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Non-Newtonian flow in pipes of non-circular cross section
Computers & Fluids, Vol. 2, pp. 317-338. Pergamon Press, 1974. Printed in Great Britain
NON-NEWTONIAN FLOW IN PIPES OF NON-CIRCULAR CROSS SECTION A. G. DODSON,* P. TOWNSEND t and K. WALTERS~ Department of Applied Mathematics, University College of Wales, Aberystwyth, Wales (Received 16 April 1973) Abstract--Rectilinear flow through pipes of non-circular cross section in the case of elasticoviscous liquids is not possible in general and some secondary flow in the cross section of the pipe is to be expected. The present paper contains a detailed theoretical and experimental study of the problem for pipes of square and rectangular cross section. Our main concern is the form that the secondary flows take and their influence on the flow rate through the pipe. The governing equations, which are non-linear partial differential equations, are solved numerically using finite difference methods used in conjunction with S.L.O.R. It is shown, that, for both the square and rectangular cross sections, there are eight vortices present. These have the same strength for the square cross section, but four of the eight become progressively weaker as the ratio of the sides of the rectangle is increased. The effect of a variable (shear-dependent) viscosity on the flow rate is found to be substantial, but the effect of the secondary flows on the flow rate is predicted to be small unless the fluid has some rather unusual properties. The experimental results are in general agreement with the theoretical predictions.
1. I N T R O D U C T I O N
NoN-NEWTONIAN flOW through straight pipes of arbitrary cross section has received considerable attention in the literature. The special case of Poiseuille flow (i.e. flow in a pipe of circular cross section under a constant pressure gradient) is very well documented and is widely used in viscometric measurements. The fact that the flow field in the case of pipes of non-circular cross section can be significantly modified by non-Newtonian behaviour has resulted in a substantial research effort in this area also. It is well known that for Newtonian fluids, rectilinear flow is always possible in a straight pipe of arbitrary cross section. For materials that exhibit normal-stress effects in a steady shear flow this is not possible in general and some secondary flow in the cross section of the pipe is to be expected. This has been studied from a rather general point of view by Ericksen[1] and Oldroyd[2] and with more specific rheological equations of state in mind by Oldroyd[3] and Waiters[4]. It transpires that a non-zero second normal stress difference is a necessary but not a sumcient condition for secondary flow to exist. The computation of the axial velocity component in the exceptional case where rectilinear flow is possible (or where the secondary flow is assumed to have a negligible influence on the flow rate) has been carried out by a number of workers, [5-10]. This is dependent only * Present address: British Food Manufacturing Industries Research Association. l.¢atberhead, Surrey. t Present address: Department of Computer Science, University College of Swansea. :1:K. Waiters, Prof. The University College of Wales, Applied Math. Dept. Physical Sciences Bldg., Penglais, Aberystwyth, SY 23 3BZ, Wales. 317
CAF Vol. 2 No. 3/4--F
318
A.G. DODSON, P. TOWNSEND and K. WALTERS
on the apparent viscosity function associated with the liquid and involves the solution of a non-linear partial differential equation. For realistic viscosity functions, the use of a high speed computer is indispensible in this connection. Green and Rivlin[l l] were the first to calculate the form of the secondary flow in the case of pipes of elliptic cross-section. They considered a special case of the Reiner-Rivlin fluid [cf. 12, 13]. The work was extended by Rivlin and Langlois[14-16] for a more general class of fluids. It transpires that if U is a characteristic velocity, 2. a characteristic relaxation time of the fluid, and L a characteristic length, the secondary flow stream function $ is related to the primary flow velocity w along the pipe by a relation of the form[17] ~=-
w.
(1)
The cube power in (1) implies that any secondary flow in the plane of the pipe is not likely to be strong. A number of workers[17-21] have considered the form of the secondary flow to be expected in pipes of various cross sections, with Pipkin[17] basing his ideas on symmetry considerations. We shall show later than such considerations do not give the whole story, a point which Pipkin himself acknowledged. Very few workers have given detailed consideration to the effect of the secondary flow on the axial velocity component (and hence on the flow rate). Green and Rivlin[11] and Wheeler and Wissler[8] have discussed the problem for a Reiner-Rivlin fluid, the latter authors also present some sample experimental results for a pipe of square cross-section. In contrast to the extensive theoretical literature on secondary flow in pipes of noncircular cross section, there is very little experimental verification of the predicted flow. There is a now celebrated private communication attributed to E. Kearsley which is quoted by many theoreticians to support the existence of secondary flows. In addition, there is experimental evidence of secondary flow in pipes of elliptic cross section for aqueous solutions of polyacrylamide[22] and polymer melts[23]. In the present paper, we give detailed theoretical and experimental consideration to the flow of elastico-viscous liquids through pipes of square and rectangular cross section, the choice of cross-section being governed partly by theoretical considerations (since the resulting partial differential equations are solved numerically) and partly for such experimental considerations as the ease of manufacture of the pipes and of carrying out flow visualization studies. 2. THEORY We refer all physical quantities to a set of Cartesian coordinates (x, y, z), the z axis being along the axis of the pipe. We take the boundary of the rectangular pipe to be given by x = +a, y = +b. The motion is assumed to be generated by a constant pressure gradient in the z direction and, neglecting end effects, a suitable form for the velocity vector (v~, vy, v,) is vx = u(x, y), vy = v(x, y), v, = w(x, y).
(2)
From the equation of continuity div v = 0, we define a stream function ~b given by oq,
o~,
(3)
Non-Newtonian flow in pipes of non-circular cross section
319
If the pressure is denoted by p and the stress tensor by P a , the equations of motion reduce to
(~g,a2~, a~q P ~yy~-~y
(
P P
ap':, ap
@'::
~xxdy2/=~'x
+ ~y
tgx'
a+:+
ap ~y'
~yTx~ + ~ o x o y : = ~ + ~ 0x
ax
= --fix + - - ~ y + ~ '
(4)
(5) (6)
where p is the density and P'~k ( = P~k + pdi~) is the extra stress tensor. Eliminating the pressure from (4) and (5) we obtain (~ ~ ~ ~ a~p'== ~ , ~ , Opxy dpx~ ~2p,y, P \ a y ax ax ~ = axa-----~+ - dy - 2 (7) dX 2 dxt~y , where a2~ a2~ '¢' = ~ + o--~-" (8) Equations (6) and (7), together with suitable rheological equations of state, have to be solved subject to the boundary conditions
w(a, y) = w(x, b) = O, dw t3w oyT--(x, O) = ~x (0, y) = O,
d~
dx
dff=0onx
dy
(9)
O,x=a,y=O,y=b.
It is well known (see, for example [2]) that when secondary flows are absent, the primary flow velocity w satisfies t~ dw
~x(t/(,)~w) + ~y(Pl(,)-~y)=-'
(lO)
where
~=
7x
(ll)
+tOy/ J
and the relevant stress distribution is t~w t~w p,3 = ~(~) ~x,P23 = .(~) ~ay, P33 - Pll = ~:2[NI(Y)- N~(V)I +
~
P33 - - P 2 2 = y2[NI(y) - N2(y)] + \ d y ] ""
- " "
:
-
dwawN2(~), dx ~y
Pl; . . . .
N~(~,), N2('Y)'
(12)
320
A.G.
DODSON, P. TOWNSEND a n d K. WALTERS
r/(?) being the apparent viscosity and NI(7)]~ 2 and - N2(7)~'2 are the first and second normal stress differences usually associated with a steady simple shear flow [2, 24, 25]. It can be shown[2] that for rectilinear flow to be possible, the second normal stress difference has to satisfy (?(M*, w) -
-
=0,
O(x, y)
(13)
where
]
M* = ~x ~x N2(r) + -dy -
]
N2(7) •
(14)
A zero second normal stress difference is clearly a sufficient condition to ensure rectilinear flow. It is not a necessary condition, however. For example, (13) is satisfied if N2 is a constant multiple of q[2]. For the particular flow situation under consideration, the stress distribution (12) can be written in the alternative form (cf. [26]) P , i*
=
2r/(7)eik ( l )
- ~ (2) + 4[N1(7) - N2(7)]ei/tl)ej~ (1) , - Nl(7)ei~
(15)
where eik(") is the n th rate of strain defined by Oldroyd[27]*. This is essentially a "secondo r d e r " equation with variable coefficients. If we now generalize (15) by writing
P'ik = 2rl(12)eik 0) -- Nl(12)eik (2) + 4[N1(12) - Nz(12)]e~/t)eik (l),
(16)
where we take the second flow invariant 12 in the form I2
=
[(dw) 2 (8wt2 - ~x + \0y /
+
(gv ~xx +
gu) 2
4~u gv.]l '
( ! 7)
we have a generally valid equation of state which can be used under all conditions of motion and stress and which is exact for the particular flow situation under consideration (i.e rectilinear flow through a pipe of non-circular cross section). In this paper, we use (16) in the non-viscometric flow situation represented by (2) (i.e. flow in a pipe of non-circular cross section with secondary flow). The hope is that an equation which is exact for viscometric flows may also be (approximately) valid for nearly viscometric flows[29-31]. Pipkin[30] rightly points out that (16) is far from being the only possible generalization of (I 5), but it is certainly the simplest and it may repay a close study, especially if the influence of the higher order accelerations elk (n) (n >t 3) [which are not present in (16)] is not excessive. The solution of (6) (7) and (16) subject to the boundary conditions (9) still represents a formidable mathematical problem. To render the mathematics tractible we make use of the experimental observation that although the second normal stress difference - N 2 ) ,2 is not zero as originally thought [32], it is nevertheless usually small. We therefore take N 2 to be a small constant, so that the second normal stress difference has a quadratic dependence on the shear rate "t- Further, we use N 2 as a perturbation parameter. We will not restrict the form of the viscosity and first normal stress functions in the analysis which follows, although we shall of course need to make their forms explicit in the numerical computation of the flow field. * The ntb Rivlin-Ericksen[28] tensor ATkis given by ATk= 2e?,.
Non-Newtonian flow in pipes of non-circular cross section
321
The perturbation solution carried out in this paper has some similarities withtheanalytic work undertaken by Green and Rivlin[l l] for a pipe of elliptic cross section. However, in their work for a Reiner-Rivlin fluid, the first normal stress difference Nl(~ 2) was necessarily zero and q was taken to be a constant. It is convenient at this stage to introduce the following non-dimensional variables:
q* = q/qo, N* = Nxpa/qo z, N~ = Nzpa/qo 2, p* = p~a3/qo z, w* = Wqo/~a 2, ~b* = d/qo/(~a ),
(18)
x* = x/a, y* = y/a, Pi*k= Pik/(Pa), where t/o is the limiting viscosity at small rates of shear. We shall immediately drop the star notation, although it is still implied. We expand the variables in powers of N 2 in the following way: a01 U = N2 --~-y,
v = - N~ -~-x '
(19)
W = Wo(X , y ) + N 2 w t ( x , y),
12 = Yo(X,Y) + N2 Yt(x, y), and neglect terms of order N22. Substituting (19) into (6), (7) and (16) and equating terms independent of N2, we obtain*
ax J + ~y
ay J = - 1 '
where 7o--
\ ax ! + \ ay !
(20)
/
'
which is to be solved for some prescribed viscosity function, subject to the boundary conditions (9). wo represents the general solution in the absence of secondary flow. When the apparent viscosity is a constant, (20) reduces to an equation of the Poisson-type. Equating terms involving Nz, we obtain the following equation for the secondary flow stream function ~Ol
a'o, [ Ox4 + 2 ~
-t- -~-f]
+~
2 ~ ~,~y~ + axay~! + 2 ~
+ \~
ax 2 ] \ a y 2
d~qf//a~o]
' -
ax21 + 4 -axay axayj
(O~o]:][a~qJ, o'~,,.] \ax/
+ ~[[\--~y/
~,ax~ + axayV
j[ ay2
O oO oa' , l
ax2 ] + 4 ax ay Oxayj
Owo o [O'wo :wol Owo o_iO o O'wol -
Ox Oy~ Ox2 + Oy2~
ay Ox[ Ox2 + Oy2 J"
* An equation similar to (20) has been discussed in a general way by Oldroyd [34, 35].
(22)
322
A.G. DODSON,P. TOWNSENDand K. WALTERS
We note that when (20) reduces to Poisson's equation (i.e. when the viscosity is a constant), the forcing function in (22) vanishes and there is no secondary flow in consequence. The corresponding equation for the perturbed axial velocity component can be shown to be
[.Oqq OWo P [ Oy Ox
o~,, ~°I
ra~w, ~,I
d~ Faro Owl
O~oeW,
"-+ Oy2 j +-~yo [ Ox --~x + ~'y m,y
~x a y ] = r / [ O ~
O2Wo + Yl I Ox2 + OY2
l [awoOTo t?WoaYo'~l
+ - -l
iywo~wl ~woO'w, llawo1 - - + awoO'w, - - + - - - -O'wo~w, + -~
+ - -l
ta'WoaW,+ awoa2wl a'Woa.,, a.oa2~,/tawoll ----+ -----f-
ro [ ax 2 0 x
Ox Ox2
Yo IOxOy Ox
+
ax axay
r lOwo ,o
Oxay ay
Oy2 ay
oy O-x~yJ(-~-xj
~ ~
j}-f}-y j]
wo :otl
dY02 [Yl ~-'~-x T x + --@-y c y / i
- NI_ I Oy ~x \ Ox2 + ~y2 ]
021//1 (a2Wo +~
02Wo'~ l low 0 a (6~21/,/1 021]/1'~ OWo 0 ~6~21P1 021/,/i'~I Ox~ ] + 2 [ Oy -~x \ Ox2 + Oy2 ]
t Oy"
a'Wol.O~,, a~Oql +~
Ox Oy ~ Ox2 + Oy2]}
t Ox~
Oy'lJ-~
d N , [ a y o [ l a O , O~Wo lO~,O~Wo t Ox t2 Oy ax ~ 2 Ox OxOy
3 o'O, aWo a201 aWo l a'~, awq 2 OxOy Oy
Oy2 ~
Ox Oy \ Ox2 + Oy2 ]j
Oro [1 oq,, O~Wo
4 2 0 x 2 Oy ] + Oy ~,2 Oy OxOy
1 a~b, a2Wo 3 OzO, aw o 2 ~x" ay 2 + 2 OxOy Oy
1 a2Oi OWo a2~b, aWo~] 2 Oy2 ~---~+ 'c?x ' ' T Ox]]'
(23)
where
I [aWoaW, aWo a,¥, 1 ~'1 = ~-~ L ax a-; + 07 ay
J"
<24)
For a Reiner-Rivlin fluid, the first normal stress difference is zero (i.e. N1 = 0) and the forcing function in (23) is dominated by inertia. For the more realistic equations of state (16) which give rise to (23) we note that although the secondary flow is critically dependent on the existence and magnitude of the second normal stress difference ( - N2 72), the effect the secondary flow has on the axial velocity component w I (and hence on the flow rate) is dependent also on thefirst normal stress difference (Nly 2) and how this varies with shear rate. We would anticipate that the effect of the secondary circulation on the flow rate would become more marked as the first and second normal stress differences become larger in comparison with the shear stress, i.e. as the fluid becomes more "elastic." Equations (20), (22) and (23) depend on the viscometric functions and how these vary with the shear rate, so that even the use of the relatively simple equations of state (16) indicates possible difficulties when quantitative agreement between theory and experiment is
Non-Newtonian flow in pipes of non-circular cross section
323
sought. The viscometric functions themselves are not easy to measure (see, for example,[33]) and reducing experimental error to the point where the derivatives of these functions can be relied on is a formidable problem. An analytic solution to the governing equations (20)--(23) is out of the question, basically on account of the non-linear nature of (20). We resort therefore to a numerical technique using finite differences. We impose a square mesh over the region R ( x > 0, y > 0) defined by x=ih,
i=O,l
.... M
(25) y = j h , j = O, 1. . . . N
where h = 1 / M = b/(aN). To solve (20), we replace the continuous functions Wo(X, y) and r/(V) by discrete approximations wt~ and qij and using standard finite differences, equation (20) becomes [?]i+l,j
-- t l t - l , j
--
4 q t , ~ ] w ~ - t , j + [ q t - t , i - rh+ t , j - 4 q i . j ] W i +
+ [r/i,j+l - rli, j - i - 4qt,j]wi, i - t
+ [qt,j-1
-
qi,j+l
-
l,j
4rh, i]wi, j+l
+ 16qi, j w~,j = - 4 h 2.
(26)
To solve this equation, following Young and Wheeler[6], an initial velocity distribution w~j given by wtj = Sin ½n(a - x) Sin ½n(b - y), (27) is adopted. This gives a crude approximation to the solution and improves convergence. The viscosity r/ij is then calculated from the given function r/()qj), when ~j is given by ~j = [(8[w~+1,~ -
wi_l,~]
- w~+~.~ + w~_2,j) ~
+ (8[wi,i+ t - w i , j _ t ] - wi.j+ 2 + wi, j_2)2]/144h 2.
(28)
Values of w,j outside R which are required in (28) are obtained using Newton-Gregory extrapolation formulae. To calculate a new distribution w~ from (26), the finite difference equations are solved iteratively using successive over-relaxation. (Both point and line iteration were initially tried, the latter proving to be significantly better). A new viscosity ~/ij is now calculated from the new w~j and this outer iterative process continued until both velocity and viscosity fields are compatible with the governing equations. For the range of parameters considered there was no difficulty with convergence of this iteration. We have applied this technique for analytic forms of the viscosity function ~/(~) as well as for viscosity/shear rate relations deduced from experiments in a pipe of circular cross-section. In the latter case it is necessary to fit to the viscosity/shear rate data a smooth continuous curve. In general, the form of the viscosity/shear rate curve is of a difficult nature to approximate, particularly for rapidly shear-thinning fluids. A number of techniques were considered, the most satisfactory proving to be to smooth out obvious ' scatter' visually and fit to the smoothed data a cubic spline using Hermite interpolating polynomials. The accuracy of this technique was checked with experiments on two pipes of circular cross-section and of different radii. Viscosity/shear rate data calculated from one pipe was used to predict flow in the other, with excellent results. Having solved (20), equations (22) and (23) are simpler to handle, since they are basically linear. Both are replaced by standard finite difference approximations, (22) requiring a ninepoint representation of the higher derivatives. In both cases the finite difference equations are solved iteratively using successive over-relaxation by lines.
324
A.G.
DODSON, P. TOWNSEND a n d K . WALTERS
Finally the flow rate in the pipe is calculated from a
b
using Simpson's rule. A knowledge of the viscosity function ~(7) is needed before equation (20) for the basic axial flow can be solved. For most elastico-viscous systems, ~ falls from a zero-shear value ~o to a "second Newtonian" value. A suitable form for the viscosity function for our purpose is given by (cf. [3, 4]) t/('),) = r/o
+ ¢--~ j,
a, > a 2
(30)
which has the essential features of observed behaviour (cf. [36]). It is of interest to plot the pressure gradient/flow rate results (in the absence of secondary flow) in the following way. If QR denotes the flow rate through a pipe of rectangular cross section and Qc is the flow rate in a pipe of circular cross section (with the same cross sectional area) under the same operating conditions, we define the percentage loss I in flow rate due to the rectangular boundary to be
Qc- QR 100 Q~
1= - - .
(31)
For a Newtonian liquid, it is not difficult to deduce that I is independent of the pressure gradient ,~ and depends only on the dimensions of the pipe. In the case of non-Newtonian liquids, on the other hand, this is not the case. Figure 1 contains (I, p) results based on the
60
/
~
a :2cm
I
=
56 . . . . . . . . . . . . . . . . .
301
~ ~
"
"
Newtonion o:b=Icrn.
250
500
I~ (dynes/cc )
750
F i g . 1. T h e o r e t i c a l (I,/~) c u r v e s f o r ~o = 10 P, (72 = 0"01 sec z, a l = 0 - 0 2 sec 2.
viscosity function given by (30). It will be seen that the decrease in flow rate due to the non-circular cross section of the pipe is exaggerated for the non-Newtonian materials. At high pressure gradients, we would expect I to return to its initial " N e w t o n i a n " value, since the governing equation is ultimately dominated by the second Newtonian region of the viscosity curve.
Non-Newtonian flow in pipes of non-circular cross section
325
Before we undertook any computation o f the secondary flow in the cross section o f the pipe, we speculated as to the likely form o f the streamlines in the square and rectangular cross sections. Symmetry considerations indicated a minimum o f 8 vortices in the square and a minimum o f 4 in the rectangle (cf. [17]). However, having the minimum n u m b e r in the rectangle would have implied a discontinuity in the secondary flow as soon as two opposite sides o f a square were made slightly larger than the others. The computation resolved the question, as we shall now indicate. To compute the secondary flow from (21) it was necessary to specify r/and the first normal stress difference Nly 2. We took (30) as a suitable form for r / a n d for N 1 (cf. [3]) Nl=r/o
[(21 -- ;'2)-J-('~'10"2- A20"1))/21
L
1 + crly2
J
(32)
where 21 and 22 are constants having the dimensions of time. Figure 2 contains representaJ
III/
~ i
/
II/
Fig. 2. Streamline projections in the cross section of the pipe for 7/o = 1 P, At =0"0l sec, )~2= 0"005 sec, ~1 =0.000125 sec 2, or2 =0.0000625 sec2, p = 1 g cm -~, ]0=200 dyne cm -3, N2 = --0-01 dynes sec2 cm -2. ab = 1 cm 2. tive streamline patterns for fixed values of the material parameters and different a/b ratios.* In the case o f the square cross section, the two vortices present in a ,quadrant are symmetrical about the diagonal as one would expect (cf. [8]) As a/b departs from 1, one o f the vortices increases in size with a corresponding decrease in the other, and for a/b = 0.5 the smaller vortex is very weak indeed. The patterns clearly demonstrate that symmetry considerations can only supply a lower bound to the number of vortices present. * We have taken a sign for N2 which corresponds to a positive second normal stress difference although there is growing support for a negative second normal stress difference, for concentrated polymers solutions at least (cf. [33, 37]) The reason for this choice will become apparent in section 3.
326
A.G. DOBSON,P. TOWNSENDand K. WAL~RS
The directions of the streamlines given in Fig. 2 correspond to a positive second normal stress difference. Their direction would be reversed for a negative second normal stress difference. Contours of constant wt for the same values of the material parameters as those used to compute Fig. 2 are given in Fig. 3. It will be seen that, due to the presence of secondary
Fig. 3. C o n t o u r s of constant w~ (on an arbitrary scale) for To : 1 P, A~- 0'01 sec, Az = 0.005 sec, o~ =0.000125 sec2, o2 =0.0000625 sec2, p = 1 gcm -3, if=200 dyne cm -3, N2 = --0.01 dynes scc2 cm- 2. ab = 1 cm2.
flows, the axial velocity component can be increased over part of the cross section and diminished over the remainder (cf. [8]). Our main concern is the overall effect of the secondary flows on the mean flow rate and this is demonstrated in Figs. 4-11. Figure 4 contains curves of the three viscometric functions for specific values of the material constants. We feel that the behaviour is fairly representative for the type of polymer solution used in the experimental work to be described in section 3. Figure 5 contains corresponding (Is, p) results for various a/b ratios, the area of the cross section of the pipe being kept constant; I s being that part of the percentage change in flow rate which is due to the influence of the secondary flows. We note that the secondary flows result in a change in flow rate but that this change is very small. It is not possible to project the curves to higher values of the pressure gradient p, since the corresponding viscometric functions then become unrealistic. Figure 6 illustrates the effect of varying the a/b ratio. The (Is , a/b) curve quickly reaches a maximum and then falls to very low values of I~ as the cross section becomes thinner. A similar situation apparently exists in the case of an elliptic cross section as the eccentricity is varied (H. Giesekus, private communication). The relative importance of the variable-viscosity effect and the secondary-flow effect on the percentage change in mean flow rate is given in Fig. 7 for the conditions described by Fig. 4.
N o n - N e w t o n i a n flow in pipes o f non-circular cross section
/
I00C
J ~t no rrr~l str~..~---e~difference
500 u~
~00
2OO
shear rate {seC"I) Fig. 4. Viscometric f u n c t i o n s for r/o = 1 P, AI = 0"01 sec, A2 = 0-005 sec, a l = 0"000125 sec 2, (Tz =0.0000625 sec 2, N'z = - 0 . 0 0 5 dynes sec z c m . - "
1"0
~x3
0"5
O-C
Fig. 5. Theoretical (I,, : ) curves for various a/b ratios. 7/0 = I P, Ax = 0.01 sec, ,~2 = 0.005 sec, a l = 0 - 0 0 0 1 2 5 sec 2, ( 7 , = 0 . 0 0 0 0 6 2 5 sec 2, p = 1 g c m -3, Nz=--0.005 dynes sec z c m -z.
0.5
/,
Fig. 6. Theoretical (Io, a/b) results for ~9o = 1 P, &l = 0"01 sec, Az = 0.005 sec, a l = 0.000125 sec 2, ~ , = 0 . 0 0 0 0 6 2 5 sec 2 p = 1 g c m -3, N2= --0.005 dynes scc 2 c m -2, . ~ = 2 5 0 dyne cm. - 3
327
328
A.G.
D o o s o ~ , P. TOWNSEND a n d K . WALTERS
60 58"
Newtonlol:1
56. 0
100
2'00 ~ {dynes/cc)
14. 13" 12" Fig. 7. (1, ,~) results for 7/o = 1 P, AI = 0.01 sec, ,~2 = 0-005 sec, al = 0.000125 secz, a z = 0.0000625 sec=, p = 1 g cm -3, N, = -0.005 dynes sec 2 cm -2. Cross sectional a r e a = 4 sq cm Broken line--rectilinear flow; full line--with secondary flow.
In an attempt to maximize the effect of the secondary flows on I, we have taken various forms for the viscometric functions a n d by a n d large we have needed w h a t might be r e g a r d e d as " u n r e a l i s t i c " functions to o b t a i n m e a s u r a b l e changes in flow rate. F o r example, the viscometric functions given in Fig. 8, give rise to the decreases in flow rate shown in Fig. 9. The results are expressed graphically in an alternative form in Fig. 10 to enable direct c o m p a r i s o n with the e x p e r i m e n t a l results given in section 3. The viscometric functions given in Fig. 8 might a p p e a r s o m e w h a t bizarre to a rheologist,
u)
500
difference
0
100
200 3 ~ shear rate ( s ~ 1] Fig. 8. Viscomctric functions for 7/o = I P, ~ = 0.01 sec,/~2 = 0.005 sec, ~ a z = 0.0000625 sec 2, N2 = -0.01 dynes secz cm -2.
=
0"000125 scc2.
Non-Newtonian flow in pipes of non-circular cross section
329
2.0 I,
1"0
dynes/cc Fig. 9. Theoretical (I,, ,~) results for 7/o= 1 P, A~=0"01 sec, A2 =0.005 sec, a~ =0.000125 sec2, o2 = 0.0000625 sec2 a = 2 cm, b = 0"5 cm, p = 1 g c m - a, Nz = -0.01 dynes seca c m - 2.
300
2
0
0
~
E
(clynes,/cm3J I00
0
tO0
200O(cc/sec)300
Fig. 10. Predicted (b, Q) for "t/o= 1 poise, A1 =0"01 sec, A2 = 0.005 sec, cq =0.000125 sec 2, u, = 0.00000625 secz, p = I g cm -3, N2 = --0"01 dynes sec" cm -z. Pipe B of rectangular cross-section a = 2 cm, b = 0.5 cm Pipe E of circular cross-section having same cross-sectional area as pipe B. Broken line--predicted curve for pipe B neglecting secondary flows. but detailed e x p e r i m e n t a l knowledge o f the viscometric functions is o n l y available f o r rather c o n c e n t r a t e d p o l y m e r solutions a n d there is some evidence to believe t h a t projecting a similar b e h a v i o u r to m o r e dilute p o l y m e r solutions m a y not be entirely correct (of. [37, 41]). W h a t we can say is t h a t the effect o f the s e c o n d a r y flow on the flow rate is likely to be very small unless the fluid has some rather unusual properties. 3. E X P E R I M E N T (i)
Apparatus
The e x p e r i m e n t a l set-up is illustrated schematically in Fig. 1 I. It consisted o f a peristaltic p u m p A which created the flow a n d s m o o t h i n g bottles B which d a m p e d o u t the pressure fluctuations a n d c r e a t e d a s m o o t h c o n t i n u o u s flow. The pipes used in the test section were t r a n s p a r e n t glass pipes o f circular cross section a n d similar pipes having square a n d rectang u l a r cross sections (supplied by Plowden a n d T h o m p s o n Ltd). F o u r different pipes o f this sort were available with dimensions as given in T a b l e I.
330
A.G. Do,sot4, P. TOWNSEr~Dand K. WALTEaS
pipe A
pipe E
Fig. I I. Schematic diagram of the apparatus.
Table I. Dimensions of the pipes
Pipe
2b (cm)
2a (cm)
b:a
A B C D
0.84 1'19 1.45 1-68
0.84 0.59 0'49 0.42
1: 1 2:1 3: 1 4:1
A pipe of circular cross section having the same cross sectional area was also available. We shall refer to this as pipe E. The pressure gradient was determined by inserting pressure probes in the wall of the tubes. These were modified syringe needles which were inserted flush with the inner wall of the tubing. They were connected to a differential pressure transducer T a n d hence to a U.V. recorder. The flow rate was measured in the conventional way by placing a graduated flask under the tap above the reservoir R for a specified time. In the experiments, the pipe of non-circular cross section was placed in series with a pipe o f circular cross section of comparable dimensions. The pressure gradient in both pipes could then be measured at the same flow rate. Such an exercise was thought to be indispensible since it was necessary to compute numerically the flow rate in the non-circular pipe using the viscosity function deduced from experiments in a pipe of circular cross section. We anticipated that any secondary flow in the pipe would lead to small changes and an accurate assessment of predicted and observed flow rate in the pipe of non-circular cross section was therefore essential. Flow visualization experiments were carried out at various flow rates and for different test solutions in both square and rectangular cross-section pipes. The dye used was a mixture of the test fluid and some strong colouring agent. In these experiments, the test pipes were illuminated from below by means of a Barnes Transilluminated tracing table. This produced an evenly illuminated surface which could be photographed from above. The Newtonian liquids used in the experiments were water and aqueous solutions of glycerol, with viscosities ranging from 0"1 to 14 P. The non-Newtonian liquids employed were aqueous solutions of polyacrylamide (P250, supplied by Cyanamide of Great Britain) In addition, a soap solution was also investigated. This was a mixture of an aqueous solution
Non-Newtonian flow in pipes of non-circular cross section
331
of (~) cetyltrimethyl ammonium bromide and (fl) an aqueous solution of toluene-4-sulphonic acid sodium salt in the ratio 2a:l/. Small amounts of salt were also added. This had the effect of altering the micelle concentration and hence the viscosity and elasticity of the solution. The soap was found to have an elasticity/viscosity ratio which was an order of magnitude higher than a comparable aqueous solution of polyacrylamide. This is demonstrated in Fig. 12 which contains dynamic viscosity r/' and dynamic rigidity G' results at different frequencies.* At the higher frequencies, the values of G' are comparable, but the soap solution has a viscosity which is an order of magnitude lower than the polymer solution. The soap solution was in fact the most "elastic" liquid we could find and was used in the hope of maximizing the effect of the secondary flow on the flow rate. In the flow visualization experiments the test section was mounted between two meter rulers. These served the dual role of measuring the distance along the pipe at which effects become apparent and defining the walls of the rectangular test section. The dye was either injected at the mid point of the top face (J) or at the mid points of the two vertical faces (K and K'). Fig. 13 contains the path of a dye thread after it had been injected at point J of a pipe of square cross section into a 1.75 percent aqueous solution of polyacrylamide. Here, there is an expansion of the dye thread and at a distance of approxiriaately 30 cm along the pipe two distinct dye streams are seen near the side walls. These tend to move back again towards the centre and at a distance of approximately 100 cm along the pipe they nearly touch at the centre. Visual inspection through the top and side faces of the pipe confirmed
\ \
/'
\ \
/
/~,
"~
G' cm"2
/
\/
/
/ /
0"
eb
3b
.~ s e c -1
Fig. 12. Experimental (r/', /2) and (G', f2) for a 2-0% aqueous solution of polyacrylamide (broken line) and a 2.0% soap solution (full line). * The dynamic viscosity and dynamic rigidity are associated with a small amplitude oscillatory deformation and are functions of the frequency of oscillation fL They may be taken as convenient measures of viscosity and elasticity, respectively (cf. [38]).
A.G. DODSON,P. TOWNSENDand K. WALTERS
332
F
I
Fig. 13. Typical flow visualization study for a 1.75°o aqueous solution of polyacrylamide. Injection point J in pipe with square cross section (pipe A). Q -- 1.22 cm 3 sec.that the dye threads moved out over the top face and then in along the diagonals, in agreement with the theoretical prediction. The observed direction o f the streamlines was consistent with the test solution possessing a positire second normal stress difference, as far as our simplified analysis is concerned. It would be wrong however to make definitive conclusions on this point since a constant multiple of r/ can be added to N 2 without affecting the secondary flow[2]. In Fig. 14, the situation is repeated except that the pipe now has a rectangular cross section. In this case, the dye threads stay at the side walls after moving out along the top face. Fig. 15 completes the picture for the rectangular pipe, since the points o f injection are now K and K'. Here, the dye filaments move towards the centre. Figures 14 and 15 indicate that while a weak secondary vortex o f the sort predicted in section 2 may have been present, it was certainly not strong enough to observe in our flow visualization experiments and to all intents and purposes only four vortices were present in the pipe o f rectangular cross section.
Fig. I4. Typical flow visualization study lor a 1-75% aqueous solution of polyacrylamide. Injection point J in pipe with rectangular cross section (pipe B). Q - - 1.72 cm a scc. -1
Fig. 15, Typical flow visualization study for a 1,75~o aqueous solution of polyacrylarmd¢. Injection points K and K ' in pipe of rectangular cross section (pipe B). Q = 0"49 cm ~ sec. 333 CAF VoL 2 No. 3/4--G
334
A.G. DoDsos, P. TOWNSENDand K. WALTERS 60 0
0
0
0
0
D
0
O - -
0--0
0 ~--~-~-
1.0 I%
I
0
0
n
0
0
0
B
0__0
~
20 C~
0
2bo
0
Re
0
c,bo
0bo
vbo
Fig. 16. Percentage decrease in flow rate/Reynolds number results for Newtonian liquids. Full lines--theory. Circles--experimental results.
The streamlines for the soap solutions showed the same general features as those for the polymer solutions but the movements took place more quickly for comparable flow rates. However, it was difficult to quantify the difference, since it depended to some extent on the thickness of the initial dye filament and its exact position on injection. Before embarking on pressure gradient/flow rate measurements for the elastico-viscous liquids, we checked the theoretical and experimental techniques for Newtonian liquids. Here, an analytic solution to the theoretical problem was also available for comparison [39] Figure 16 illustrates the agreement between theory and experiment, which is entirely satisfactory. In the case of the non-Newtonian liquids we adopted the following procedure. The (Qc,/3) results for the pipe of circular cross section were first used to determine the apparent viscosity r/(~) of the test fluid. This was then used in conjunction with the numerical techniques of section 2 to compute the flow rate in a pipe of square or rectangular cross section on the assumption that secondary flows were absent. We then took the difference between the predicted (QR, P) results for the pipes of non-circular cross section and the corresponding experimental results as a measure of the effect of the secondary flow on the flow rate. Figures 17 and 18 contain experimental (QR, P) results for two pipes for a 1"5~o and a 1.75 ~o aqueous solution of polyacrylamide. Also included are the theoretical curves based on the absence of secondary flows. The close agreement between the experimental data and the theoretical predictions implies that, for these particular solutions, the "inelastic'" approximation is sufficient to predict the flow, which in turn implies that the secondary flows, although in evidence in the flow visualization experiments, were not strong enough to significantly affect the flow rate. In contrast, experiments with the highly elastic soap solutions did indicate some secondaryflow influence on the flow rate. Experimental (Q~,/3) results for a 1-5 ~o soap solution in a pipe of square cross section are given in Fig. 19. Also included are the (Qc,/3) results for a pipe of circular cross section of the same cross sectional area together with predicted (QR,/3) results for the square pipe based on the numerical techniques of section 2 for rectilinear
Non-Newtonian flow in pipes of non-circular cross section
335
800
A
6OO
.~ 400
0
2
4 6 8 ORc.c/sec
10
Fig. 17. Experimental (Qa,/~) results for a 1"75~. aqueous solution of polyacrylamide (circles and triangles). Theoretical curves based on absence of secondary flow--full lines.
ipe D
400 tJ O~ Q,I
,0..
2OO
i
o
Fig. 18. Experimental (Qa, P) results for a 1"5 ~o aqueous solution of polyacrylamide (circles and triangles). Theoretical curves based on absence of secondary flow--full lines. flow. A similar situatiorr is shown in Fig, 20 for a 2.0 ~o s o a p solution in a p i p e o f r e c t a n g u l a r cross section. It will be observed t h a t in both cases, the theoretical predictions i n d i c a t e d a higher flow rate than was observed; the difference we associate with the influence o f the
A.G. DODSON,P. TOWNSENDand K. WALTERS
336
600
E 400 I/i
~ 200
Fig. 19. (Q,/~) results for a 1"5% soap solution. Full lines--experimental results. Broken line--predicted curve for pipe A assuming no secondary flow.
/ ,a.
200
0
4 Q cc~ se~"18
12
Fig. 20. (Q,)6) results for a 2"0% soap solution. Full lines--experimental results. Broken line--predicted curve for pipe B assuming no secondary flow. secondary flows. The effect is further demonstrated in Fig. 21 which contains (/,/~) results for a 1.5 per cent solution. The rather complicated shape of the theoretical and experimental curves in this figure from that given in Fig. 1 and found for the polymer solutions[40] would suggest that the soap solutions do possess the rather unusual material properties required to produce measureable secondary-flow influences.
Non-Newtonian flow in pipes of non-circular cross section
337
tO
O
3O I%
2O
/,
7
\
/ J
,/7
Newton~an i
1C
i
l
2 ' 0 0 dynes crr~3~00
|
C~O
Fig. 21. (1, p) results for a 1'5 % soap solution (pipe A). Circles--experimental points. Broken line--predicted curve assuming no secondary flow.
Acknowledgements--We have benefited from several discussions with Dr. H. A. Barnes and Dr. J. M. Broadbent. The experimental work described in this paper was carried out under a contract from the Department of Trade and Industry. The paper is published by permission of the Director of the Department's National Engineering Laboratory. REFERENCES Ericksen J. L., Quart Applied Math. 14, 318 (1956). Oldroyd J. G., Proc. Roy. Soc. A 283, 115 (1965). Oldroyd J. G., Proc. Roy. Soc. A 245, 278 (1958). Waiters K. Quart. J. Mech. appl. Math. 15, 63 (1962). Schechter R. S., A.LCh.E.J. 7, 445 (1961). Young D. M. and M. F. Wheeler, Non Linear Problems in Engineering, Academic Press, New York (1964). 7. Wheeler J. A. and E. H. Wissler, A.LCh.E.J. !1,207 (1965). 8. Wheeler J. A. and E. H. Wessler, Trans. Soc. Rheol. 10, 353 (1966). 9. Arai I. and H. Toyoda H., Proc. 5th Int. Congr. on Rheology. (University Park Press) 1970 Vol. 4, 461. 10. Mitsuishi N. and Y. Aoyagi, Chem. Enging. Sci. 24, 309 (1969). 1 i. Green A. E. and R. S. Rivlin, Quart. appl. Math. 14, 299 (1956). 12. Reiner M., Am. J. Math. 67, 3.50 (1945). 13. Rivlin R. S., Proc. Roy. Soc. A 193 260 (1948). 14. Rivlin R. S., Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, Oxford 668 (1964). 15. Rivlin R. S. and W. E. Langlois, Brown Universtiy Tech. Report D.A. 4725/3 1959. 16. Rivlin R. S. and W. E. Langlois, Rend. Mat. 22, 169 (1963). 17. Pipkin A. C., Proc. 4th Int. Congr. on Rheology (Interscience) Vol. 1,213 (1963). 18. Jones J. R.,J. de Mecanique 3, 79 (1964). 20. Jones J. R., and R. S. Jones, J. de Mecanique 5, 375 (1966). 21. Jones R. S., J. de Mecanique 6, 443 0967). 22. Giesekus H., Rheol. Acta. 4, 299 (1965). 23. Semjonow V., Rheol. Acta. 6, 171 (1967). 24. Coleman B. D., H. Markovitz, and W. Noll, Viscometric Flows of Non-Newtonian Fluids. SpringerVerlag, Berlin (1966). 25. Lodge A. S., Elastic Liquids, Academic Press, New York (1964). 26. Criminale W. O. Jr., J. L. Ericksen, and G. L. Filbey, Arch. Rat. Mech. Anal. 1,410 (1958). 27. Oldroyd J. G., Proc. Roy. Soc. A 200, 523 (1950). 28. Rivlin R. S. and J. L. Ericksen, J. Rat. Mech. Anal. 4, 323 (1955). 29. Tanner R. I., Trans. Soc. Rheol. 11, 39 (1967). 1. 2. 3. 4. 5. 6.
338
A . G . DODSON, P. TOWSSESDand K. WALTEV.S
30. Pipkin A. C., Tech. Report No. 2, Non-linear phenomena in Continua, Brown University, R. l., U.S.A. 31. Waiters K., Progress in Heat and Mass Transfer Vol. 5, Pergamon, Oxford, 217 (1972). 32. Roberts J. E.,Proc. 2ndlnt. Cong. on Rheol., Oxford, 91 (1953). 33. Brindley G. and J. M. Broadbent, Rheol. Acta 12, 48 (1973). 34. Oldroyd J. G. Proc. Cam. Phil. Soc. 45, 595 (1949). 35. Oldroyd J. G., Proc. Camb. Phil. Soc. 47, 410 (1951). 36. Jones T. E. R. and K. Waiters, J. Phys. A 4, 85 (1971). 37. Kuo Y. and R. I. Tanner, To appear in Rh¢ol. Acta. 1974. 38. Waiters K., Basic Concepts and Formulae for the Rheogoniometer Sangamo Controls Ltd (1968). 39. Yardoff O., C. R. Acad. Sci. Paris 223, 192 (1946). 40. Dodson A. G., Ph.D. thesis (Univ. of Wales) (1971). 41. Moore C. A. and J. R. A. Pearson, To appear in Rheol. Acta 1974.
NON-NEWTONIAN FLOW IN PIPES OF NON-CIRCULAR CROSS SECTION A. G. DODSON,* P. TOWNSEND t and K. WALTERS~ Department of Applied Mathematics, University College of Wales, Aberystwyth, Wales (Received 16 April 1973) Abstract--Rectilinear flow through pipes of non-circular cross section in the case of elasticoviscous liquids is not possible in general and some secondary flow in the cross section of the pipe is to be expected. The present paper contains a detailed theoretical and experimental study of the problem for pipes of square and rectangular cross section. Our main concern is the form that the secondary flows take and their influence on the flow rate through the pipe. The governing equations, which are non-linear partial differential equations, are solved numerically using finite difference methods used in conjunction with S.L.O.R. It is shown, that, for both the square and rectangular cross sections, there are eight vortices present. These have the same strength for the square cross section, but four of the eight become progressively weaker as the ratio of the sides of the rectangle is increased. The effect of a variable (shear-dependent) viscosity on the flow rate is found to be substantial, but the effect of the secondary flows on the flow rate is predicted to be small unless the fluid has some rather unusual properties. The experimental results are in general agreement with the theoretical predictions.
1. I N T R O D U C T I O N
NoN-NEWTONIAN flOW through straight pipes of arbitrary cross section has received considerable attention in the literature. The special case of Poiseuille flow (i.e. flow in a pipe of circular cross section under a constant pressure gradient) is very well documented and is widely used in viscometric measurements. The fact that the flow field in the case of pipes of non-circular cross section can be significantly modified by non-Newtonian behaviour has resulted in a substantial research effort in this area also. It is well known that for Newtonian fluids, rectilinear flow is always possible in a straight pipe of arbitrary cross section. For materials that exhibit normal-stress effects in a steady shear flow this is not possible in general and some secondary flow in the cross section of the pipe is to be expected. This has been studied from a rather general point of view by Ericksen[1] and Oldroyd[2] and with more specific rheological equations of state in mind by Oldroyd[3] and Waiters[4]. It transpires that a non-zero second normal stress difference is a necessary but not a sumcient condition for secondary flow to exist. The computation of the axial velocity component in the exceptional case where rectilinear flow is possible (or where the secondary flow is assumed to have a negligible influence on the flow rate) has been carried out by a number of workers, [5-10]. This is dependent only * Present address: British Food Manufacturing Industries Research Association. l.¢atberhead, Surrey. t Present address: Department of Computer Science, University College of Swansea. :1:K. Waiters, Prof. The University College of Wales, Applied Math. Dept. Physical Sciences Bldg., Penglais, Aberystwyth, SY 23 3BZ, Wales. 317
CAF Vol. 2 No. 3/4--F
318
A.G. DODSON, P. TOWNSEND and K. WALTERS
on the apparent viscosity function associated with the liquid and involves the solution of a non-linear partial differential equation. For realistic viscosity functions, the use of a high speed computer is indispensible in this connection. Green and Rivlin[l l] were the first to calculate the form of the secondary flow in the case of pipes of elliptic cross-section. They considered a special case of the Reiner-Rivlin fluid [cf. 12, 13]. The work was extended by Rivlin and Langlois[14-16] for a more general class of fluids. It transpires that if U is a characteristic velocity, 2. a characteristic relaxation time of the fluid, and L a characteristic length, the secondary flow stream function $ is related to the primary flow velocity w along the pipe by a relation of the form[17] ~=-
w.
(1)
The cube power in (1) implies that any secondary flow in the plane of the pipe is not likely to be strong. A number of workers[17-21] have considered the form of the secondary flow to be expected in pipes of various cross sections, with Pipkin[17] basing his ideas on symmetry considerations. We shall show later than such considerations do not give the whole story, a point which Pipkin himself acknowledged. Very few workers have given detailed consideration to the effect of the secondary flow on the axial velocity component (and hence on the flow rate). Green and Rivlin[11] and Wheeler and Wissler[8] have discussed the problem for a Reiner-Rivlin fluid, the latter authors also present some sample experimental results for a pipe of square cross-section. In contrast to the extensive theoretical literature on secondary flow in pipes of noncircular cross section, there is very little experimental verification of the predicted flow. There is a now celebrated private communication attributed to E. Kearsley which is quoted by many theoreticians to support the existence of secondary flows. In addition, there is experimental evidence of secondary flow in pipes of elliptic cross section for aqueous solutions of polyacrylamide[22] and polymer melts[23]. In the present paper, we give detailed theoretical and experimental consideration to the flow of elastico-viscous liquids through pipes of square and rectangular cross section, the choice of cross-section being governed partly by theoretical considerations (since the resulting partial differential equations are solved numerically) and partly for such experimental considerations as the ease of manufacture of the pipes and of carrying out flow visualization studies. 2. THEORY We refer all physical quantities to a set of Cartesian coordinates (x, y, z), the z axis being along the axis of the pipe. We take the boundary of the rectangular pipe to be given by x = +a, y = +b. The motion is assumed to be generated by a constant pressure gradient in the z direction and, neglecting end effects, a suitable form for the velocity vector (v~, vy, v,) is vx = u(x, y), vy = v(x, y), v, = w(x, y).
(2)
From the equation of continuity div v = 0, we define a stream function ~b given by oq,
o~,
(3)
Non-Newtonian flow in pipes of non-circular cross section
319
If the pressure is denoted by p and the stress tensor by P a , the equations of motion reduce to
(~g,a2~, a~q P ~yy~-~y
(
P P
ap':, ap
@'::
~xxdy2/=~'x
+ ~y
tgx'
a+:+
ap ~y'
~yTx~ + ~ o x o y : = ~ + ~ 0x
ax
= --fix + - - ~ y + ~ '
(4)
(5) (6)
where p is the density and P'~k ( = P~k + pdi~) is the extra stress tensor. Eliminating the pressure from (4) and (5) we obtain (~ ~ ~ ~ a~p'== ~ , ~ , Opxy dpx~ ~2p,y, P \ a y ax ax ~ = axa-----~+ - dy - 2 (7) dX 2 dxt~y , where a2~ a2~ '¢' = ~ + o--~-" (8) Equations (6) and (7), together with suitable rheological equations of state, have to be solved subject to the boundary conditions
w(a, y) = w(x, b) = O, dw t3w oyT--(x, O) = ~x (0, y) = O,
d~
dx
dff=0onx
dy
(9)
O,x=a,y=O,y=b.
It is well known (see, for example [2]) that when secondary flows are absent, the primary flow velocity w satisfies t~ dw
~x(t/(,)~w) + ~y(Pl(,)-~y)=-'
(lO)
where
~=
7x
(ll)
+tOy/ J
and the relevant stress distribution is t~w t~w p,3 = ~(~) ~x,P23 = .(~) ~ay, P33 - Pll = ~:2[NI(Y)- N~(V)I +
~
P33 - - P 2 2 = y2[NI(y) - N2(y)] + \ d y ] ""
- " "
:
-
dwawN2(~), dx ~y
Pl; . . . .
N~(~,), N2('Y)'
(12)
320
A.G.
DODSON, P. TOWNSEND a n d K. WALTERS
r/(?) being the apparent viscosity and NI(7)]~ 2 and - N2(7)~'2 are the first and second normal stress differences usually associated with a steady simple shear flow [2, 24, 25]. It can be shown[2] that for rectilinear flow to be possible, the second normal stress difference has to satisfy (?(M*, w) -
-
=0,
O(x, y)
(13)
where
]
M* = ~x ~x N2(r) + -dy -
]
N2(7) •
(14)
A zero second normal stress difference is clearly a sufficient condition to ensure rectilinear flow. It is not a necessary condition, however. For example, (13) is satisfied if N2 is a constant multiple of q[2]. For the particular flow situation under consideration, the stress distribution (12) can be written in the alternative form (cf. [26]) P , i*
=
2r/(7)eik ( l )
- ~ (2) + 4[N1(7) - N2(7)]ei/tl)ej~ (1) , - Nl(7)ei~
(15)
where eik(") is the n th rate of strain defined by Oldroyd[27]*. This is essentially a "secondo r d e r " equation with variable coefficients. If we now generalize (15) by writing
P'ik = 2rl(12)eik 0) -- Nl(12)eik (2) + 4[N1(12) - Nz(12)]e~/t)eik (l),
(16)
where we take the second flow invariant 12 in the form I2
=
[(dw) 2 (8wt2 - ~x + \0y /
+
(gv ~xx +
gu) 2
4~u gv.]l '
( ! 7)
we have a generally valid equation of state which can be used under all conditions of motion and stress and which is exact for the particular flow situation under consideration (i.e rectilinear flow through a pipe of non-circular cross section). In this paper, we use (16) in the non-viscometric flow situation represented by (2) (i.e. flow in a pipe of non-circular cross section with secondary flow). The hope is that an equation which is exact for viscometric flows may also be (approximately) valid for nearly viscometric flows[29-31]. Pipkin[30] rightly points out that (16) is far from being the only possible generalization of (I 5), but it is certainly the simplest and it may repay a close study, especially if the influence of the higher order accelerations elk (n) (n >t 3) [which are not present in (16)] is not excessive. The solution of (6) (7) and (16) subject to the boundary conditions (9) still represents a formidable mathematical problem. To render the mathematics tractible we make use of the experimental observation that although the second normal stress difference - N 2 ) ,2 is not zero as originally thought [32], it is nevertheless usually small. We therefore take N 2 to be a small constant, so that the second normal stress difference has a quadratic dependence on the shear rate "t- Further, we use N 2 as a perturbation parameter. We will not restrict the form of the viscosity and first normal stress functions in the analysis which follows, although we shall of course need to make their forms explicit in the numerical computation of the flow field. * The ntb Rivlin-Ericksen[28] tensor ATkis given by ATk= 2e?,.
Non-Newtonian flow in pipes of non-circular cross section
321
The perturbation solution carried out in this paper has some similarities withtheanalytic work undertaken by Green and Rivlin[l l] for a pipe of elliptic cross section. However, in their work for a Reiner-Rivlin fluid, the first normal stress difference Nl(~ 2) was necessarily zero and q was taken to be a constant. It is convenient at this stage to introduce the following non-dimensional variables:
q* = q/qo, N* = Nxpa/qo z, N~ = Nzpa/qo 2, p* = p~a3/qo z, w* = Wqo/~a 2, ~b* = d/qo/(~a ),
(18)
x* = x/a, y* = y/a, Pi*k= Pik/(Pa), where t/o is the limiting viscosity at small rates of shear. We shall immediately drop the star notation, although it is still implied. We expand the variables in powers of N 2 in the following way: a01 U = N2 --~-y,
v = - N~ -~-x '
(19)
W = Wo(X , y ) + N 2 w t ( x , y),
12 = Yo(X,Y) + N2 Yt(x, y), and neglect terms of order N22. Substituting (19) into (6), (7) and (16) and equating terms independent of N2, we obtain*
ax J + ~y
ay J = - 1 '
where 7o--
\ ax ! + \ ay !
(20)
/
'
which is to be solved for some prescribed viscosity function, subject to the boundary conditions (9). wo represents the general solution in the absence of secondary flow. When the apparent viscosity is a constant, (20) reduces to an equation of the Poisson-type. Equating terms involving Nz, we obtain the following equation for the secondary flow stream function ~Ol
a'o, [ Ox4 + 2 ~
-t- -~-f]
+~
2 ~ ~,~y~ + axay~! + 2 ~
+ \~
ax 2 ] \ a y 2
d~qf//a~o]
' -
ax21 + 4 -axay axayj
(O~o]:][a~qJ, o'~,,.] \ax/
+ ~[[\--~y/
~,ax~ + axayV
j[ ay2
O oO oa' , l
ax2 ] + 4 ax ay Oxayj
Owo o [O'wo :wol Owo o_iO o O'wol -
Ox Oy~ Ox2 + Oy2~
ay Ox[ Ox2 + Oy2 J"
* An equation similar to (20) has been discussed in a general way by Oldroyd [34, 35].
(22)
322
A.G. DODSON,P. TOWNSENDand K. WALTERS
We note that when (20) reduces to Poisson's equation (i.e. when the viscosity is a constant), the forcing function in (22) vanishes and there is no secondary flow in consequence. The corresponding equation for the perturbed axial velocity component can be shown to be
[.Oqq OWo P [ Oy Ox
o~,, ~°I
ra~w, ~,I
d~ Faro Owl
O~oeW,
"-+ Oy2 j +-~yo [ Ox --~x + ~'y m,y
~x a y ] = r / [ O ~
O2Wo + Yl I Ox2 + OY2
l [awoOTo t?WoaYo'~l
+ - -l
iywo~wl ~woO'w, llawo1 - - + awoO'w, - - + - - - -O'wo~w, + -~
+ - -l
ta'WoaW,+ awoa2wl a'Woa.,, a.oa2~,/tawoll ----+ -----f-
ro [ ax 2 0 x
Ox Ox2
Yo IOxOy Ox
+
ax axay
r lOwo ,o
Oxay ay
Oy2 ay
oy O-x~yJ(-~-xj
~ ~
j}-f}-y j]
wo :otl
dY02 [Yl ~-'~-x T x + --@-y c y / i
- NI_ I Oy ~x \ Ox2 + ~y2 ]
021//1 (a2Wo +~
02Wo'~ l low 0 a (6~21/,/1 021]/1'~ OWo 0 ~6~21P1 021/,/i'~I Ox~ ] + 2 [ Oy -~x \ Ox2 + Oy2 ]
t Oy"
a'Wol.O~,, a~Oql +~
Ox Oy ~ Ox2 + Oy2]}
t Ox~
Oy'lJ-~
d N , [ a y o [ l a O , O~Wo lO~,O~Wo t Ox t2 Oy ax ~ 2 Ox OxOy
3 o'O, aWo a201 aWo l a'~, awq 2 OxOy Oy
Oy2 ~
Ox Oy \ Ox2 + Oy2 ]j
Oro [1 oq,, O~Wo
4 2 0 x 2 Oy ] + Oy ~,2 Oy OxOy
1 a~b, a2Wo 3 OzO, aw o 2 ~x" ay 2 + 2 OxOy Oy
1 a2Oi OWo a2~b, aWo~] 2 Oy2 ~---~+ 'c?x ' ' T Ox]]'
(23)
where
I [aWoaW, aWo a,¥, 1 ~'1 = ~-~ L ax a-; + 07 ay
J"
<24)
For a Reiner-Rivlin fluid, the first normal stress difference is zero (i.e. N1 = 0) and the forcing function in (23) is dominated by inertia. For the more realistic equations of state (16) which give rise to (23) we note that although the secondary flow is critically dependent on the existence and magnitude of the second normal stress difference ( - N2 72), the effect the secondary flow has on the axial velocity component w I (and hence on the flow rate) is dependent also on thefirst normal stress difference (Nly 2) and how this varies with shear rate. We would anticipate that the effect of the secondary circulation on the flow rate would become more marked as the first and second normal stress differences become larger in comparison with the shear stress, i.e. as the fluid becomes more "elastic." Equations (20), (22) and (23) depend on the viscometric functions and how these vary with the shear rate, so that even the use of the relatively simple equations of state (16) indicates possible difficulties when quantitative agreement between theory and experiment is
Non-Newtonian flow in pipes of non-circular cross section
323
sought. The viscometric functions themselves are not easy to measure (see, for example,[33]) and reducing experimental error to the point where the derivatives of these functions can be relied on is a formidable problem. An analytic solution to the governing equations (20)--(23) is out of the question, basically on account of the non-linear nature of (20). We resort therefore to a numerical technique using finite differences. We impose a square mesh over the region R ( x > 0, y > 0) defined by x=ih,
i=O,l
.... M
(25) y = j h , j = O, 1. . . . N
where h = 1 / M = b/(aN). To solve (20), we replace the continuous functions Wo(X, y) and r/(V) by discrete approximations wt~ and qij and using standard finite differences, equation (20) becomes [?]i+l,j
-- t l t - l , j
--
4 q t , ~ ] w ~ - t , j + [ q t - t , i - rh+ t , j - 4 q i . j ] W i +
+ [r/i,j+l - rli, j - i - 4qt,j]wi, i - t
+ [qt,j-1
-
qi,j+l
-
l,j
4rh, i]wi, j+l
+ 16qi, j w~,j = - 4 h 2.
(26)
To solve this equation, following Young and Wheeler[6], an initial velocity distribution w~j given by wtj = Sin ½n(a - x) Sin ½n(b - y), (27) is adopted. This gives a crude approximation to the solution and improves convergence. The viscosity r/ij is then calculated from the given function r/()qj), when ~j is given by ~j = [(8[w~+1,~ -
wi_l,~]
- w~+~.~ + w~_2,j) ~
+ (8[wi,i+ t - w i , j _ t ] - wi.j+ 2 + wi, j_2)2]/144h 2.
(28)
Values of w,j outside R which are required in (28) are obtained using Newton-Gregory extrapolation formulae. To calculate a new distribution w~ from (26), the finite difference equations are solved iteratively using successive over-relaxation. (Both point and line iteration were initially tried, the latter proving to be significantly better). A new viscosity ~/ij is now calculated from the new w~j and this outer iterative process continued until both velocity and viscosity fields are compatible with the governing equations. For the range of parameters considered there was no difficulty with convergence of this iteration. We have applied this technique for analytic forms of the viscosity function ~/(~) as well as for viscosity/shear rate relations deduced from experiments in a pipe of circular cross-section. In the latter case it is necessary to fit to the viscosity/shear rate data a smooth continuous curve. In general, the form of the viscosity/shear rate curve is of a difficult nature to approximate, particularly for rapidly shear-thinning fluids. A number of techniques were considered, the most satisfactory proving to be to smooth out obvious ' scatter' visually and fit to the smoothed data a cubic spline using Hermite interpolating polynomials. The accuracy of this technique was checked with experiments on two pipes of circular cross-section and of different radii. Viscosity/shear rate data calculated from one pipe was used to predict flow in the other, with excellent results. Having solved (20), equations (22) and (23) are simpler to handle, since they are basically linear. Both are replaced by standard finite difference approximations, (22) requiring a ninepoint representation of the higher derivatives. In both cases the finite difference equations are solved iteratively using successive over-relaxation by lines.
324
A.G.
DODSON, P. TOWNSEND a n d K . WALTERS
Finally the flow rate in the pipe is calculated from a
b
using Simpson's rule. A knowledge of the viscosity function ~(7) is needed before equation (20) for the basic axial flow can be solved. For most elastico-viscous systems, ~ falls from a zero-shear value ~o to a "second Newtonian" value. A suitable form for the viscosity function for our purpose is given by (cf. [3, 4]) t/('),) = r/o
+ ¢--~ j,
a, > a 2
(30)
which has the essential features of observed behaviour (cf. [36]). It is of interest to plot the pressure gradient/flow rate results (in the absence of secondary flow) in the following way. If QR denotes the flow rate through a pipe of rectangular cross section and Qc is the flow rate in a pipe of circular cross section (with the same cross sectional area) under the same operating conditions, we define the percentage loss I in flow rate due to the rectangular boundary to be
Qc- QR 100 Q~
1= - - .
(31)
For a Newtonian liquid, it is not difficult to deduce that I is independent of the pressure gradient ,~ and depends only on the dimensions of the pipe. In the case of non-Newtonian liquids, on the other hand, this is not the case. Figure 1 contains (I, p) results based on the
60
/
~
a :2cm
I
=
56 . . . . . . . . . . . . . . . . .
301
~ ~
"
"
Newtonion o:b=Icrn.
250
500
I~ (dynes/cc )
750
F i g . 1. T h e o r e t i c a l (I,/~) c u r v e s f o r ~o = 10 P, (72 = 0"01 sec z, a l = 0 - 0 2 sec 2.
viscosity function given by (30). It will be seen that the decrease in flow rate due to the non-circular cross section of the pipe is exaggerated for the non-Newtonian materials. At high pressure gradients, we would expect I to return to its initial " N e w t o n i a n " value, since the governing equation is ultimately dominated by the second Newtonian region of the viscosity curve.
Non-Newtonian flow in pipes of non-circular cross section
325
Before we undertook any computation o f the secondary flow in the cross section o f the pipe, we speculated as to the likely form o f the streamlines in the square and rectangular cross sections. Symmetry considerations indicated a minimum o f 8 vortices in the square and a minimum o f 4 in the rectangle (cf. [17]). However, having the minimum n u m b e r in the rectangle would have implied a discontinuity in the secondary flow as soon as two opposite sides o f a square were made slightly larger than the others. The computation resolved the question, as we shall now indicate. To compute the secondary flow from (21) it was necessary to specify r/and the first normal stress difference Nly 2. We took (30) as a suitable form for r / a n d for N 1 (cf. [3]) Nl=r/o
[(21 -- ;'2)-J-('~'10"2- A20"1))/21
L
1 + crly2
J
(32)
where 21 and 22 are constants having the dimensions of time. Figure 2 contains representaJ
III/
~ i
/
II/
Fig. 2. Streamline projections in the cross section of the pipe for 7/o = 1 P, At =0"0l sec, )~2= 0"005 sec, ~1 =0.000125 sec 2, or2 =0.0000625 sec2, p = 1 g cm -~, ]0=200 dyne cm -3, N2 = --0-01 dynes sec2 cm -2. ab = 1 cm 2. tive streamline patterns for fixed values of the material parameters and different a/b ratios.* In the case o f the square cross section, the two vortices present in a ,quadrant are symmetrical about the diagonal as one would expect (cf. [8]) As a/b departs from 1, one o f the vortices increases in size with a corresponding decrease in the other, and for a/b = 0.5 the smaller vortex is very weak indeed. The patterns clearly demonstrate that symmetry considerations can only supply a lower bound to the number of vortices present. * We have taken a sign for N2 which corresponds to a positive second normal stress difference although there is growing support for a negative second normal stress difference, for concentrated polymers solutions at least (cf. [33, 37]) The reason for this choice will become apparent in section 3.
326
A.G. DOBSON,P. TOWNSENDand K. WAL~RS
The directions of the streamlines given in Fig. 2 correspond to a positive second normal stress difference. Their direction would be reversed for a negative second normal stress difference. Contours of constant wt for the same values of the material parameters as those used to compute Fig. 2 are given in Fig. 3. It will be seen that, due to the presence of secondary
Fig. 3. C o n t o u r s of constant w~ (on an arbitrary scale) for To : 1 P, A~- 0'01 sec, Az = 0.005 sec, o~ =0.000125 sec2, o2 =0.0000625 sec2, p = 1 gcm -3, if=200 dyne cm -3, N2 = --0.01 dynes scc2 cm- 2. ab = 1 cm2.
flows, the axial velocity component can be increased over part of the cross section and diminished over the remainder (cf. [8]). Our main concern is the overall effect of the secondary flows on the mean flow rate and this is demonstrated in Figs. 4-11. Figure 4 contains curves of the three viscometric functions for specific values of the material constants. We feel that the behaviour is fairly representative for the type of polymer solution used in the experimental work to be described in section 3. Figure 5 contains corresponding (Is, p) results for various a/b ratios, the area of the cross section of the pipe being kept constant; I s being that part of the percentage change in flow rate which is due to the influence of the secondary flows. We note that the secondary flows result in a change in flow rate but that this change is very small. It is not possible to project the curves to higher values of the pressure gradient p, since the corresponding viscometric functions then become unrealistic. Figure 6 illustrates the effect of varying the a/b ratio. The (Is , a/b) curve quickly reaches a maximum and then falls to very low values of I~ as the cross section becomes thinner. A similar situation apparently exists in the case of an elliptic cross section as the eccentricity is varied (H. Giesekus, private communication). The relative importance of the variable-viscosity effect and the secondary-flow effect on the percentage change in mean flow rate is given in Fig. 7 for the conditions described by Fig. 4.
N o n - N e w t o n i a n flow in pipes o f non-circular cross section
/
I00C
J ~t no rrr~l str~..~---e~difference
500 u~
~00
2OO
shear rate {seC"I) Fig. 4. Viscometric f u n c t i o n s for r/o = 1 P, AI = 0"01 sec, A2 = 0-005 sec, a l = 0"000125 sec 2, (Tz =0.0000625 sec 2, N'z = - 0 . 0 0 5 dynes sec z c m . - "
1"0
~x3
0"5
O-C
Fig. 5. Theoretical (I,, : ) curves for various a/b ratios. 7/0 = I P, Ax = 0.01 sec, ,~2 = 0.005 sec, a l = 0 - 0 0 0 1 2 5 sec 2, ( 7 , = 0 . 0 0 0 0 6 2 5 sec 2, p = 1 g c m -3, Nz=--0.005 dynes sec z c m -z.
0.5
/,
Fig. 6. Theoretical (Io, a/b) results for ~9o = 1 P, &l = 0"01 sec, Az = 0.005 sec, a l = 0.000125 sec 2, ~ , = 0 . 0 0 0 0 6 2 5 sec 2 p = 1 g c m -3, N2= --0.005 dynes scc 2 c m -2, . ~ = 2 5 0 dyne cm. - 3
327
328
A.G.
D o o s o ~ , P. TOWNSEND a n d K . WALTERS
60 58"
Newtonlol:1
56. 0
100
2'00 ~ {dynes/cc)
14. 13" 12" Fig. 7. (1, ,~) results for 7/o = 1 P, AI = 0.01 sec, ,~2 = 0-005 sec, al = 0.000125 secz, a z = 0.0000625 sec=, p = 1 g cm -3, N, = -0.005 dynes sec 2 cm -2. Cross sectional a r e a = 4 sq cm Broken line--rectilinear flow; full line--with secondary flow.
In an attempt to maximize the effect of the secondary flows on I, we have taken various forms for the viscometric functions a n d by a n d large we have needed w h a t might be r e g a r d e d as " u n r e a l i s t i c " functions to o b t a i n m e a s u r a b l e changes in flow rate. F o r example, the viscometric functions given in Fig. 8, give rise to the decreases in flow rate shown in Fig. 9. The results are expressed graphically in an alternative form in Fig. 10 to enable direct c o m p a r i s o n with the e x p e r i m e n t a l results given in section 3. The viscometric functions given in Fig. 8 might a p p e a r s o m e w h a t bizarre to a rheologist,
u)
500
difference
0
100
200 3 ~ shear rate ( s ~ 1] Fig. 8. Viscomctric functions for 7/o = I P, ~ = 0.01 sec,/~2 = 0.005 sec, ~ a z = 0.0000625 sec 2, N2 = -0.01 dynes secz cm -2.
=
0"000125 scc2.
Non-Newtonian flow in pipes of non-circular cross section
329
2.0 I,
1"0
dynes/cc Fig. 9. Theoretical (I,, ,~) results for 7/o= 1 P, A~=0"01 sec, A2 =0.005 sec, a~ =0.000125 sec2, o2 = 0.0000625 sec2 a = 2 cm, b = 0"5 cm, p = 1 g c m - a, Nz = -0.01 dynes seca c m - 2.
300
2
0
0
~
E
(clynes,/cm3J I00
0
tO0
200O(cc/sec)300
Fig. 10. Predicted (b, Q) for "t/o= 1 poise, A1 =0"01 sec, A2 = 0.005 sec, cq =0.000125 sec 2, u, = 0.00000625 secz, p = I g cm -3, N2 = --0"01 dynes sec" cm -z. Pipe B of rectangular cross-section a = 2 cm, b = 0.5 cm Pipe E of circular cross-section having same cross-sectional area as pipe B. Broken line--predicted curve for pipe B neglecting secondary flows. but detailed e x p e r i m e n t a l knowledge o f the viscometric functions is o n l y available f o r rather c o n c e n t r a t e d p o l y m e r solutions a n d there is some evidence to believe t h a t projecting a similar b e h a v i o u r to m o r e dilute p o l y m e r solutions m a y not be entirely correct (of. [37, 41]). W h a t we can say is t h a t the effect o f the s e c o n d a r y flow on the flow rate is likely to be very small unless the fluid has some rather unusual properties. 3. E X P E R I M E N T (i)
Apparatus
The e x p e r i m e n t a l set-up is illustrated schematically in Fig. 1 I. It consisted o f a peristaltic p u m p A which created the flow a n d s m o o t h i n g bottles B which d a m p e d o u t the pressure fluctuations a n d c r e a t e d a s m o o t h c o n t i n u o u s flow. The pipes used in the test section were t r a n s p a r e n t glass pipes o f circular cross section a n d similar pipes having square a n d rectang u l a r cross sections (supplied by Plowden a n d T h o m p s o n Ltd). F o u r different pipes o f this sort were available with dimensions as given in T a b l e I.
330
A.G. Do,sot4, P. TOWNSEr~Dand K. WALTEaS
pipe A
pipe E
Fig. I I. Schematic diagram of the apparatus.
Table I. Dimensions of the pipes
Pipe
2b (cm)
2a (cm)
b:a
A B C D
0.84 1'19 1.45 1-68
0.84 0.59 0'49 0.42
1: 1 2:1 3: 1 4:1
A pipe of circular cross section having the same cross sectional area was also available. We shall refer to this as pipe E. The pressure gradient was determined by inserting pressure probes in the wall of the tubes. These were modified syringe needles which were inserted flush with the inner wall of the tubing. They were connected to a differential pressure transducer T a n d hence to a U.V. recorder. The flow rate was measured in the conventional way by placing a graduated flask under the tap above the reservoir R for a specified time. In the experiments, the pipe of non-circular cross section was placed in series with a pipe o f circular cross section of comparable dimensions. The pressure gradient in both pipes could then be measured at the same flow rate. Such an exercise was thought to be indispensible since it was necessary to compute numerically the flow rate in the non-circular pipe using the viscosity function deduced from experiments in a pipe of circular cross section. We anticipated that any secondary flow in the pipe would lead to small changes and an accurate assessment of predicted and observed flow rate in the pipe of non-circular cross section was therefore essential. Flow visualization experiments were carried out at various flow rates and for different test solutions in both square and rectangular cross-section pipes. The dye used was a mixture of the test fluid and some strong colouring agent. In these experiments, the test pipes were illuminated from below by means of a Barnes Transilluminated tracing table. This produced an evenly illuminated surface which could be photographed from above. The Newtonian liquids used in the experiments were water and aqueous solutions of glycerol, with viscosities ranging from 0"1 to 14 P. The non-Newtonian liquids employed were aqueous solutions of polyacrylamide (P250, supplied by Cyanamide of Great Britain) In addition, a soap solution was also investigated. This was a mixture of an aqueous solution
Non-Newtonian flow in pipes of non-circular cross section
331
of (~) cetyltrimethyl ammonium bromide and (fl) an aqueous solution of toluene-4-sulphonic acid sodium salt in the ratio 2a:l/. Small amounts of salt were also added. This had the effect of altering the micelle concentration and hence the viscosity and elasticity of the solution. The soap was found to have an elasticity/viscosity ratio which was an order of magnitude higher than a comparable aqueous solution of polyacrylamide. This is demonstrated in Fig. 12 which contains dynamic viscosity r/' and dynamic rigidity G' results at different frequencies.* At the higher frequencies, the values of G' are comparable, but the soap solution has a viscosity which is an order of magnitude lower than the polymer solution. The soap solution was in fact the most "elastic" liquid we could find and was used in the hope of maximizing the effect of the secondary flow on the flow rate. In the flow visualization experiments the test section was mounted between two meter rulers. These served the dual role of measuring the distance along the pipe at which effects become apparent and defining the walls of the rectangular test section. The dye was either injected at the mid point of the top face (J) or at the mid points of the two vertical faces (K and K'). Fig. 13 contains the path of a dye thread after it had been injected at point J of a pipe of square cross section into a 1.75 percent aqueous solution of polyacrylamide. Here, there is an expansion of the dye thread and at a distance of approxiriaately 30 cm along the pipe two distinct dye streams are seen near the side walls. These tend to move back again towards the centre and at a distance of approximately 100 cm along the pipe they nearly touch at the centre. Visual inspection through the top and side faces of the pipe confirmed
\ \
/'
\ \
/
/~,
"~
G' cm"2
/
\/
/
/ /
0"
eb
3b
.~ s e c -1
Fig. 12. Experimental (r/', /2) and (G', f2) for a 2-0% aqueous solution of polyacrylamide (broken line) and a 2.0% soap solution (full line). * The dynamic viscosity and dynamic rigidity are associated with a small amplitude oscillatory deformation and are functions of the frequency of oscillation fL They may be taken as convenient measures of viscosity and elasticity, respectively (cf. [38]).
A.G. DODSON,P. TOWNSENDand K. WALTERS
332
F
I
Fig. 13. Typical flow visualization study for a 1.75°o aqueous solution of polyacrylamide. Injection point J in pipe with square cross section (pipe A). Q -- 1.22 cm 3 sec.that the dye threads moved out over the top face and then in along the diagonals, in agreement with the theoretical prediction. The observed direction o f the streamlines was consistent with the test solution possessing a positire second normal stress difference, as far as our simplified analysis is concerned. It would be wrong however to make definitive conclusions on this point since a constant multiple of r/ can be added to N 2 without affecting the secondary flow[2]. In Fig. 14, the situation is repeated except that the pipe now has a rectangular cross section. In this case, the dye threads stay at the side walls after moving out along the top face. Fig. 15 completes the picture for the rectangular pipe, since the points o f injection are now K and K'. Here, the dye filaments move towards the centre. Figures 14 and 15 indicate that while a weak secondary vortex o f the sort predicted in section 2 may have been present, it was certainly not strong enough to observe in our flow visualization experiments and to all intents and purposes only four vortices were present in the pipe o f rectangular cross section.
Fig. I4. Typical flow visualization study lor a 1-75% aqueous solution of polyacrylamide. Injection point J in pipe with rectangular cross section (pipe B). Q - - 1.72 cm a scc. -1
Fig. 15, Typical flow visualization study for a 1,75~o aqueous solution of polyacrylarmd¢. Injection points K and K ' in pipe of rectangular cross section (pipe B). Q = 0"49 cm ~ sec. 333 CAF VoL 2 No. 3/4--G
334
A.G. DoDsos, P. TOWNSENDand K. WALTERS 60 0
0
0
0
0
D
0
O - -
0--0
0 ~--~-~-
1.0 I%
I
0
0
n
0
0
0
B
0__0
~
20 C~
0
2bo
0
Re
0
c,bo
0bo
vbo
Fig. 16. Percentage decrease in flow rate/Reynolds number results for Newtonian liquids. Full lines--theory. Circles--experimental results.
The streamlines for the soap solutions showed the same general features as those for the polymer solutions but the movements took place more quickly for comparable flow rates. However, it was difficult to quantify the difference, since it depended to some extent on the thickness of the initial dye filament and its exact position on injection. Before embarking on pressure gradient/flow rate measurements for the elastico-viscous liquids, we checked the theoretical and experimental techniques for Newtonian liquids. Here, an analytic solution to the theoretical problem was also available for comparison [39] Figure 16 illustrates the agreement between theory and experiment, which is entirely satisfactory. In the case of the non-Newtonian liquids we adopted the following procedure. The (Qc,/3) results for the pipe of circular cross section were first used to determine the apparent viscosity r/(~) of the test fluid. This was then used in conjunction with the numerical techniques of section 2 to compute the flow rate in a pipe of square or rectangular cross section on the assumption that secondary flows were absent. We then took the difference between the predicted (QR, P) results for the pipes of non-circular cross section and the corresponding experimental results as a measure of the effect of the secondary flow on the flow rate. Figures 17 and 18 contain experimental (QR, P) results for two pipes for a 1"5~o and a 1.75 ~o aqueous solution of polyacrylamide. Also included are the theoretical curves based on the absence of secondary flows. The close agreement between the experimental data and the theoretical predictions implies that, for these particular solutions, the "inelastic'" approximation is sufficient to predict the flow, which in turn implies that the secondary flows, although in evidence in the flow visualization experiments, were not strong enough to significantly affect the flow rate. In contrast, experiments with the highly elastic soap solutions did indicate some secondaryflow influence on the flow rate. Experimental (Q~,/3) results for a 1-5 ~o soap solution in a pipe of square cross section are given in Fig. 19. Also included are the (Qc,/3) results for a pipe of circular cross section of the same cross sectional area together with predicted (QR,/3) results for the square pipe based on the numerical techniques of section 2 for rectilinear
Non-Newtonian flow in pipes of non-circular cross section
335
800
A
6OO
.~ 400
0
2
4 6 8 ORc.c/sec
10
Fig. 17. Experimental (Qa,/~) results for a 1"75~. aqueous solution of polyacrylamide (circles and triangles). Theoretical curves based on absence of secondary flow--full lines.
ipe D
400 tJ O~ Q,I
,0..
2OO
i
o
Fig. 18. Experimental (Qa, P) results for a 1"5 ~o aqueous solution of polyacrylamide (circles and triangles). Theoretical curves based on absence of secondary flow--full lines. flow. A similar situatiorr is shown in Fig, 20 for a 2.0 ~o s o a p solution in a p i p e o f r e c t a n g u l a r cross section. It will be observed t h a t in both cases, the theoretical predictions i n d i c a t e d a higher flow rate than was observed; the difference we associate with the influence o f the
A.G. DODSON,P. TOWNSENDand K. WALTERS
336
600
E 400 I/i
~ 200
Fig. 19. (Q,/~) results for a 1"5% soap solution. Full lines--experimental results. Broken line--predicted curve for pipe A assuming no secondary flow.
/ ,a.
200
0
4 Q cc~ se~"18
12
Fig. 20. (Q,)6) results for a 2"0% soap solution. Full lines--experimental results. Broken line--predicted curve for pipe B assuming no secondary flow. secondary flows. The effect is further demonstrated in Fig. 21 which contains (/,/~) results for a 1.5 per cent solution. The rather complicated shape of the theoretical and experimental curves in this figure from that given in Fig. 1 and found for the polymer solutions[40] would suggest that the soap solutions do possess the rather unusual material properties required to produce measureable secondary-flow influences.
Non-Newtonian flow in pipes of non-circular cross section
337
tO
O
3O I%
2O
/,
7
\
/ J
,/7
Newton~an i
1C
i
l
2 ' 0 0 dynes crr~3~00
|
C~O
Fig. 21. (1, p) results for a 1'5 % soap solution (pipe A). Circles--experimental points. Broken line--predicted curve assuming no secondary flow.
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