30 in 50-50s water-glycerol mixture was also examined. The viscometric characteristics of the above fluids are summarized in Fig. 2. In addition, a solution of 200 p.p.m.w. of Separan AP-30 in corn syrup was used. This solution shows properties of a second-order fluid, i.e. a constant viscosity through a wide range of shear rates, while exhibiting substantial normal-stress differences. For completeness, a 0.05% toluene--sulphonic acid and Cetil-trimethylammonium solution in water was tested. This dilute “soap” solution has been shown to possess high elastic properties in other non-viscometric flow situations [4].
0
0.6%
A 1.5 Y.
AP-30
IN
DISTILLED
AP-30
IN DISTILLED
SHEAR
Fig. 2. Viicometric
WATER WATER
RATE
o’hd
properties of the experimental
fluids.
430
Results Following the notation used in previous articles [l-3], the increase in flow rate Z from the stationary to the oscillating case is defined as: z (%) =
Q
“““e- Qstatx 100%) stat
(1)
where Q,, and Qstitatare the flow rates corresponding to the oscillatory and stationary (Poiseuille) conditions respectively. Fig. 3a shows the percentage increase in flow rate (Z) plotted as a function of the Reynolds number (Re) for the 0.6% aqueous Separan AP-30 solution. The Reynolds number is based on the viscosity value calculated from an experimental curve of flow rate vs. pressure gradient under Poiseuille flow condi-
005
b
Re
0.6 % Separan AP-30 in Hz0 d = 0.26 cm w = 75 Rad/Sec a-A=l.3cm o-A= km o -A = 065cm
431
I
D
>
0
DOL
I
.Ol
I
I
I
.02
.03
.04
RI
Fig. 3. Increase in flow rate (I) vs. Reynolds number (Re)for a 0.6% Separan AP-30 solution in distilled water for various values of amplitude, frequency and pipe diameter,
432
WI
cl I.6 % SEPARAN d 80.26cm A -1.3 cm A-W=7l O-W860 o-w 8 45 +-w= 30
IO0
W-30
in
Il.0
100 -
200 -
100 -
o-
(2 +
#CC ,005
1 +-+-
!
-
.olo
o 1;; -
--,015
0 km~/mc)
b-A 81.3 em 0-Aal.Ocm
I
I
.ca
.OlO
I ,015
o*mV=l
433
1.5 X SEPAltAN d = 0.52 cm A=L3cm
&-0 g --
-+-
e
I
.Ol
0
+ t+ .02
l-3
+
:
AP-30
in Ii.<
n +
.03
0
+
.04
Olctnpc)
d w* 75
M
Rod /Set
loo0
sot
lol (
Fig. 4. Increase in flow rate (I) vs. stationary flow rate (Q& for a 1.5% Separan AP-30 solution in distilled water at various values of amplitude, frequency and pipe diameter.
434
a (WI
b W = 75 Rod/me A-A=L3cm Q-A*l.Ocm 0-A*O.63cm
a 0
a
0
a a
Q
0
”
Ei
0 I 0
. 0
I
I
I
1
I
.02
.04
.o(I
.a
h
J
435
C WI
I 0
0 + +
I .ool
+
I .003
I .Oi
I .OIS
RI
d ox
w=?G Rod /See A-A=I.3cm 0-A=1.0cm 0-A=O.SG cm
Fig. 5. Increase in flow rate (I) vs. Re for a 1% Separan AP-30 solution in 50%~50% glycerol-water for various values of amplitude, frequency and pipe diameter.
436
tions for the pipe considered. The results are for a fixed amplitude and several oscillating frequencies. It may be observed that extremely high increases in flow rate are obtained for low values of Re. The increase becomes moderately high as Re increases and finally approaches asymptotically the Newtonian zero value at higher flow rates. Fig. 3b shows a similar plot of I vs. Re for the same solution as in Fig. 3a but at fixed frequency and various amplitudes. It is clear that in the range of experimental values the maximum increase in flow rate is obtained at the maximum frequency and amplitude. This behaviour should reach a value where it ceases to be valid, but the experimental limitations did not allow us to attain such a point. The effect of increasing the pipe diameter by a factor of two is shown in Figs. 3c and 3d. This effect appears to decrease the values OfI. The behaviour of a different solution, 1.5% Separan AP-30 in distilled water under the same experimental conditions is shown in Figs. 4a-d. Although the same general results are obtained it should be noted that in this case the larger pipe diameter yields a higher increase in flow rate. It may also be observed that at sufficiently low values of the Reynolds number, the increase in flow rates becomes smaller, returning to the origin for zero Re. Results for another solution, 1% AP-30 in 50-50% glycerol-water are shown in Figs. 5a-d. These results follow the same general lines as the ones before but lower values of I are attained when compared to the above solutions. At this point it was thought that the striking increases in flow rate should be due to a combination of elastic and viscous effects in the highly non-viscometric flow situation under consideration; therefore, two more fluids were examined: a dilute soap solution and a second-order type fluid (200 p.p.m.w. Separan AP-30 in corn syrup). The first solution showed no increase in flow rate when compared to the stationary case making it obviously clear that dilute solutions do not present the effect, which is not surprising. Subsequently, the effect of elasticity upon the flow was examined for the 200 p.p.m.w. Separan AP-30 in corn syrup solution; again, no increase in flow rate was presented under the experimental range considered. It is of interest to examine the dependence of the increase in flow rate upon the product amplitude X frequency of the oscillations. This is shown in Fig. 6 for the 0.6% solution. Initially the increase in flow rate I varies as a quadratic function of the product oA and for higher increases, say beyond 400% the dependance is a logarithmic function of wA. Similar results are applicable for the other solutions examined as may be seen in Fig. 7 for the 1% AP-30 in 50-50% glycerol-water solution. Flow visualization
experiments
In order to examine closely the nature of the flow field inside the oscillating
437
I’ h.7 cwO-4 w!imolo’
Ro*3.3xIO“
IO’
_/--_-47 */-+-
l/f
R.. I.Rx lo-8 IO’
a’
.’
/’
I
/3
./ ./’
Fig. 6. Dependence of the increase in flow rate (I) on the parameters OA and (uA)~ several values of the Reynolds number (0.6% aqueous Separan AP-30 solution).
for
pipe, a series of flow visualization experiments were performed. A dye of the same density as the test fluid was injected upstream and photographed,first under unsteady conditions and then once the oscillatory steady state had been reached. The velocity profiles inside the pipe for a purely viscous glycerol solution are shown in Fig. 8 for the stationary case and for an oscillatory flow of 75 rad/sec with an amplitude of 1.3 cm. It may be seen that both profiles are the same. This is not surprising since for a purely viscous fluid there is no increase in flow rate from the stationary to the oscillatory case. Figure 9 shows velocity profiles for the viscoelastic 0.6% Separan AP-30 in distilled water solution flowing under identical conditions as for Fig. 8. It may be seen that at 30 rad/sec the velocity profile has changed drastically suffering an elongation of the centre part. At 45 rad/sec, the change is even
438 I
I
400
200
IO0
looo 3ooo 5ooo Fig. 7. Same as Figure 6 for 1% AP-30 solution in 50%-50%
glycerol-water.
F&:9. Velocity profiles for a solution diameter d = 0.62cm, Re = 0.003.(a) 1.3 cm; (c) oscillatory, o = 46 radlsec, 1.3 cm; (e) oscillatory, o = 75 rad/sec,
of 0.6% Separan AP-30 in distilled water. Pipe Stationary; (b) oscillatory, o = 30 rad/sec, A = A = 1.3 cm; (d) oscillatory, w = 60 rad/sec, A = A = 1.3 cm.
440
more obvious and at 75 rad/sec, it bears no resemblance to the original prome. It should be noti4zed that at this point the increase in flow rate is the largest ce to the experimental results of Figs. 3c and 3d. In order to separate elastic from viscous effects, the 200 p.p.m.w. Separan AP-30 in corn syrup solution was examined under the same conditions as in Figs. 8 and 9. The results are shown in Fig. 10 where enlarged superposition of the negatives corresponding to stationary and oscillatory conditions showed no measurable effect upon the flow pattern. This is in agreement with the theoretical analysis for a linear viscoelastic fluid as will be shown later in this artiole. It would appear f&m the observation of Fig. 9 that perturbations both in the axial and radial directions are present when the oscillations are imposed; to examine this point, a series of time sequence exposures were taken at 45 rad/sec. Each photograph (Fig. 11) has a time lapse of 1 sec. It may be seen that although a radial perturbation appears to be present, it is the axial perturbation that dominates the effect. The centre part is accelerated and the flow velocity is drastically increased. A similar result is&own in Fig. 12 for a frequency of 60 rad/sec. The first three photographs (a, b, c) follow the same line as for Fig. 11 while the last exposures (d and e) were taken after the os&ations had ceased. Photograph d was taken 20 set after the oscillations had been interrupted and it took approximately 2 min for the flow to return to its initial condition (Fig. 12e).
Fig. 10. Velocity profiles for a 200 p.p.m.w. Separan A.P-30 in corn mp. Pipe diameter d = 0.52 cm, Re = 0.003. (a) Stationary; (b) oscillatory, o = 75 rad/sec, A = 1.3 cm.
441
J?ig_ll. Time sequence velocity development, 0.6% AP-80 in water, Re = 0.003, o = 45 radlsec, time inter& 1 aec, A = 1.3 cm.
Experimental
conclusions
(1) The experimental results confirm the drag reduction effects exhibited in other non-viscometric pipe flows showing at the same time very large viscoelastic effects upon the flow due to the superimposed oscillations on the pipe. Increases in flow rate of 20 times are possible for the fluids examined. The increases appear at moderate wlue~ of amplitude and frequency of the oscillations and take place while the pressure gradient along the pipe remains virtually constant. (2) For small increases in flow rate, a quadratic dependence on the product amplitude X frequency (oA) is observed, but as the increases become larger, the dependence becomes a logarithmic function of oA. (3) From the experimental data taken for a second-order type fluid together with a visual examination of the flow pattern inside the pipe, it becomes appar-
442
Fig. 12. Time sequence velocity development, 0.6% AP-30 in water, Re = 0.003, w = 60 rad/sec, A = 1.3 cm (the last two photographs are for 20 set (d) and 2 min (e) after cessation of flow).
443
ent that the dominating factor in the oscillating flow of viscoelastic liquids in pipes is the shear dependent viscosity. The elastic properties of the liquid are of secondary importance in the phenomenon. II. Theoretical considerations Linear viscoelastic
theory
Although a linear viscoelastic model is not capable of predicting increases in flow rate, it does provide information regarding the influence of elasticity upon the flow field under consideration; therefore, a simple analysis will be presented which includes the Newtonian solution as a limiting case [ 61. All physical quantities are referred to cylindrical polar coordinates (I; 8, z), where the z direction is along the axis of the pipe. The pipe has a circular cross section of radius a. The flow is generated by a constant pressure gradient G in the z direction; in addition an oscillatory motion is imposed on the wall r = a. This motion is of the form u = a! exp(int), where the real part is implied and a! represents the product of the amplitude and frequency of the oscillations. It may be shown that the velocity gradually becomes a periodic function of time with the same frequency as the velocity of the boundary. Only this steady periodic state will be considered and any transient phenomena disregarded. The physical components of the velocity vector under such a situation are U, = 0, u = 0, U, = u(r, t) which satisfy the equation of continuity identically. The relevant stress equation of motion is in the z direction and for an incompressible, homogeneous liquid is given by
where p is the density of the liquid, G is the generating constant pressure gradient and p’,, is the relevant component of the extra stress tensor. Equation (2) is to be solved subject to the boundary conditions: 24= (Yexp(int)
at r = a,
au -= ar
at r = 0 for all t .
(3) 0
The viscoelastic liquid is characterized by equations of state of the form: Pik
=
-Pgik
+e’k
I
9
(4)
t lJ& = 2
J -00
$(t - t’) e$)(zc, t’) dt’,=
(5)
where Jl(p--t’)=
s Ep 0
exp(-(t
- t')/T}
dr .
(6)
444
Following the usual notation Pik is the stress tensor, p an arbitrary isotropic pressure, gik is the metric tensor of a fixed coordinate system xi, e#) is the rate of strain tensor and N(T) is the relaxation spectrum. The liquid represented by eqns. (4)-(6) contains as a special case Oldroyd’s B liquid by setting x1---2
N(r) = rlo 2
S(r) + 7)o 7
(7)
Q~-h), 1
1
where q, is the zero shear rate viscosity, X1 and X2 are the relaxation and retardation times respectively and 6 represents a Dirac delta function. It is easily shown that the relation between P,, and the velocity component in the direction of flow u,, reduces to:
au, p:, = ar
m J $(x)
exp(inx) d.r .
(8)
0
Equation (2) can be made homogeneous using as dependent variable the departure of the velocity from the steady state value G(a2 - r2)/4 rlo, i.e. o(r, t) = u(r, t) - G(a2 - r2)/4 q.
(9)
Equation (2) then becomes
aa a2w +;a? -=- q. at p ar2 r ar 1
(
(10)
with appropriate boundary conditions. The solution to eqn. (10) is then,
JoW
w(r, t) = QIJoW)
exp(int)
,
(11)
where, using Equation (7), k is given by
(12) It is easily seen that the purely viscous Newtonian solution is obtained by setting Xi = X2 and the Maxwell fluid solution for A2 = 0. The effect of elasticity upon the velocity profile is shown in Fig. 13 at different times for several values of (n, Xl) and (n, A,). The difference in profile from the purely viscous case is seen to depend on the values of the elastic parameters X, and X2 as expected, but in any case the profiles are not very different. (The velocity profile in this analysis corresponds to Fig. 10 of the flow visualization experiments). Inelastic theory The previous analysis and the experimental evidence seem to indicate that the presence of elasticity in the fluid has a minor effect upon the flow rate.
445
t = - 0.4
t=o
t = 0.4
Fig. 13. Velocity profiles at various times for different values of the elastic parameters nhI and n&. The dashed line is the Newtonian solution.
It is therefore of interest to analyze the oscillating pipe problem for an inelastic model. For the purpose of this analysis the fluid may be characterized by the following equation of state: Pik =-pSik
+ 2 Tj(II1, II,, IIS) et:),
(13)
where U1, II, and 11s are the three principal invariants of the first rate of strain tensor elk) and q is the apparent viscosity of the fluid. It is easily seen that II, = II, = 0 and II, = : (a u/a r)2. Hence the viscosity may be regarded as a function of the rate of shear only. That is, 77= V(1), where + = au/ar.
(14)
The relevant equation of motion is identical to eqn. (2) of the previous section, i.e. (15) with p’,, = ~TJ(+) and the same boundary conditions as in the previous analysis, i.e. u=Awcoswt
atr=a,
au/at-
atr=Oforallt.
= 0
(16)
The flow rate Q at any instant will be given by Q = 2 r p r(u -A
cosot)dr.
(17)
0 The
time average flow rate Q,,,
once the flow has reached a steady oscil-
446
latory flow is then, 2njw =Q osc
"J 2lr
Qdt.
(18)
0
For simplicity a power-law model, i.e. Pi, = KT” was used. The constants K = 18.2 and n = 0.36 were determined from experimental stress-shear rate data. The solution of eqn. (15) was effected using a standard finite difference method.
w=4G
w=30 IO
/ w=o (EXPERIMENTAL)
IO
IO’
/
Y’
I
W-O
IO< PRESSURE
GRADIENT
POWER LAW SOLUTION
IO' G(dynnlcmz~
Fig. 14. Numerical solution of flow rate (Q) vs. pressure gradient (G) for a 0.6% Separan AP-30 solution in distilled water for various frequencies of oscillation at an amplitude of 1.3 cm and a pipe diameter of 0.26 cm. The solid line represents the theoretical zero-frequency curve and the dashed line is the experimental curve.
447
The results are shown in Fig. 14 as a plot of flow rate Q vs. pressure gradient G along the pipe. The solid line represents the stationary numerical solution while the points represent the experimental data for zero frequency. It is seen that the agreement is good until the power-law approximation ceases to be valid. This is shown by the dashed line. The lines corresponding to 30 rad/ set and 45 rad/sec are also shown as obtained from the numerical method for the 0.6% solution at an amplitude of 1.3 cm and a pipe diameter of 0.26 cm. The results for the 30 and 45 rad/sec show very large increases in flow rate when compared to the zero-frequency curve (more than an order of magnitude). This increase exceeds the experimental results of previous sections although the qualitative behaviour is preserved. The region of the maximum increase in flow rate may be predicted by the inelastic approximation, since the rates of shear involved are sufficiently high, but not so for the eero-frequency curve in that region where the shear rates are too small for the approximation to be valid. Therefore it becomes apparent that a more general model is necessary to improve the theoretical results. That is, a model which will include a shear rate dependent viscosity as well as elastic effects. This will involve more sophisticated numerical schemes for the solution of the equations. Theoretical conclusions (1) A linear viscoelastic model predicts small differences in the flow pattern when compared to a purely viscous model. This is in agreement with experimental results. Nevertheless due to the obvious limitations of the model, no increase in flow rate can be obtained. (2) An inelastic “power-law type” model predicts large increases in flow rate for the regions of small pressure gradient in qualitative agreement with the experimental data. Nevertheless quantitative agreement is not possible and is different in some cases by an order of magnitude. (3) It becomes apparent that although the elastic properties of the fluid might be of secondary importance in the flow through an oscillating pipe, it is not possible to neglect them completely for an accurate theoretical description of the flow problem. Some interesting applications The interesting effects exhibited by the flow of viscoelastic solutions through oscillating pipes suggest numerous possibilities for industrial applications. At present, a project is being undertaken to examine the problem for polymer melts, with the purpose of modifying and improving the extrusion process. Although the programme is still at an early stage it is expected to obtain useful results in the near future.
443
Acknowledgements The authors wish to acknowledge the invaluable photographic aid and comments of Dr. Rati Valenzuela of the Centro de InvestigacGn de Materiales, Mexico. Helpful discussions with Prof. A.B. Metzner of the University of Delaware are also acknowledged. The interest in this work was pioneered and encouraged by Prof. K. Walters of the University College of Wales, to whom we are deeply grateful.
1 B. Mena and 0. Manero, in C. Klaxon and J. Kubat (Edr.), Proc. VIIth. Int. Congr. on Rheology, Gothenburg, Sweden, 1976, p. 400. 2 0. Manero and B. Mena, Rheol. Acta, 16(6) (1977) 673. 3 0. Manero, B. Mena and R. Valenzuela, Rheol. Acta, 17 (in press). 4 K. Walters, Rheometry, Chapman and Hall, London, 1975. 5 R. Gunn, B. Mena and K. Walters, Z. Angew. Math. Phys., 25 (1974) 591. 6 A. Kaye and B. Mena, Rev. Mex. Fis., (in prers).