79
Shearing effects for the flow of surfactant and polymer solutions through a packed bed of spheres J. Vorwerk and P.0, Brunn *
(Received May 3, 1993; in revised form August 11, 1993)
Abstract The behavior of surfactant solutions through a porous medium is solely due to shear. Despite pronounced rheopexy, one rheological property that can uniquely be correlated with porous medium flow data is the shear stress z. A correlation of the same type allows us to account for the effects of variable viscosity in the case of polymeric solutions. The onset of increased resistance, which remains after this viscosity correction, occurs at the very same stress z”, at which the first normal stress difference Ni equals z in viscometric flows. This stress 2^is a universal constant, characteristic of the type of polymer solution. From these results it follows (a) that the Deborah number concept emerges in the c + 0 limit (infinitely dilute solution), (b) that outside this limit (finite c) the Deborah number concept cannot be applied and (c) that the non-viscous behavior in porous medium flow is a normal stress effect, i.e. it is elastic in origin. It can also be described by the concept of shear waves. The onset criterion, N,/r w 1, allows upscaling. Keywords: onset criterion; polymer solutions; porous medium; surfactant effects
solutions; viscous
1. Introduction The flow through a packed bed of spheres is of interest to many fields of engineering and science. If the fluids used contain additives, which cause more or less pronounced deviations from the Newtonian behavior of the solvent, rather strange effects have been reported [l-4]. Tertiary oil recovery is an example of a process which relies on such effects [S]. Since the flow * Corresponding
author.
0377-0257~94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDf 0377-0257(93)01192-7
80
J. Vorwerk and P.O. Brunn /J. Non-Newtonian Fluid Mech. 51 (1994) 79-95
within a concepts, extension study to
porous medium is rather complex it is quite clear that simple like the ones based on model simulations for steady uniaxial [4], have to be considered with care. It is the purpose of the present show how the experimental observations are affected by shear.
2. General results for porous medium flow of Newtonian fluids Identical spheres (diameter d) packed randomly in a cylindrical container (diameter D) comprise for d/D 4 1 an isotropic porous medium of porosity E and hydraulic radius &,
de K ’
Rh= 6(1 -E)
(1)
(2) being the wall effect factor (K --) 1 as d/D + 0). Dimensional reasoning reveals that the fundamental dimensionless parameters characterizing the flow of Newtonian fluids through this packed bed are (i) the Reynolds number Re Re = 6ptiRh/q
(3)
where p is the mass density of the fluid, V is the average velocity, related to the superficial velocity o via 0= %
(4)
and q is the shear viscosity, and (ii) the friction coefficient A
with Ap/L the pressure drop per unit length. For creeping motion it follows by definition that A has to be constant (Darcy regime). In the opposite extreme (highly turbulent flow) the tube flow model hints at a linear relation between A and Re (Burke-Plummer regime [6]). In our case, described in detail in Ref. 7, the correlation A = 181 + 2.01Re0.96
(6)
describes the experimental results rather well over almost seven decades (5 x lop5 < Re < 500). This is shown in Fig. 1. In subsequent figures this correlation will appear as a solid line, relative to which deviations from the behavior of a Newtonian fluid manifest themselves.
J. Vorwerk and P.O. Brunn 1 J. Non-Newtonian Fluid Mech. 51 (1994) 79-95
81
Fig. 1. The A-Re correlation for Newtonian fluids. The solid line represents the correlation given by eqn. (6).
3. Surfactant solutions
The cationic surfactant solutions used were equimolar mixtures of hexadecyltrimethylammonium bromide (Cl 6TMA-Br) and the salt sodium salicylate (Na-Sal) in deionized water. The concentrations c listed in this paper are based on the concentration of pure surfactant C16TMA-Sal. These surfactant solutions have been studied extensively [8], notably with respect to their drag-reducing effectiveness in turbulent pipe flow. It is known that above a certain critical concentration, termed c,, the surfactant molecules form cylindrical micelles [g]. This concentration increases with increasing temperature T.For concentrations c larger than c, (but not so large that liquid crystals form) strange rheological effects have been reported [9,10]. Starting from a Newtonian base line of q. x Q (region I), the apparent viscosity Q, increases almost instantaneously at some characteristic shear rate y* ( re gi on II). Then, after a transition region having an almost constant value q, for q (region III), it decreases after exceeding some shear rate j2 (see Fig. 2). This decrease occurs (on a log q, vs. log j plot) under an angle of -45”, implying an almost instantaneous decrease of qa at some characteristic shear stress r * (region IV). These results become more pronounced the higher c is (constant T) or the lower T is (contant c). On the basis of what has been said about c,, this was to be expected. Interpreting these results (obtained in the rotational Couette viscometer HAAKE RV 2O/CV 100) in terms of results characteristic for capillary viscometers (capillary radius R) has the following consequences (recall
82
J. Vorwerk and P.O. Brunn If. Non-Newtonian Fluid Mech. 51 (1994) 79-95 10 -*
ClGTMA-Sal ,
I ; 10 +:
’
AI
I
G .w
-s
Id
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III
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I I
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I
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IA
I
I
I
i
I
A+
I ,
I
I
I
I
1
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AA AAA
I
II II
3
Fig. 2. The apparent shear viscosity qs as a function of the apparent shear rate of a 750 ppm by wt. equimolar mixture of the surfactant C16TMA-Sal and NaBr in aqueous solution.
10 -l
I :’
Cl GTMA-Sal 1000 w-ppm
+ NaBr 35°C
Fig. 3. The apparent shear viscosity qa for a 1000 ppm by wt. equimolar mixture of Cl6TMA-Sal and NaBr aqueous soiution as a function of apparent shear rate f, for various sweep times.
3, = 4v/7cR3, with P the volumetric flow rate and z = qafa = R/2 Ap/L with Ap/L the pressure drop per unit length, ‘c denoting the wall shear stress): o regions I and III: a linear increase of 6’ with Apex; o region II: almost constant 3 despite an increase in pressure drop; o region IV: an increase in P although the pressure drop is (almost) constant.
J. Vorwerk and P.O. Brunn 1 J. Non-Newtonian Fluid Mech. 51 (1994) 79-95 10
83
J
ClGTMA-Sal 1000 wppm
+
NaBr 35°C
10 I
1
10 -’
I
1
7
in
Pal’-’
Fig. 4. The data of Fig. 3 redrawn to show q, as a function of shear stress 2. Note that the drop at the characteristic shear stress z * ( z 1.3 Pa) is independent of the sweep time.
,
ClGTMA-Sal
I
1
+ NaBr
II ’
10 1: ’
c”
III
@%cOowoo I
-
$1 I
IO1
10 3:
500
I I
I
I
IO
I
I
lo 0
1 ,
I
I
IV
’ ’ 9 ,
I
spDmogb
wppm
00
35°C
( I I
00 “00
I 00
I
0 00
I 0
I
I
I
I
I
I
Fig. 5. Typical porous medium flow curve of an equimolar mixture of C16TMA-Sal NaBr in aqueous solution (c = 500 ppm by wt.).
and
Although all these features (four characteristic regions) remain unchanged irrespective of how the measurements are taken, the details change rather drastically. The reason being the rather pronounced rheopexy of these solutions [lo]. Figure 3 (taken in the rotational Couette viscometer HAAKE RV20/CV 100 with double gap measuring system DA45 for various sweep times), reveals that all characteristic parameters, namely 3*, q, as well as j2 depend strongly upon the way in which the data are
J. Vorwerk and P.O. Brunn 1 J. Non-Newtonian Fluid Mech. 51 (1994) 79-95
84 1
ClGTMA-Sal
+ NaBr
600
wppm
10 -’
m loa 2 .z 10 -’ > 10 J
10 4
10 -’
I’
Ap/L
in
Pa/m
Fig. 6. Data of Fig. 5 replotted as superficial velocity vs. pressure drop.
acquired. Rather surprisingly, the shear stress r* (region IV) is independent of any of these details (Fig. 4). For the flow of these solutions through a packed bed of (identical) spheres, the correlations of AS vs. Re, (the index s emphasizing the fact that the Newtonian viscosity qSof the solvent has been used (note qSM q. in our case) in the definitions (3) and (5) respectively), show a striking similarity to the rheological qa-fa correlations reported above (see Fig. 5). For c > c,, A, increases from its Newtonian base line (region I) at some characteristic Reynolds number Re* (region II). Then, after a transition region at an almost constant value Am for AS (region III), it decreases after exceeding a Reynolds number Re,. This decrease occurs on a log ASvs. log Re, plot under an angle of -45”, implying an almost instantaneous decrease of AS at some characteristic value for ASRe,, called ( ARe) *; region IV. The results become more pronounced the higher c is (constant T) or the lower T is (constant c). For c < c, no effect is detected. Thus, there seems to be a relation between AS and Q, as well as between Re, and 3,. On the basis of the actually measured data, P vs. Ap (see Fig. 6), this becomes even more apparent: v increases linearly with Ap in regions I and III, stays almost constant in region II despite an increase in Ap (region II), while it increases almost instantaneously in region IV despite an essentially constant pressure drop. Thus, there can be no doubt that the behavior of surfactant solutions in the flow through a porous medium is due to shear [ 111. Even some kind of rheopexy can be detected in porous medium flow. Since d/6 corresponds to a characteristic time of this flow, a change in d should lead to different results; Fig. 7 shows this rather clearly.
J. Vorwerk and P.O. Brunn / J. Non-Newtonian Fluid Mech. 51 (1994) 79-95
10’1 10”
1 ““‘“I
10-l
8 ““‘*‘I
1
’ “““‘I
10
’ “““‘I
10 *
u ‘aim
85
1 I’
Re, Fig. 7. The effect of sphere size on an aqueous solution of a 500 ppm by wt. equimolar mixture of C16TMA-Sal and NaBr. Note that the wall effect factor K (eqn. ( 1)) is included in the definitions of A, and Re, (D = 43 mm).
This figure also reveals that the intuitive correlations ja + Re, and 1, + A, between rheological and porous medium flow data are wrong. If they were true then the shear stress z would have to correlate with &Re,. However, the characteristic shear stress z* is independent of the details of the flow curve (rheopexy), while (ARe)* is not (see Region IV of Fig. 7). The quantity that is independent of particle size in region IV is AsRes/d2, which is proportional to R,Ap/L. Thus, As as a function of R,Ap/L becomes independent to the details of the flow field in region IV, where R,Ap/L reaches some critical value (RhAp/L)*. These results imply that the fundamental role played by r for surfactant solutions in viscometric flows is replaced by the fundamental role of R,Ap/L, which occurs in porous medium flows. Since the behavior in porous medium flow is due to shear, a correlation of the form r = kR,AplL has to prevail. For surfactant solutions, the constant of proportionality k=
(7) k,
z*
(8) RP* ( hL > is independent of any of the details of the flow field. (In our case, k M 0.1.) The fact that surfactant solutions show rheopexy no longer matters. It does matter, however, as far as the other two pairs of characteristic quantities (3*
J. Vorwerk and P.O. Brunn 1 J. Non-Newtonian Fluid &tech. 51 (1994) 79-95
86
and qrn and, respectively, Re* and A& are concerned, the exception being flow conditions for which qrn/qs w &.,/A,, and simultaneously, f&j* x Re,/ Re*. In this case the constant k alone suEices. This exceptional (and rather singular) case, which allows, with a single constant k, substitution of rheological data by data obtained in porous medium flow and vice versa, has been treated before [ 121. 4. Polymer solutions 4.1 Viscosity eflects for polymeric solutions For infinitely dilute polymer solutions, the shear viscosity q differs very little from qS, the shear viscosity of the solvent. Thus, dramatic effects found in porous medium flow have to be of different origin. The idea that they could be due to elongation seems natural [4]. For higher concentrations the shear viscosity q will differ substantially from qS, with rather pronounced shear thinning. Figure 8 shows a typical example for aqueous solutions of HPG (hydroxypropyl guar), a polysaccharide derivative (in our case with an average molecular weight of 2.4 x lo6 Kg kmol) of interest to the oil industry (cracking fluid). It does not come as a surprise that these solutions in porous medium flow display a behavior quite different from the one which characterizes a Newtonian fluid (eqn. (6)). Figure 9 shows the experimental results. Accounting for viscosity effects implies that q instead of Q has to be used (the corresponding dimensionless parameters will be denoted by A and Re,
1 j
1 HPG
25°C
ooooo lop
wpppm
00000
wpppm
500
AI&M 00000 800 wppm 2600 lrppm -
5000
wppm
0
0
Fig. 8. The shear viscosity of aqueous HPG solutions as a function of shear stress of various concentrations.
J. Vorwerk and P.O. Brunn /J. Non-Newtonian Fluid Mech. 51 (1994) 79-95
Fig. 9. Porous medium flow results for HPG solutions. direction of shift for the viscosity correction.
The dashed line indicates
87
the
respectively). As far as Fig. 9 is concerned, this is a shift along the dashed line. Since q is variable this is a point by point shift. If the behavior in porous medium flow were solely due to shear, the A-Re correlation would have to be governed by eqn. (6). We do not expect this to be the case, since infinitely dilute solutions, with no shear effects (q M qS) essentially follow eqn. (6) only for small values of Re, [ 131. To acomplish the viscosity correction, qS+ q, we rely on eqn. (6). For each point of Fig. 9 the data Ap/L and P have been measured. For each point measured, that value of q can be determined such that the A- Re correlation, based upon this q-value would follow eqn. (6). Especially for the first few points (low Re, values) this value for q is associated with the viscometrically determined flow curve, q = q(z) (or its extension via master curves [ 131). As a consequence, the corresponding value for the shear stress z can be deduced. Utilizing eqn. (7), the constant k results. Repeating this procedure for a couple of data points in the Re, + 1 range without noticing any change in the value for k, reveals that the correct value for k has been found. Figure 10 shows a typical result for this correction. While for the HPG solutions a k value of 1 sufhces, the fluid Al, a 2 wt.% solution of PIB (the polyisobutylene Oppanol B200) in decalin, requires the value k = 0.78 (Fig. 11). These results reveal that the influence of variable viscosity can be accounted for by eqn. (7) with a k value of order one. 4.2 The influence of viscometric data for the onset point of polymeric solutions Having estimated the influence of viscosity for the behavior of polymeric solutions in porous medium flow it seems rather natural to concentrate-as
Fig. 11. Porom ~~~~~ flow restits for the fitid Al, c1, ~~~~~, k = 0.78
corrected for shear. The Newtonian correlation h = 181 + 2.01 Rc’.~~.
far as the
ad 0, k = 1; +, base line (solid line) refers to the A-&
departure of A from its Nebulas base line is co~~~~-o~ other types of Gow, notably elong8tion. As a matter of fact, for ~~n~~ly dilute polymeric solutions (essentially no shear effects) the concept of infinitely dilute FENE dumbbell model solutions in steady uniaxial elangation seems to work rather well [4]. This concept implies that the elongationat viscusity qe esscnt~al~~jumps at a Deborah number Lie of 0.5 from its ~ewton~~ base va.Jue to some other plateau value, which will remain ~~cha~g~ fur all values & r 0.5. Assuciatiug A with qe and neglecting
J, Vorwerk and P.O. Brunn /J. Non-Newtonian
~0~0~ 500 AAAAA800 00000 2500 **et* 5000
Fluid Mech. 51 (1994) 79-95
89
wppm wppm wppm wppm
Fig. 12. The data of Fig. 10 replotted as A/( 181 + 2.01 x Re0.96) as a function of strain rate i = v/d.
constants of proportionality, this implies a jump of A from its (Newtonian) value of 181 to some other plateau value at a specific strain rate 1 cc d/u. Serious theoretical objections against this (idealistic) concept have been advanced [ 141. Our own experimental results [ 131 reveal three objections to this result (see Fig. 12): (a) the onset strain rate is by no means a characteristic constant, but depends strongly upon concentration; (b) the increase of A from its Newtonian base line is not sudden but rather gradual (the more so the more concentrated the solutions are); (c) there is no ultimate plateau value for A, but rather some maximum value, after which A decreases. This decrease has absolutely nothing to do with polymeric degradation, since repeating the experiments with the same solution reproduces the results. If we maintain the concept that the A-Re correlation is (mainly) due to elongation we have to abandon the notion of FENE dumbbell models in steady uniaxial extension. As a matter of fact, we have to abandon the idea that results characteristic for any type of steady elongational flow will carry over for porous medium flow. This is not unexpected if we realize that inside a porous medium, flow into a pore will have to be followed by flow out of a pore. This would-in the simplest kind of interpretation-correspond to an elongational type of flow followed by a compression type of flow. Since the transit time t* for a given type of flow would-for a given porous medium-vary inversely with u, while the elongation rate < would be proportional to u, the product
90
J. Vorwerk and P.O. Brutm / J. Non-Newtonian
Fluid Mech. 51 (1994) 79-95
strain rates (1 is too small), while for large strain rates the transit time t* would be too small for the polymer molecules to react. A A = A(<) curve passing through some maximum would be the consequence. Our results (Fig. 12) show this behavior. Furthermore, they reveal that the A maximum always occurs at the same strain rate. This implies that for all HPG solutions there exists one characteristic time constant L, which governs the behavior in porous medium flow. For completeness we should point out that the ideas advanced have been tested quantitatively in the case of the fluid Al [ 151. There the increase of A with Re is so steep that it is possible to describe it by the behavior of an upper convected Maxwell model fluid in a periodic step change succession of elongation-compression type of flow. Thus, while it seems clear that the behavior after onset in porous medium flow is due to extensional types of flow, it is the onset itself which can be predicted by results obtained in viscometric flows. Playing around with the data reveals that the onset is characterized by a specific wall stress, which in the case of HPG solutions is in the range l-5 Pa (see Fig. 13). This implies that there is a characteristic stress z^which governs the onset, irrespective of concentration. For completeness, we note that at the very same shear stress z^drag reduction in turbulent tube flow sets in [16]. In viscometric flows, there, too, exists the notion of some characteristic shear stress z”.This is based on experimental findings which reveal that the flow curves q = q(i() for one type of polymer solution all follow one universal master curve, irrespective of temperature, concentration, etc. This
c
10 =
~0~~0 500 wppm AAAAA800 wppm
00000 *****
2500 5000
wppm wppm
a-
0 a
Fig. 13. The data of Fig. 10 replotted as A as a function of the wall shear stress 2.
J. Vorwerk and P.O. Brtmn f J. Non-New~o~~nF&id Mech. 51 (1994) 79-95
91
10’3
10
-9
00000
7
aooun
N1
t 1
HPG 25'C 5000
,
10
_3
in s-’
wppm
1
10*
Fig, 14. The shear stress T and the first normal stress difference N, as a function of shear rate i for a 5000 ppm by wt. HPG solution. The point of intersection is denoted by z”. fact has been utilized for HPG solutions to extend the q = q(z) range experimentally accessible to us [ 131. An analogous universal law seems to hold for the first normal stress difference NE as well [ 17. Plotted as a function of shear rate j, the N, curve will intersect the z curve, at some value ? (see Fig. 14). Since Ni as well as r can be described by universal laws, 2’has to be universal too, i.e. independent of temperature, concentration, and so on [ 171. The value for z”is characteristic for the type of polymer solution. For the HPG solution studied by us, z”was always found to be around l-5 Pa (see Fig. 14). Comparison with the results from porous medium flow suggests that z”is identical to z^.What this imples is that-after correction for viscosity-the onset in porous medium flow (which corresponds to the onset of drag reduction in turbulent tube flow) occurs at a characteristic stress f. This .stress is characteristic for the type of solution (irrespective of con~ntration~ and coinicdes with the characteristic stress f, which originates from viscometric studies. Thus, rheologkal data will sufhc~ to reveal the point of onset of increased resistance in porous medium flow (and, correspondingly, drag reduction in turbulent tube flow). This fact seems fundamental for any scale-up problems. Since after onset N1/z 2 1, it follows that the (non-viscous) behavior in porous medium flow is a normal stress effect, i.e. it is purely elastic in origin. For the fluid Al the onset occurs at a critical wall shear stress, z^* 3 Pa (see Fig. 11). In view of the experimental Al results reported in Ref. 18 for viscometric flows, it seems that the critical stress ?, where N, starts to exceed z, also is of that order. Thus, again we find z^= 2”(or N1/z), as the onset criterion.
92
J. Vorwerk and P.O. Brm
1 J. Non-Newto~~n Fluid Me&
51 (1994) 79-95
For infinitely dilute solutions of polyacrylamide (PAM), the rather sudden increase of AS observed at some specific onset Reynolds numbers Re,, has been associated with the behavior of isolated FENE dumbell model macromolecules in steady uniaxial extensional flow [4]. Although conceptually this simple idea cannot be correct [ 151, it is remarkable that it does a pretty good job in predicting the onset of infinitely dilute polymer solutions. To do so, it utilizes the Deborah number De = &6,
(9)
with
(10) a characteristic time constant ([qlo is the intrinsic viscosity, T the absolute temperature and M the molecular weight) and i the strain rate of the uniaxial elongational flow. Dimensional reasoning requires, for porous medium flow, 1 to be proportional to v/d,
i=/$
(11)
d’
with kl a constant of proportion~ity. This is dete~ined by the requirement, that at onset the Deborah number (termed Bee) equals 0.5, i.e, De0 = 0.5= &klz
(12)
at onset. For infinitely dilute PAM solutions the constant kl w 8 seems to work rather well [4]. For HPG solutions we have & = 1.44 x 10m3s [ 131. Looking at Fig. 10 reveals that the condition (12) could be valid, if kl is allowed to depend strongly upon concentration. Formally using eqn. (12) leads to the results of Fig. 15, This is in contradiction to the Deborah number concept, which requires a constant kl, independent of con~ntration. To show this, we have to realize that up to the onset the (shear-corrected) values follow Darcy’s law for the solvent, A = 181. Using eqn. (7) (with k = 1) and recalling eqns. (9) and (1 l), shows that
1,
h&C 181 (1 -E)~~ -r e2KkI ’ 4% = 36R,,=6 For K = 0.98 and E = 0.375 [ 131 we have De
kl = 136.5 (/l,,q,)r
-
(13)
(14)
J. Vorwerk and P.O. Brunn 1 J. Non-Newtonian Fluid Mech. 51 (1994) 79-95
93
)O
Fig. 15. The “constant” k, of eqn. (12) as a function of concentration. has been taken from eqn. (15).
Evaluating the right-hand side at onset for HPG 2^M 3 Pa, qs = 0.89 mPa s) leads to k, x 14.
The c = 0 limit of k,
solutions
(De = 0.5, (15)
Since eqn. (15) shows no dependence on c, the Deborah number concept will work only in the c + 0 limit (infinitely dilute solutions). Realizing that an onset point can not be determined experimentally with absolute certainty, we conclude (a) that the Deborah number concept can be applied to porous medium flow of infinitely dilute solutions of industrial polymers and (b) that it requires one unique value for k, of order 10. For PAM solution, k, values in the range 4-9.1 have also been reported by the authors of Ref. 4. Thus, for c + 0 our results, onset at one specific stress z*,is synonymous with the Deborah concept, De = 0.5 at onset. For higher concentrations? the Deborah number concept cannot be applied. 5. Conclusions In this paper, we have studied the effect of shear for the flow through a packed bed of spheres of surfactant solutions and polymeric solutions. The results can be summarized as follows. (A) Surfactant solutions For surfactant solutions the behavior in flow through a packed bed of spheres is solely due to shear. Correlating the effets quantitatively is, in
J. Vorwerk and P.O. Brunn 1 J. Non-Newtonian Fluid Mech. 51 (1994) 79-95
94
general, not possible, since the solutions show pronounced rheopexy. No rheopexy is found in the essentially sudden decrease of q at some specific shear stress z*. An analogous behavior is displayed in porous medium flow, where AS suddenly decreases at constant R,Ap/L, irrespective of any details of the packed bed. The fundamental relation, given by eqn. (7) is the consequence. (B) Polymeric solutions
The influence of viscometric data on the flow of polymeric solutions through packed beds is two-fold. (a) To account for the variable viscosity q = q(r). This amounts to replacing AS by A and Re, by Re. There, too, the fundamental relation, eqn. (7), is utilized. The constant of proportionality k is determined by requiring that for very small Reynolds numbers Re, 4 1; the behavior should be governed by correlations characteristic for a Newtonian fluid (Darcy law). All of the solutions tested furnished a k factor of order one. (b) To obtain the onset point. The onset point can be predicted from viscometric data, the reason being that for one type of polymer solution the point of onset is characterized by some specific stress z^(independent of concentration!), which is identical to that stress where the rheological curves z = r(j) and JVr= N1(f) intersect. The importance of these results for industrial practice (scale-up) should be emphasized, the more so since for the polymers studied the onset of increased resistance in porous medium flow coincides with the onset point for drag reduction in turbulent tube flow. The increased (non-viscous) resistance after onset corresponds to N1/ z > 1. Thus, it is a normal stress effect and therefore purely elastic in origin. It is worth while to look at these results from an entirely different point of view. In viscoelastic fluids there exists the possibility that disturbances propagate the means of waves, the shear waves. For a linearized Maxwell model fluid (time constant &, shear viscosity qO), the shear wave velocity turns out to be us=
Jf-&
(16)
Applying the tube flow model to the flow through a porous medium (recall also eqn. (7)) reveals that the wall shear stress velocity u,, u,=
JI, P
(17)
95
J. Vorwerk and P.O. Brunn 1 J. Non-Newtonian Fluid Mech. 51 (1994) 79-95
will play a fundamental role. For low Reynolds number flow of our model fluid, shear waves should exist for u, 2 u,, i.e. for
(18) This shows that waves account for the non-viscous behavior. While this different way of interpreting the data does not enhance our physical insight (save to say that shear waves are elastic in origin) it does point out the fundamental role of z, at onset. Finally we have shown that for c + 0 (infinitely dilute polymer solution) our results imply the Deborah number concept and that this concept cannot be applied to moderately concentrated polymeric solutions. Acknowledgment This work was supported in part by Deutsche Forschungsgemeinschaft and the authors gratefully acknowledge this support. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
R.J. Marshall and A.B. Metner, Ind. Eng. Chem. Fundam., 6 (1967) 393. E.H. Wissler, Ind. Eng. Chem. Fundam., 10 (1971) 411. H. Michele, Rheol. Acta, 16 ( 1977) 413. R. Haas and F. Durst, Rheol. Acta, 21 (1982) 150. J. Holweg, P.O. Brunn and F. Durst, in 4th European Symp. on Enhanced Gil Recovery, Hamburg, 1987, p. 1007. R.B. Bird, W.E. Steward and E.N. Lightfoot, Transport Phenomena, John Wiley, New York, 1960, p. 196. H. Holweg, Dissertation, University of Erlangen, 1989. D. Ohlendorf, W. Interthal and H. Hoffmann, Rheol. Acta, 25 (1986) 468. H. Rehage, I. Wunderlich and H. Hoffmann, Prog. Colloid. Polym. Sci., 72 (1986) 51. A.M. Wunderlich and P.O. Brunn, Colloid Polym Sci., 267 (1989) 627. P.O. Brunn and J. Holweg. in D.R. Oliver (Ed.), Third European Rheology Conference, Elsevier, London, 1990, p. 78. P.O. Brunn and J. Holweg, J. Non-Newtonian Fluid Mech., 30 (1988) 317. S. Chakrabarti. B. Seidl, J. Vorwerk and P.O. Brunn, Rheol. Acta, 30 (1991) 114. R.K. Gupta and T. Sridhar, Rheol. Acta, 24 (1985) 148. J. Vorwerk and P.O. Brunn, J. Non-Newtonian Fluid Mech., 41 (1991) 119. S. Chakrabarti, B. Seidl, J. Vorwerk and P.O. Brunn, Rheol. Acta, 30 (1991) 124. C. Biihme, Strijmungsmechanik nicht-Newtonscher Fluide, Teubner, Stuttgart, 1981. N.E. Hudson and T.E.R. Jones, J. Non-Newtonian Fluid Mech., 46 (1993) 69.