Viscoelastic flow in a periodically constricted tube: The combined effect of inertia, shear...

Viscoelastic flow in a periodically constricted tube: The combined effect of inertia,shear thinning, and elasticityStergios Pilitsis, Athanassios Souvaliotis, and Antony N. Beris

Citation: Journal of Rheology (1978-present) 35, 605 (1991); doi: 10.1122/1.550183 View online: http://dx.doi.org/10.1122/1.550183 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/35/4?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Shear thinning effects on blood flow in straight and curved tubes Phys. Fluids 25, 073104 (2013); 10.1063/1.4816369 Interplay of Inertia and Elasticity, Enhanced Heat Transfer and Change of Type of Vorticity in Noncircular TubeFlow of Nonlinear Viscoelastic Fluids AIP Conf. Proc. 1027, 78 (2008); 10.1063/1.2964848 Polymer-induced drag reduction: Effects of the variations in elasticity and inertia in turbulent viscoelastic channelflow Phys. Fluids 15, 2369 (2003); 10.1063/1.1589484 Effect of free surface and inertia on viscoelastic parallel plate flow J. Rheol. 38, 151 (1994); 10.1122/1.550509 Laminar Flow in Tubes with Constriction Phys. Fluids 15, 1700 (1972); 10.1063/1.1693765

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Viscoelastic ft.ow in a periodically constrictedtube: The combined effect of inertia, shear

thinning, and elasticity

Stergios Pilitsis,a) Athanassios Souvaliotis, and Antony N. Beris

Department of Chemical Engineering and Center forComposite Materials, University of Delaware, Newark,

Delaware 19716

(Received 7 August 1990; accepted 11 February 1991)

Synopsis

The flow of a shear-thinning viscoelastic fluid (Separan AP-30) through aperiodically constricted tube (peT) is simulated using a mixed pseudospectralfinite difference numerical method and a variety of constitutive equations. Theresults obtained with viscoelastic models show, at best, a small increase in theflow resistance beyond the value computed for a purely viscous inelastic fluid.The largest increase was observed with the modified Phan-Thien-Tanner model.This model, in addition to fitting the standard viscometric data (viscosity andfirst normal stress difference), predicts a nonzero second normal stress difference and allows for dynamic behavior characterized by nonaffine deformationand stress overshoot in start-up of shear flow. All the calculated values, although they show the correct tendency, are well below the experimentally measured values of Deiber and Schowalter (1981).

I. INTRODUCTION

Viscoelastic flow through a periodically constricted tube (PCT) is ofimportance for a number of reasons: for the evaluation of constitutiveequations (Pilitsis and Beris, 1989; 1990a; James et al., 1990), for thetesting of numerical methods in viscoelastic flow calculations (Burdetteet al., 1989; Zheng et al., 1990; Crochet et al., 1990), and also for theunderstanding of viscoelastic effects in flow through porous media(Deiber and Schowalter, 1981; Zick, 1983).

aJpresent address: Dow Plastics, Application Engineering Development, CH-8274 Tagerwilen, Switzerland.

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606 PILITSIS, SOUVALlOTlS, AND BERIS

The analysis to the problem of viscoelastic flow through a PCT isconsiderably more difficult than the generalized Newtonian one (Lahbabi and Chang, 1986), although this has just been recognized. Thetheoretical investigations reported over the last few years have usedOldroyd-type fluid models and perturbation (Ahrens et al., 1987; Pi1itsis and Beris, 1989) or numerical techniques in order to determine theeffect of fluid elasticity on the flow resistance (Pilitsis and Beris, 1989;1990a; Burdette et al., 1989; James et al., 1990; Zheng et al., 1990;Crochet et al., 1990). For a comprehensive review of the literature, theinterested reader is referred to our previous work (Pilitsis and Beris,1989; 1990a; James et al., 1990). In that work, the analysis has concentrated on constant viscosity viscoelastic models, such as the Maxwelland the Oldroyd-B fluids.

The key to the success of our previous investigations has been the useof an efficient and highly accurate numerical technique which is basedon a mixed pseudospectrallfinite difference approximation of the variables (PSFD and PCFD methods, Pilitsis and Beris, 1989). These newmethods in conjunction with very fine discretizations allowed the computation of converged with mesh refinement solutions at high Weissenberg numbers. The calculations showed no substantial increase in theflow resistance with either the Maxwell or the Oldroyd-B fluid models.These numerical predictions were later confirmed by the finite elementcalculations of Burdette et af. (1989) and, more recently, by the boundary element predictions of Zheng et al. (1990) and the finite elementresults of Crochet et af. (1990).

Numerical predictions using the Oldroyd-B fluid model were foundto be in semiquantitative agreement with the experimental data of Jameset al. (1990), obtained with the M1 Boger fluid. However, both calculations and the above experiments, involving viscoelastic fluids modeledby either the upper convected Maxwell or the Oldroyd-B constitutiveequations, have failed to show for the PCT flow any substantial increasein the flow resistance, which, as experimental evidence indicates, is acharacteristic of viscoelastic flow through porous media (Marshall andMetzner, 1967;James and McLaren, 1975; Mannheimer, 1983; Durst etaf., 1987). This failure may be attributed to an inherent inadequacy ofthe PCT geometry, as a model pore geometry, to capture the characteristics of the porous medium paths or it may be considered a singularfeature of Oldroyd-B type fluids.

Deiber and Schowalter (1981) conducted both experiments and numerical calculations for the flow of a shear-thinning viscoelastic fluidthrough a PCT. For dilute aqueous polyacrylamide solutions (Separan

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SINUOUS TUBE FLOWS 607

AP-30), the experiments showed an increase in the flow resistance withincreasing flow rate. Although these findings were obtained under inertial flow conditions, they seemed to justify the use of the PCT as a modelpore geometry for porous media flows. Their numerical results, obtainedfor both an upper convected Maxwell fluid and a White-Metznermodel, showed no substantial deviations from the Newtonian flow resistance. However, the elasticity values investigated were small due tonumerical difficulties associated with the use of low-order finite difference approximations. On the basis of their numerical predictions forflow of a power-law fluid through the PCT, Deiber and Schowalterconcluded that the increase in the flow resistance is due to elastic effects.

The numerical method used by Deiber and Schowalter employed alow-order finite difference approximation of the derivatives. This numerical scheme attempted to solve the problem for a particular set ofgeometrical parameters proceeding in terms of iterations with respect tothe wavelength of the corrugation. This method has been later severelycriticized by Lahbabi and Chang (1986) and by Ralph (1987). In bothcases, enough evidence was presented that the predictions of Deiber andSchowalter (1979) for Newtonian flow through a corrugated tube usingthe same numerical method were also not accurate (see also Sec. III A).In addition, for the flow of a power-law fluid, Deiber and Schowalter( 1981) did not perform any mesh refinement calculations in order tosupport the validity of their results. Finally, Sheffield and Metzner(1976), who studied the same flow problem, concluded that inertialeffects may arise at very low Reynolds numbers in the case of highlynon-Newtonian fluids.

Given all the above considerations, it is not even clear whether theincrease in the flow resistance observed in the experiments of Deiber andSchowalter (1981) is solely due to elastic effects. It may very well bedue to inertial or to a combination of inertial and elastic effects. Itshould be emphasized here that experimental findings for the flow resistance of shear-thinning fluids expressed in terms of dimensionlessflow resistance parameters, such as the one used by Deiber and Schowalter, must be interpreted with caution, particularly when conclusionsconcerning the significance of elastic effects are to be drawn on the basisof such data. The reason is that there is no guarantee that a particularnondimensionalization of the flow resistance (which essentially impliesa certain functional relationship between pressure drop-flow rate) canreliably compensate for nonlinear effects such as shear thinning andinertia (see also Sec. III C). The only way to assess the importance ofelastic effects is by normalizing the data against the predictions obtained

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608 PILITSIS, SOUVALIOTIS, AND BERIS

for an inelastic fluid with a matching viscosity. Alternatively, the difference between the prediction of a detailed simulation of the flow usingan inelastic fluid model and the experimental observations needs to beconsidered.

The present investigation has two parts. First, the flow of a purelyviscous, but nonlinear inelastic (generalized Newtonian) fluid is examined, using a power-law model for the viscosity. Second, the effects ofviscoelasticity are considered using a variety of shear-thinning viscoelastic fluid models. The objective of the numerical calculations is to elucidate the cause of the increase in the flow resistance observed in theexperiments of Deiber and Schowalter (1981).

The computational vehicle which is used in this investigation is thesuccessfully tested pseudospectral cylindrical/finite-difference (PCFD)technique (Pilitsis and Beris, 1989; 1991a; James et al., 1990). Occasionally, for comparison purposes, a standard Galerkin finite elementtechnique (Pilitsis, 1990) and a full pseudospectral method (Pilitsis andBeris, 1991b) are also employed.

This paper is organized as follows. In Sec. II, the problem formulation is outlined. In Sec. III, the results from the investigation of theinertial flow of an inelastic fluid through a PCT are presented. SectionIV contains a description of the results obtained for inertial viscoelasticflow corresponding to a number of different shear-thinning viscoelasticfluid models. Finally, in the last section, Sec. Y, the conclusions fromthis work are presented.

II. PROBLEM FORMULATION

A. Flow geometry

The flow geometry under consideration is shown in Fig. 1. The localradius of the undulating tube along the z axis is given by

,w=R[I-acos(27TZ/L)], (2.1)

where R is the average radius of an equivalent straight tube and a, L arethe dimensionless amplitude of the undulation and the wavelength, respectively. In addition to a, another characteristic dimensionless number is the aspect ratio N, related to the dimensionless wave number k by

N=R/L=k/2rr. (2.2)

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SINUOUS TUBE FLOWS

\7 'J \J \J

FIG. 1. Flow geometry of the periodically constricted tube (Pf'T). The wall of the tubeis described by Eq. (2.1). The shaded area represents a unit computational cell.

B. Viscoelastic fluid models

In the present investigation, a variety of differential constitutiveequations are being used. These include both purely viscous fluids (suchas the Newtonian and the power-law fluids) and viscoelastic models. Inall these models, the stress of the polymeric fluid rp is given by thegeneric differential constitutive equation

tlrTrp +..i( 1') 5; = 211p( y)[ I + ~(2 - ~)A.(1')2f] D. (2.3)

Here, 0 is the deformation rate tensor defined as

D=(Vv + Vv T)I2, (2.4)

and A.( 1'), l1p(r> are the relaxation time and the viscosity, respectively,which are allowed to be functions of the generalized shear rate l'(square root of the second invariant of the rate of deformation D),

1'= [2 tr(D·D) J112.

The scalar function Tin Eq. (2.3) is given by

T=l + E.~.(y)tr(rp)/l1pO'

where E, as well as ~ in Eq. (2.3), are model parameters (see below),and l1pO is the viscosity function l1p(y) in the limit of vanishing deformation rate (y-O). Finally, the operator 5/& is the Johnson-Segalmanobjective time derivative defined as a linear combination of lower andupper convected derivatives (Gordon and Schowalter, 1972)

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610 PILITSIS. SOUVALIOTIS, AND BERIS

8Tp aTp T-= - + V-VTp - Vv 'Tp - Tp'VV + t(D'Tp + Tp'D), (2.7)8t at

where S= 0 corresponds to the upper convected derivative, t = 1 to thecorotational derivative, and S= 2 to the lower convected derivative(Bird et al., 1987).

The generic model given by Eq. (2.3) may be considered with thepresence of a purely viscous component. When present, the purely viscous component is usually taken as Newtonian. Then, the total stress isgiven as

T=Tp + Ts'

where T s is the solvent (Newtonian) component defined by

Ts= 211sD,

where 11s is the solvent viscosity.The generic model of Eq. (2.3) is a particular variant of the modified

Phan-Thien and Tanner (MPTT) fluid (Phan-Thien, 1984), of the typeproposed recently by Baer and Finlayson (1988). The difference between the original MPTT fluid and the one proposed by Baer andFinlayson is that the relaxation time and the viscosity of the lattermodel are allowed to be independent functions of the generalized shearrate [Eq. (2.5)]. In that sense, this fluid can be thought of as a WhiteMetzner generalization of the MPTT fluid (MPTT/WM fluid).

Table I lists the fluids which result by selective choice of the parameters of the MPTTIWM fluid. A wide variety of responses in standardrheological flows can be obtained, depending on the values of the parameters (Bush et af., 1985; Larson, 1988).

c. Governing equations and dimensionless parameters

For steady-state flow of a MPTT/WM fluid with a purely viscouscomponent present, the equations of motion can be written as

pv-Vv=-VP+V'Tp + 11s'i/2v, (2.10)

where Tp is the viscoelastic extra stress of the MPTT/WM fluid given byEq. (2.3). These equations are to besolved together with the continuityequation V·v = 0 subject to the periodic boundary conditions

awr=O: u=- =0,ar (2.11a)

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TABLE I. Limiting cases of the generic constitutive model given by Eq. (2.3)(MPTTIWM fluid).

Parameters

Constitutive equation f ~ A TIp TI, Comments

Newtonian 0 0 0 TIp 0 TI,may be finitePower law 0 0 0 Tlp(Y) 0Upper convected Maxwell(UCM) 0 0 A TIp 0Oldroyd-B 0 0 A TIp 1/,White-Metzner (WM) 0 0 A(Y) 1/p(Y) 0 1/, may be finitePhan-Thien and Tanner (PIT)" E ~ A TIp 1/, TIs may be zeroModified Phan-Thien and TannerlWhite-Metzner (MPTTIWM) e ~ A('y) 1/p(Y) 0 TI,may be finite

'With the PTT fluid. the parameter ~ in the right-hand side ofEq. (2.3) has always beentaken in this work as zero.

r=rw: u=w=O,

v(z+L)=v(z); P(z+L)=P(z) _K2L;

(2.11b)

Tp(z+L)=Tp(Z), (2.11c)

where U and ware the components of the velocity vector v in the randz directions, respectively, and - ~ is the average pressure gradient.Also note that all the higher derivatives of the variables are axiallyperiodic with the same period L.

Alternatively, instead of solving the continuity equation, v can beexpressed for axisymmetric flow in terms of the streamfunction t/!.

1 at/!u=-;az'

1 at/!w--

- r ar'

(2.12a)

(2.12b)

Due to the use of the streamfunction in the numerical solution of theproblem. the appropriate characteristic quantities for the nondimensionalization of the variables are expressed in terms of the flow rate Q: R isused as characteristic length, Q/R2 as characteristic velocity, andiipQIR 3 as characteristic stress, where Tip is a characteristic viscosity. Ifa power-law model is used for the viscosity

07:13:19

7Jp= m [Y J(n- 1)I2, (2.13 )

then, the characteristic viscosity is taken as T7p=m(QIR3)n - l . Iftheviscosity is constant, T7p=1Jp- Using the above normalization, the upperlimit for the dimensionless streamfunction t/J is Q*, where Q*=1I21T.

When a purely viscous stress component is present, viscous (solvent)effects are weighted by the quantity {3=1Js1T7p- On the other hand, theimportance of elastic effects is measured either by a dimensionless shearrate, the Weissenberg number, defined as

We=)."QIR3, (2.14 )

or by the ratio of the relaxation time to a characteristic time for theflow, the Deborah number. The most straightforward definition of theDeborah number involves as a characteristic flow time the quantityL1TR2IQ. This corresponds to a Deborah number defined asDe=WeNI1T. This formulation is appropriate because it is proportional to A. and Q, yet it is not fully satisfactory primarily because theamplitude a is not a factor. An alternative form has been proposedrecently by James et al. (1990), which overcomes this deficiency. In thisform, the characteristic time for the flow is the inverse of a characteristic rate of extension E. The value for Eis estimated from the differencebetween the maximum and minimum average velocities, divided by halfthe wavelength. This corresponds to

2Q [1 I]E= L1T R2(l - a)2 - R 2(l + a)2 .

Then, the new Deborah number denoted by De" is given as

8N a--"'2- 2'De*=We --;- (l - a )

(2.15 )

(2.16 )

In the above form, De" has the expected behavior, namely De" is proportional to a for small a and also De" -0 as a -0. Notice also that inboth the We and the De" numbers, the relaxation time)." can be afunction of the generalized shear rate y.

The quantity of primary importance in the present investigation isthe flow resistance. For a fluid having a power-law viscosity and in theabsence ofa viscous component ('I7s = 0), the flow resistance is denotedby fRen and it is defined as (Deiber and Schowalter, 1981)

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(2.17)mL(!

fRe = 2nrrnt1PR3n

+ In-----

where I:..P is the pressure drop per unit cell, I:..P=tlL. The Reynoldsnumber Re, is defined as

Ren=2nrrn- 2pfi - nR3n - 4/ m. (2.18)

The above definition for the flow resistance partially compensates forthe influence of the shear thinning, in the sense that it takes into consideration the nonlinear pressure drop-flow rate relationship corresponding to the flow of a power-law fluid in a straight tube. Therefore,for straight Poiseuille flow, fRen is constant with respect to the Reynolds number (it depends only on the power-law index), However, forvalues of the undulation different than zero, the compensation of shearthinning effects is not complete (see Sec. III).

Finally, notice that due to the periodicity assumption [Eq. (2.11c)],the governing equations need to be solved in a unit computational cell,as shown by the shaded area in Fig. 1.

D. Implementation of the numerical method

The principal numerical method used in this work is thepseudospectral/finite-difference method implemented in a stretched cylindrical coordinate system fitting the PCT geometry (PCFD method).It is a mixed collocation technique where finite differences are used inorder to approximate the radial dependence of the variables and a truncated Fourier series in order to approximate the periodic (axial) one. Inorder to accurately resolve the steep boundary layers which are presentat high Weissenberg numbers, occasionally, the nodal points were distributed nonuniformly in the radial direction using an arctan transformation (Pilitsis and Beris, 1989; Pilitsis, 1990),

For comparison purposes, two other numerical techniques were alsoused. The first is a standard Galerkin finite element technique (GFE)using bilinear basis functions for the velocities and a penalty approximation for the pressure (Pilitsis, 1990). The second method is a fullpseudospectral technique (Fourier Chebyshev Collocation, FCC)which is very similar to the PCFD method except that it employs atruncated series of Chebyshev polynomials in lieu of finite differences forthe approximation of the radial dependence of the variables (Pilitsis,1990). For more information about the numerical methods formulation,

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TABLE II. Row resistance fRe for Newtonian flow. Parameters are: We = 0.0, a = 0,3,N=0.16.

Lahbabi and FCC a GFEb

Re Chang (1986) Nx = 16, Nc = 33 N,= 40, Nz=4O

0.0 26.4 26.4484 26.41930.012 26.4 26.4484 26.4193

12.0 27.1 27.1791 27.091122.6 28.5 28.5536 28.443351.0 31.7 31.7484 31.698473.0 33.4 33.4488 33.4039

132.0 36.7 36.5264 36.5392207.4 38.9 38.9607 38.9330264.0 39.7 40.2446 40.1544397.2 40.6 42.3479 42.1112783.0 41.2 45.5828 45.0734

"Nx, Nc are the number of Fourier. Chebyshev modes in the axial, stretched radialdirections, respectively.

bN" Nz are the number of elements in the r, z directions, respectively.

the interested reader is advised to consult our previous work (Pilitsisand Beris, 1989; 1991a; 1991b).

III. INELASTIC INERTIAL FLOW

The purpose of the calculations reported in this section is twofold.First, it is intended to validate the numerical algorithms used in thiswork by comparing the predictions among independent techniques. Second, it is intended to evaluate the importance of viscoelastic effects inthe experimental data of Deiber and Showalter (1981) by comparingtheir experimental data for the flow of the polyacrylamide solutionthrough the PCT against the results of a numerical simulation using aninelastic power-law viscosity model.

A. Inertial Newtonian flow

In order to understand the effect of inertia on the flow, it is instructive to first examine the inertial flow of a Newtonian viscous fluidthrough a PCT. Two independent numerical techniques have been usedfor the calculations in order to provide a reliable evaluation of the flowresistance and also in order to validate the numerical methods used inthis work. As Table II indicates, the results between the two techniques

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SINUOUS TUBE FLOWS

10 [ ""I "J

'l-- 10

Co-WUco

-WU·rl

"tl+e+~

+0*. i-+~<r':

102 1 , '" ,I , , , " ,I " ,I1 2 3 4

10 10 10 10Re

FIG. 2. Comparison of the friction factor f between the numerical predictions using theFCC method (see Table II), and the experimental data of Deiber and Schowalter (1979),as a function of the Reynolds number Re for flow of a glycerin solution through the PCT.The symbols correspond to: (0) numerical predictions, ( + ) experimental data. Parameters are: a = 0.3, N = 0.16, We = O.

(FCC and GFE) are in excellent agreement. They also agree very wellwith the best available results in the literature (Lahbabi and Chang,1986), except at very high Re numbers where the results of Lahbabi andChang differ from this work by as much as 9%. Since the results of theFCC method agree very well with those of the independently developedGFE method, this discrepancy should be attributed to the coarseness ofthe spectral expansion used by Lahbabi and Chang (see also Pilitsis andBeris, 1989; 199Ib).

Figure 2 shows a comparison of the friction factor f between thenumerical predictions listed in Table II and the experimental data ofDeiber and Schowalter (1979) obtained with a number of glycerinwater solutions (corresponding to different concentrations), for different values of the Reynolds number. As shown, the agreement is excellent until the point where the flow becomes turbulent and the numericalsolution is not realized physically. This point is identified with capital Tin Fig. 2 and corresponds to a Reynolds number Re - 580. In Fig. 3 the

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50 1 """ '"'''' "'" "I '"'' ""'"o

illa:"-

19000000

10 10 10o 1

10 10Re

20 1 '" ,,.' "'II"! "lI11,1 , .. I "'!I"I "llW'

FIG. 3. Flow resistance IRe as a function of the Reynolds number using the FCC methodfor the conditions in Table II.

numerical predictions for the flow resistance fRe. tabulated in Table II,are depicted as a function of Re. It is instructive to relate the behaviorshown in Fig. 3 to the computed flow fields. This is done in Fig. 4 wherestreamlines for different Re numbers are shown. For small Re, thestreamlines follow closely the geometrical shape [Fig. 4(a)). For theparticular set of geometrical parameters, recirculation starts atRe:::::45.6 upstream from the large cross section of the tube [Fig. 4(b».As Re increases, the vortex grows with the center of it moving downstream [Fig. 4(c)]. Finally, at high enough Re numbers, the flow becomes potential-like and the fluid flows through an idealized streamtube formed by the two narrow cross sections of the tube. The rest of thearea is occupied by the by now large vortex, the center of which hasmoved substantially downstream [Fig. 4(d)].

The above study shows that the increase in fRe with increasing Renumber observed in Fig. 3 is due to inertial nonlinear effects not necessarily linked to the recirculatory flow pattern. This can be seen fromFig. 4(a) where the flow resistance corresponding to the shown flowfield (no recirculation present) is already about 13% higher than theone corresponding to the Stokes limit. It should be emphasized here that

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SINUOUS TUBE FLOWS

Ae-32.6

I~ JJ:7e?qRe-45.6

F~ 7?t:nAe-73

c:f7f~S7tnAe-397.2

gff517EzszJ

FIG. 4. Streamlines for Newtonian flow through the PCT, obtained using the FCCmethod tNx = 32. Nc = 33). for the geometrical parameters in Table II for variousvalues of Reynolds number: (a) Re = 32.6, streamfunction values are (from the centerline to the wall): 0.02,0.06.0.1,0.14,0.156,0.159. (b) Re = 45.6, streamfunction valuesare: 0.02. 0.06, 0.1, 0.14, 0.157, 0.159156. (e) Re = 73.0, streamfunction values are: 0.02,0.06,0.1,0.14,0.156, 0.1592, 0.16. (d) Re = 397.2, streamfunction values are: 0.02, 0.06,0.1,0.14,0.1592,0.162,0.167. Flow is from left to right.

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after the point indicated with T in both Figs. 2 and 3, the numericalpredictions are not in agreement with the available experimental data;as mentioned above, after this point the flow becomes turbulent and thecomputed steady-state solution is not realized physically. Mathematically, it has been demonstrated by Lahbabi and Chang (1986) that thesteady-state solution loses its linear stability to nonaxisymmetric (azimuthal) disturbances at Re:::: 200, and, therefore, it becomes irrelevantbeyond this point. However, the axial periodicity assumption appears toremain valid for all Re numbers in Fig. 3. This conclusion follows fromnumerical calculations performed imposing periodicity every two unitcells which have not revealed any bifurcations on the one-cell periodicsolution family, which would have caused a change in the sign of thedeterminant of the Jacobian matrix (Brown et al., 1980).

B. Inertial flow of a power-law fluid

Deiber and Schowalter (1981) conducted experiments for four aqueous solutions of polyacrylamide corresponding to 0.02%, 0.05%, 0.1%,and 0.53% polyacrylamide concentrations by weight. The experimentaldata for the flow resistance fRen as a function of the ReI! number exhibited similar behavior for all these solutions. Therefore, comparisonbetween numerical predictions and experiments will be done only forthe 0.05% by weight solution. The viscosity of this solution was foundto follow a power-law behavior for shear rates between I and 103 s - 1.

The power-law parameters are given by Deiber and Schowalter (1981)as n = 0.54 and m = I PsI! - 1 [see also Eq. (2.13)].

The numerical predictions are obtained using the PCFD method,which is the method of choice when elasticity is incorporated into theconstitutive model (see Sec. IV). Moreover, the predictions for thepower-law fluid using the PCFD method are subsequently confirmed bythe calculations of the GFE method. Finally, these results are comparedagainst both the theoretical and the experimental findings of Deiber andSchowalter (1981).

Numerical calculations with nonlinear power-law-type inelastic constitutive equations are known to be difficult due to the shear-thinningform of the viscosity function which results in the formation of steepboundary layers near solid boundaries (Karagiannis et al., 1988). Thedifficulty in the calculations increases as the power-law index n decreases, which results in slower convergence in Newton's method (Crochet et al., 1984). In order to obtain reliable results with the minimumcomputational effort, an investigation was carried out for the purposes

07:13:19

TABLE III. Flow resistance IRe n for flow of a power-law fluid using the PCFD method.Parameters are: a = 0.3, N = 0.1592, n = 0.54, m = I P sn - I.

Reynolds numberRen

0.01.528

12.48421.58136.91250.43062.90585.934

Mesh UNINx= 8, Np=61

9.10389.12229.8422

10.369410.969911.369711.666212.0753

Mesh UN3Nx = 16, Np= 100

9.10529.12409.8508

10.388511.008311.398811.687612.1067

of evaluating the performance of several numerical discretizations. Fora particular point in the parameter space (Re, = 50.43), several discretizations were tested involving anywhere from 8 to 32 Fourier modes(Nx) and 61 to 141 finite difference points (Np). Furthermore, both auniform mesh and a graded mesh towards the solid boundaries (resulting from either a bilinear or an arctan-type transformation) were used.Although the agreement among all the predictions was within 0.4%(Pilitsis, 1990), a uniform discretization appeared to give more accurateresults and, as a consequence, was selected for the "production" runs ofthis work.

Table III lists the predictions of the PCFD method using two different uniform discretizations (UNI and UN3) for all the Reynoldsnumbers Re; examined in this work. As seen, the maximum deviationbetween the two predictions is always less than 0.4%. Table IV lists thepredictions of the GFE method using three different discretizations forthe same Reynolds numbers of Table III. The agreement between thenumerical predictions of these two totally independent methods is excellent. Note that the results of the finite element method tend to matchthe PCFD predictions upon refinement of the discretization.

Figure 5 compares the numerical predictions of this work against thetheoretical as well as the experimental findings of Deiber and Schowalter (1981). The numerical results of Deiber and Schowalter are shownwith the solid line. As seen, these predictions are not very accurate evenin the limit of creeping flow. Furthermore, they underestimate thepurely inelastic contributions to the flow resistance. Our calculationsclearly indicate that the shear-thinning behavior, combined with inertia,accounts for part of the increase observed in the dimensionless flow

07:13:19

TABLE IV. Flow resistance IRe. for flow of a power-law fluid using the GFE method.Parameters are: a = 0.3, N = 0.1592, n = 0.54, m = 1 P s" ~ I.

Reynolds number Re, Nr=Nz=20 N r=Nz=40 N r=Nz=6O

0.0 9.0902 9.1015 9.10371.528 9.1060 9.1184 9.1211

12.484 9.8008 9.8306 9.838121.581 10.3395 10.3739 10.380736.912 10.9476 10.9924 11.000050.430 11.3252 11.3792 11.388662.905 11.6037 11.6647 11.675485.934 12.0043 12.0768 12.0894

resistance. This shows the danger of reaching erroneous conclusions inreference to the role of viscoelasticity, if the importance of viscoelasticity is solely judged upon changes in the flow resistance defined as in Eq.(2.17).

On the other hand, although the trends are correct, the above calculations underpredict by as much as 40% the experimentally measuredvalues for the flow resistance fRe of the polyacrylamide solution. Thus,it has been definitely shown that other effects are important for thepolyacrylamide flow within a peT and they have prompted the modeling using a viscoelastic constitutive equation which could account forelastic effects in addition to the shear-thinning ones. Such a study isundertaken in the next section.

IV. VISCOELASTIC FLOW

A. White-Metzner (WM) fluid

The White-Metzner (WM) constitutive equation is a modification ofthe Maxwell model where the relaxation time and the viscosity dependon the rate of deformation tensor, more specifically on its second invariant (White and Metzner, 1963). The relaxation time and the viscosity functions are usually obtained empirically by fitting viscometricdata. The viscosity, for example, can be fitted using any inelastic modelsuch as the power-law or the Carreau model (Bird et ai., 1987). Therelaxation time may be calculated from the relationship

).,(y) =N1(y)/2yT 12(y), (4.1 )

07:13:19

SINUOUS TUBE FLOWS

,'" '"

5.00. 1

'"",'"

",,,,'" '" 1Il

1:>6 ",'" 11M"'''''''

'"tt:,,'"

<IllIl<ll

FlG. S. Flow resistance fRen for a power-law fluid. Parameters are: a = 0.3, N = 0.1592.The symbols correspond to: (\l) experimental data of Deiber and Schowalter ( 1981) forflow of a Separan AP·30 aqueous solution through the PCT, ( + ) PCFD predictions(Nx= 16, Np= 100, UN3), (0) GFEpredictions (N, = oo,Nz = (0). The solid (-)line represents the numerical predictions of Deiber and Schowalter.

where N\> T12 are the first normal stress difference and shear stress,respectively, under simple shear flow. This relationship is derived whensimple steady-shear flow kinematics are inserted into the WhiteMetzner constitutive equation [Eq. (2.3)1 with the comments of TableI].

The WM model, despite its flexibility, has not enjoyed widespreaduse among researchers in the field of numerical simulation of nonNewtonian flows. It has been reported that numerical calculations withthe WM model become increasingly difficult as the flowelasticity grows(Dhahir and Walters, 1989) and that the calculations cannot be validated with mesh refinement (Baird et af., 1988).

An explanation for the potential numerical difficulties involved inusing the WM model is its nonevo1utionarity (Joseph et al., 1985; Joseph and Saut, 1986). It has been shown by Dupret and Marchal(1986a) and by Verdier and Joseph (1989) that the WM model maylose its evolution even in simple shear flow, depending on the shear rate

07:13:19

622 PILITSIS, SOUVALIOTIS, AND BEAlS

dependence of the viscosity function; a shear-thickening fluid neverloses evolution. Most polymeric fluids, however, are shear thinning and,therefore, the evolutionary character of the equations is expected todisappear for high enough shear rates. In the present work, we used theWM model as the simplest constitutive model capable of accuratelydescribing the viscometric properties of the fluid used in the experiments of Deiber and Showalter. The power-law coefficients of the viscosity function to be used in the numerical solution have been given inSec. III B. The power-law portion of the curve is truncated at a shearrate about 0.5 s - 1. This value was estimated from the viscometric dataof Leal et al. (1971) for aqueous polyacrylamide (Separan AP-30)solutions. For shear rates smaller than 0.5 s - I, the viscosity achieves aconstant value. Numerically, the following relationship is being used:

77p=m[0.25 + y) (n - 1)12, (4.2)

which permits a smooth transition from the power law to the flat regionof the curve [see Fig. 6(a»). The first normal stress difference N 1 canalso follow a power-law expression of the form

N1=m'yn'. (4.3)

For the 0.05% by weight polyacrylamide solution, the power-law coefficients for N 1 are given by Deiber and Schowalter (1981) as m' = 0.26P sn' - 1 and n' = 1.14 within the range 50 <r< 450 s - I. The corresponding first normal stress coefficient 1JI 1 used in this work is expressedas

1JI 1= m' [0.25 + y)(n'-2)12. (4.4 )

Notice that, according to Eq. (4.4), 1JI 1 starts to flatten for shear ratesless than 0.5 s - 1 [see also Fig. 6(b»). This value is the same as the oneused for the viscosity function in Eq. (4.2). Substituting Eqs. (4.2) and(4.3) into Eq. (4.1), the relaxation time takes the form

A= [m'/(2m») (0.25 + y)(n' - n - 1)12. (4.5)

Figure 6(c) shows the variation of the relaxation time with the shearrate.

As with the inelastic power-law fluid (see Sec. III B), an investigation of the performance of different discretizations was first undertakenfor a particular value of Reynolds number Re.; The maximum deviationamong all the predictions was found to be within 0.3% (Pilitsis, 1990).The prediction using the uniform mesh UN3 (Nx = 16, Np = 1(0) was

07:13:19

SINUOUS TUBE FLOWS

.,10-'

10 -z ! !!! II " ",I, 11.1, ,I!, ,,1 II! ,I

10 -s 10 -2 10 -, 1 10 10' 10'"I (sec -')

N E 10-'u

;;--.u(lJ

U1 10 -1C~

.; 10 -]

(b)10 -4 i, ",I,!! ! ",I ! , ••1 , ! ",I , ,I

10 -J 10 '10 -, 1 10 10 2 10't (S8C-

rr'0 -1 !

. (c) ,10 -a ,,,I '!II "Ii 11",1, II'!' ! .1

10 -3 10 -a 10 -, 1 10 10 1 10'-y (sec -1)

FIG. 6. Viscometric functions of the aqueous polyacrylamide solution (Separan AP·30,0.05% by weight) used in the experiments of Deiber and Schowalter (1981). (a) Viscosity '7 vs shear rate [Eq. (4.2)], (b) first normal stress coefficient IJI, vs shear rate [Eq.(4.4»), (c) relaxation time A. vs shear rate [Eq. (4.5)].

07:13:19

624 PILITSIS, SOUVALIOTIS, AND BEAlS

TABLE V. Flow resistance fRen using the PCFD method (Nx = 16, Np = 100, uniform).

Parameters are: a = 0.3, N = 0.1592, n = 0.54, m = 1 P s" - I, n' = 1.14, m' = 0.26 Psn' - 1.

Reynolds number Power-law White-MetznerRen fluid fluid

0.0 9.1052 9.1052

1.528 9.1240 9.063412.484 9.8508 9.845421.581 10.3885 10.411836.912 11.0083 11.100750.430 11.3988 11.526562.906 11.6876 11.828985.934 12.1067 12.2653

different only by 0.04% from the one obtained with the fine mesh UN6(Nx = 32, Np = 141). Furthermore, it was found that with the arctantype transformation, many more finite difference points were necessaryin order to reach the accuracy obtained with the uniform discretizations(Pilitsis, 1990).

Table V compares the PCFD predictions for fRen between the powerlaw and the WM fluid. The tabulated data are also plotted in Fig. 7. Asshown, the presence of elasticity in the mathematical modeling throughthe WM fluid does not improve substantially the nonelastic predictions(power-law fluid) with respect to the experimental data. The maximumincrease in the fRen due to elastic effects was only about 1.4%, asopposed to the experimentally measured viscoelastic effect of about55%. As a consequence, the numerical predictions using the WhiteMetzner model still underpredict significantly the experimental datathroughout most of the parameter space.

In Figs. 8 and 9 the streamlines at two different Re, numbers arecompared, obtained using the power-law and the WM models. It can beseen that elasticity causes a small shrinkage of the vortex, as was observed in the Oldroyd-B fluid calculations for large amplitudes of undulation (Pilitsis, 1990; Pilitsis and Beris, 199Ia).

From a numerical standpoint, it should be mentioned that axial oscillations were observed in most of the variables when a coarse discretization was used. However, it was found that the oscillations disappearupon increasing the number of Fourier modes, Nx. This point is bestdemonstrated by examining the improvement observed with increasingNx in the vorticity profiles; the vorticity was the variable which exhib-

07:13:19

10010Ren

5.0 ' , I , I I! II I I I'l' , I , ,!, II

FIG. 7. Comparison between numerical predictions for the flow resistance fRen as afunction of the Reynolds number Re; obtained using the PCFD method, and the experimental data of Deiber and Schowalter (1981). The symbols correspond to: (V) experimental data, ( + ) power-law fluid, mesh UN3, (0) White-Metzner fluid, mesh UN3.Parameters are: a = 0.3, N = 0.1592. The rheological parameters are shown in Table V.

ited the worst behavior, particularly close to the centerline of the tube.Indeed, Fig. 10shows that a dramatic improvement in the quality of thesolution is obtained by increasing the number of Fourier modes. Notethat similar numerical behavior was observed in the inertial flow simulation of an upper convected Maxwell fluid, under conditions of changeof type (Pilitsis and Beris, 199Ia).

B. Phan-Thien-Tanner (PIT) fluid

As a next step in order to match the numerical predictions and theexperimental data of Deiber and Schowalter (1981), a more realisticconstitutive equation has been used in the numerical modeling. For thispurpose, we have turned our attention to the PIT constitutive equation,which exhibits shear-thinning material behavior as a result of its math-

07:13:19

Pawer law fluid

FC5752i7t::JWhite-Metzner fluid

@C?¥??tnFIG. 8. Comparison of streamlines between an inelastic (power law) and a viscoelastic(White-Metzner) fluid, obtained using the PCFD method. Parameters are: a = 0.3,N = 0.1592, Re; = 36.912.The rheological parameters are shown in Table V. (a) Streamlines for a power-law fluid (mesh UN3). Streamfunction values are (from centerline towall): 0.02, 0.06, 0.1, 0.14, 0.159, 0.159157,0.159164, (b) streamlines for a White-Metzner fluid (mesh UN5, Nx = 32, Np = 1(0). Streamfunction values are (from centerline to wall): 0.02, 0.06, 0.1, 0.14, 0.159, 0.159 157. How is from left to right.

ematical structure rather than because of the fitting of parameter functions, as was the case with the White-Metzner model.

The Phan-Thien-Tanner (PIT) model is a nonlinear constitutiveequation for polymer melts and concentrated solutions derived from anetwork picture of the polymeric fluid (Phan-Thien and Tanner, 1977;Phan-Thien, 1978). It has been shown that the PTT model can providenot only an adequate prediction of viscometric properties but it can alsogive a good description of both transient and steady-state extensionalproperties (Bird et al., 1987). The model features two adjustable parameters E and (; in addition to the constant relaxation time A. and theconstant zero shear rate viscosity Tfp [see Eq. (2.3) and Table I]. Theparameter E controls the elongational behavior of the fluid. The parameter {; influences the shear behavior provided that E is small It can beobtained either from the ratio of the first and second normal stress

07:13:19

SINUOUS TUBE FLOWS

?ower law fluid

FC=~~C:~~-~??n

~hite-~etzner fluid

....-=-/~ ,;-:,-=---"'--.

FIG. 9. Same as Fig. 8 but for Re., = 62.905. (a) Streamlines for a power-law fluid.Strearnfunction values are (from centerline to wall): 0.02, 0.06, 0.1, 0.14,0.159<;1/1<;0.16with a linear variation of 0.000 25, (b) streamlines for a White-Metzner fluid. Streamfunction values are (from centerline to wall): 0.02, 0.06, 0.1, 0.14, 0.159<;1/1<;0.15975with a linear variation of 0.000 25. Flow is from left to right.

difference or from the horizontal shift between the dynamic and steadyshear viscosity curves (Phan-Thien and Tanner, 1977; Phan-Thien,1978).

In simple shear flow, the PTT model predicts that the shear stressapproaches zero at high rates of shear. This unrealistic behavior can beavoided by allowing the presence ofa purely viscous (Newtonian) component to the stress (Bush et al., 1985). This way, the total shear stressgrows monotonically with the shear rate. In the present investigation,the absence of experimental data for the extensional viscosity of the fluidused by Deiber and Schowalter ( 1981) necessitates the fit of the parameters of the PTT model from the available viscometric data. In particular, E, ,1, and 'T/p are computed from the best fit of the shear viscosityand the first normal stress difference. As solvent viscosity 'T/s' the viscosity of water ('T/s = 0.01 P) has been used.

The parameter ~ is set equal to zero for the work described in thissection (see Sec. IV C for the influence of nonzero ~ values within a

07:13:19

I:3 005t/>-....,·M

U·M....,C-o> 0

0 1.57 3.14 4.71 6.28z

:3 I>- ooor!....,·M

U'M....,C-o> 0

-0,,1 Ill)

0 1.57 3.14 4.71 6.28z

FIG. 10. Vorticity close to the centerline (at (;= r/rw = 0.05), along the axial distance z,for a White-Metzner fluid, obtained using the PCFD method. Parameters are: a = 0.3,N = 0.1592, Re; = 50.43. The rheological parameters are shown in Table V. (a) MeshUN3, (b) the broken (- -) line is obtained with mesh UN5 (Nx = 32. Np = 100) and thesolid (-) one with mesh UN7 iNx = 64. Np = 100). Row is from left to right.

similar setting). The choice of the upper convected derivative (~= 0)has been motivated by a number of reasons. Larson (1988) discussesthe disadvantages of using the Gordon-Schowalter convected derivative[see Eq. (2.7)], which are inherited by the PTT model when ~ is nonzero. One of the problems is the appearance of aphysical oscillations in

07:13:19

the viscosity 11 and the first normal stress coefficient '1'1 in start up ofsteady shear flow at large values of shear rate. Joseph, Renardy, andSaut (1985) and Dupret and Marchal (1986) found that JohnsonSegalman (Johnson and Segalman, 1977) type of models may becomeill posed (lose evolution) unless ~ = 0 or ~ = 2 (which corresponds tothe upper and lower convected derivatives, respectively). Similarly,Beris and Edwards (1990a) showed that only when an upper or lowerconvected derivative is used can the nonnegativeness of entropy production for Maxwell-type models be demonstrated for all flows.

Figure II shows a comparison between the predictions of the WhiteMetzner and the PIT models for the viscosity and the first normalstress difference. Recall that the predictions of the WM model are essentially the viscometric data for the fluid used by Deiber and Schowalter (1981). It is worth mentioning here that the parameter E of thePIT model resulting from the fit shown in Fig. II is larger (E = 0.8)than the value suggested by Phan-Thien and Tanner (1977) and PhanThien (1978). In their original derivation of the model, the authorssuggested that E is on the order 0.01, which results in reasonable extensional predictions. However, in the literature, there have been occasional uses of somewhat larger values (order 0.1-0.2), particularly inthe modeling of rheological properties of melts (Khan and Larson,1987; White and Baird, 1988).

Figure 12 (solid line) shows the predicted extensional viscosity obtained for E = 0.8 and A= 0.4 s. This extensional behavior is seen morein some polymer melts (Larson, 1988) rather than in dilute polymersolutions, for which large Trouton ratios are expected (Fuller et al.,1987;Schunk and Scriven, 1990). However, this large value of E in thesimulation is not expected to greatly affect the results, at least for moderate values of amplitude of undulation for which the extensional character of the flow is very small. This conclusion is corroborated by several pieces of evidence: First, in our previous work (Pilitsis and Beris,1989), it was shown that the normal to the flow direction stresses diminish with increasing elasticity, suggesting negligible extensional effects. Second, it was shown in the previous section that the flow resistance of the WM fluid (which predicts an increasing elongationalviscosity) was almost identical to the predictions of the power-law fluid.Furthermore, as it can be seen in the next section, the predictions of theMPIT/WM fluid with E = 0.01 and t; = a (which predicts finite extensional viscosity for all extensional rates) do not differ substantially fromthose of the WM fluid.

07:13:19

PILITSIS. SOUVALIOTIS. AND BERIS

~",....--..

~,OJ(fJ

~10 -1 ~ '" ".;

10 -21 I ""III! I I "!I"! I """.I

(0)I !,",,,I ! tl"I!!1 I III

10 -3 10 -2 10 -1 1 10 10 2 10 J

-; (sec-')

,....--.. 10 /

<, IC I»,

~ 10-' II

~ IZ 10 -2 I

I::1 "u,.,!u u•••••••

(b)" ""I' '1".1 , 'I

10 -3 10 -2 10 -1 1 10 10 2 10 J

)I (sec")

FIG. n. Comparison between the viscometric predictions of the WM and the PTImodels, for the fluid used in the experiments of Deiber and Schowalter (1981). Therheological parameters of the WM fluid are shown in Table V and the ones of the PIT inTable VI. The solid (-) line is the WM prediction and the broken (- -) line the PITprediction. (a) Viscosity 1/ vs shear rate, (b) first normal stress difference N J vs shearrate.

07:13:19

IIHI111111 IlllllIIl 1"'''''110 3 L I ''''''''111111 -

--------

~10 2QJ(f)

{::"" 10

110 -3 10 -2 10 -1 1 10 10 2 10 3

i: (sec- 1)

FIG. 12. Elongational viscosity "'lEas a function of the elongational rate E, predicted bythe PIT fluid. The solid (-) line is obtained with E = 0.8, Ii = 0.4 s and the broken (- -)line with E = 0.015, Ii = 5 s ("'Ip = 1.36 P, "'Is = 0.01 P, S= 0).

Thus, we opted using the rather high E value in order to approximately fit the observed viscometric behavior using moderate values of A(-0.4 s) and thus requiring reasonable resources to compute at therange of We numbers (We=AQ/R3) corresponding to the experimentally used flow rates. Should an E value of -0.015 had been used, Awould have been -5 s (Pilitsis, 1990) which results in a more realistic(for polymer solution) extensional viscosity (see Fig. 12, dotted line)but requires calculations up to We-250.

Finally, in order to gain additional confidence about the validity ofthe numerical calculations using E = 0.8 and A = 0.4, we also relied onrecently developed molecular interpretation and thermodynamic admissiblity criteria for the faithfulness of numerical calculations using aparticular class of differential constitutive equations (Beris and Edwards, 1990b). These criteria have their origin in the Hamiltonian formulation of viscoelastic equations through Poisson brackets. For thePIT fluid, a structural interpretation of the stress is possible only whenthe conformation tensor tr [defined as u= Tp + ('flplA) I] is positive definite. In addition, thermodynamically acceptable solutions are considered the ones for which

07:13:19

TABLE VI. Flow resistance fRen for flow of the PTf fluid using the PCFD method(Nx = 32, Np = 100, uniform). Parameters are: a = 0.3, N = 0.1592, A = 0.4 s, E = 0.8,"Ip = 1.36 P, "I, = 0.01 P.

We Ren fRen

1.3018 1.528 11.20265.4715 12.484 10.37837.9540 21.581 10.5588

11.4793 36.912 10.696412.6262 42.430 10.736914.2085 50.430 10.794816.5260 62906 10.864520.4537 85.934 10.9504

I + E.1. tr( 1"p)/1Jp>O, (4.6)

which, in principle, places no a priori restrictions on the value of E. Boththese criteria were monitored during the course of the computations andit was found that they were never violated.

Computed values for the flow resistance fRen are listed in Table VI.The values are also plotted in Fig. 13 along with the predictions of theWhite-Metzner fluid. Figure 13 shows that the PIT model does notimprove the numerical predictions of the White-Metzner fluid. In fact,it does a little worse. Notice also that for small Re, numbers the PITmodel overpredicts the experimental data. This is due to the fact that inthe limit of small relaxations, the PIT model does not reduce to ashear-thinning inelastic fluid (like the WM model which reduces to apower-law fluid). Instead, it reduces to the Maxwell (or Oldroyd-B)model which does not shear thin, and as a consequence, it exhibitshigher flow resistance.

For the calculations with the PIT model, no systematic mesh refinement validation studies were deemed necessary. Previous experiencewith other constitutive models prompted the consideration of the employed mesh as adequate enough for a reliable estimation of the flowresistance. Additional support for this conjecture came from the factthat the criterion given by Eq, (4.6) was never violated. In addition, thetensor a was always positive definite.

The positive definiteness of the tensor T defined as

T=Tp + (1J/.1.)I- pvv, (4.7)

07:13:19

SINUOUS TUBE FLOWS

20.0 I I iii j i iii Iii i j Iii Iii i i iii

""15.0 ~

" ~ x)( )C"

" 0d>10.0 ~..."

6 I;M/i.""" M "... "

5J10 1000.1 1

FIG. 13. Comparison between numerical predictions for the flow resistance fRe. as afunction of the Reynolds number Re, obtained using the PCFD method, and the experimental data of Deiber and Schowalter (1981). The symbols correspond to: (\7) experimental data, (X) P1T fluid, mesh (Nx = 32, Np = 100, uniform), (0) WM fluid, meshUN3. Parameters are: a = 0.3, N = 0.1592. The rheological parameters of the WM fluidare shown in Table V and the ones of the PIT fluid in Table VI.

was also examined throughout the course of the computations since itdetermines the change of type to hyperbolic-like behavior of the set ofequations (Dupret and Marchal, 1986b). Notice that because of thepresence of a purely viscous Newtonian component in the stress, we cannot really have a true change of type of the equations. However, there isstill a value in monitoring the hyperbolic-like character of the equations,because even if there is a viscous component present, there is no guarantee that numerical difficulties associated with the change of type couldbe avoided (compare with the advection-diffusion equation at high Peelet numbers). Figure 14 shows the evolution of the hyperbolic-likeregion with increasing flow strength (flow rate). The region starts fromthe centerline of the tube at the two narrowest cross sections, locationswhere high stresses and velocities are developed. Progressively, thehyperbolic-like region occupies all the flow domain surrounding the

07:13:19

(.J re=:= E==:=a(blC=:=~

FIG. 14. Formation of the hyperbolic-like region for flow of the PTT fluid through thePCT, as obtained with the PCFD method (Nx = 32, Np = 200, uniform). Parametersare: a = 0.3, N = 0.1592. The rheological parameters of the PIT fluid are shown in TableVI. The hyperbolic-like region is denoted by H and the elliptic-like by E. Flow is from leftto right. (a) Ren = 1.53, (b) Ren=21.58.

centerline, except a strip around the wall. Further increase in the flowrate does not change qualitatively this picture, as shown in Fig. 14(b).

Finally, Fig. 15 shows streamlines at two different Reynolds numbersRen. Recall from Fig. 8 that the flow fields of both the power law andthe White-Metzner fluid exhibit recirculation at Re, = 36.912. A similar flow pattern is not obtained with the PIT fluid. Also, notice that forthe higher Re, number the vortex has a similar appearance to the oneobserved using either the power law or the White-Metzner fluid [seeFigs. 15(b) and 9]. Recirculation with the PIT fluid appears at someRe, number between 36.912 and 42.43. This is in qualitative agreementwith the value of about 39 reported by Deiber and Schowalter (1981)on the basis of visual observations.

c. Modified Phan-Thien and Tanner/White-Metzner (MPTTIWM) fluid

Neither the White-Metzner (WM) nor the Phan-Thien and Tanner(PIT) (with ~ = 0) fluid models, examined in the previous sections,helped in resolving the discrepancy between the numerical predictionsand the experimental data. The common characteristic of both thesemodels is the prediction of a zero second normal stress difference and asimple dynamic behavior characterized, among other things, by theabsence of a stress overshoot in the start-up of shear flows. However,

07:13:19

~-~I')~~~

Ibl~gpzgFIG. 15. Streamlines for flow of the PIT fluid calculated using the PCFD method(Nx=32, Np=200, uniform). Parameters are: a=0.3, N=0.1592. The rheologicalparameters of the PTI fluid are shown in Table VI. How is from left to right. (a)Re, = 36.912, streamfunction values are (from the centerline to wall): 0.02, 0.06, 0.1,0.14, 0.15436625, 0.159, (b) Re, = 62.905, streamfunction values are (from the centerline to wall): 0.02, 0.06, 0.1, 0.14, 0.159<$<0.15949 with a linear variation of0.000 16333.

both of these features are in contrast to what it is typically observedwith polyacrylamide solutions (Tanner, 1985;Bird, 1987). Therefore, itappeared that a more flexible model was necessary in order to successfully predict the flow behavior within a peT. The dynamic shear properties could be of importance if we note that the flow within a PCTcorresponds, from a Lagrangian (i.e, comoving with the material particles) point of view to similar unsteady kinematics (mostly shear) asthose characterizing the step-up and step-down simple shear flow experiments.

The modified Phan-Thien and Tanner (MPIT) model is an empirical extension of the PIT model proposed by Phan-Thien (1984) inorder to provide for additional modeling flexibility [see Eq. (2.3) andTable I]. However, one should be cautious in using this model with ~*Oin light of the remarks made in the previous section about the use ofmixed codeformational derivatives. Alternative models, such as theGiesekus constitutive equation (Larson, 1988), might also be used andare the subject of a current investigation.

Since the MPTT model also shows unrealistic behavior in simpleshear flow (such as shear thickening) when both the parameters e and~ are finite, in the present investigation, we examined independently the

07:13:19

10 3 "TOT"milI "I1i1-.--rmTTlj--,--,rnm"'TTTT1~-""'"' """" '""'" I '"'"'' I ""'li

.......... 10 2Q)u:o0-

110 -3 10 -2 10 -, 1 10 10 2 10 3

i: (sec")

FIG. 16. Comparison of the elongational viscosity 1)Eas a function of the elongational rates, predicted by the WM and MPTT/WM fluids. The rheological parameters of both fluidsare shown in Table VII. The broken (- -) line is the WM prediction and the solid (-)line is the prediction of the MPTT/WM fluid.

effectof e and t; in the numerical predictions. In addition, the version ofthe MPTI fluid used is the one suggested recently by Baer and Finlayson (1988), where the relaxation time is allowed to be a function of thesecond invariant of the rate of deformation tensor. In particular, therelaxation time is taken to be the same one used with the WhiteMetzner constitutive equation [see Eq. (4.5)]. Similarly, the viscosityfunction is the same one used with the WM model, given by Eq. (4.2)(MPTT/WM fluid).

When the parameter (; is equal to zero and ~ is finite (but small), theversion of the MPTI/WM fluid resulting can be thought of as a generalized White-Metzner constitutive equation, having, however, onefewer problem: the e1ongational viscosity remains bounded at all finiteextensional rates. This behavior is shown in Fig. 16. However, for thefluid used in the experiments of Deiber and Schowalter, there are noexperimental data for the extensional viscosity available in order toindependently fit the parameter ~. Instead, ~ has been assigned a smallvalue (~= 0.01) so that the predictions of the MPTT/WM model in

07:13:19

'"15.0 ~

'"'t,'"c

Il0:::~ '" Illl'+-

'"'" Ii

'" ..10.0 f- '" ..

~ /lMl:J.t:.t:J.t:.A/::/i

'" """ '".... '"

l1005.00 1 1 10

FIG. 17. Comparison between numerical predictions for the ftow resistance fRen as afunction of the Reynolds number Re; obtained using the PCFD method, and the experimental data of Deiber and Schowalter (1981). The symbols correspond to: (\7) experimental data, ( X ) MPlT/WM fluid, mesh UN3, (0) WM fluid, mesh UN3. Parametersare: a = 0.3, N = 0.1592. The rheological parameters of both ftuids are shown in TableVII.

simple shear flow coincide with those of the WM fluid (and, as a consequence, with the viscometric data for the fluid under consideration).

Figure 17 shows the predictions for the flow resistance and Table VIItabulates the data. The results are almost identical with the ones obtained using the WM model. The general tendency is that fRen decreases slightly, the maximum decrease being about 0.7%. This behavior reinforces the statement in the previous section that the extensionalproperties of the fluid do not influence macroscopic quantities such asthe flow resistance fRen in the present flow problem.

Next, the influence of the parameter t; in the numerical predictions isexamined (E is set to zero). This version of the MPTT IWM fluid predicts a nonzero second normal stress difference and allows for dynamicbehavior characterized by nonaffinedeformation and stress overshoot instart-up of simple shear flow (Phan-Thien, 1984; Phan-Thien et a/.,1987). Recently, Zheng et al. (1990) tested the MPTT fluid in the

07:13:19

638 PILITSIS, SOUVALlOTlS, AND BERIS

TABLE VII. Flow resistance fRen using the PCFD method (Nx = 16, Np = 100, uniform). Parameters are: a = 0.3. N = 0.1592, n = 0.54, m = 1 P s" -1, n' = 1.14,m' =0.26 P sn'-I, <=0.01, ~= O.

Reynolds number White-Metzner MPTTIWMae, fluid fluid

1.528 9.0634 9.061412.484 9.8454 9.831621.581 10.4118 10.387936.912 11.1007 11.059150.430 11.5265 11.470562.906 11.8289 11.758885.934 12.2653 12.1765

problem of flow through a periodically constricted tube. It was foundthat the MPTT fluid with a small positive ; value gives rise to anincrease in the flow resistance, in reference to that predicted by anOldroyd-B fluid.

In order to investigate the influence of; in the numerical predictions,a knowledge of a realistic range of values for; is required. Ideally, ;should be estimated by fitting transient data for start-up of shear flow.For the particular fluid used in the experiments of Deiber and Schowalter, no such data are available. However, data for the transient viscosity of a similar solution (1.5% by weight polyacrylamide in mixtureof water and glycerin) show that there is indeed a stress overshoot instart-up of shear flow (Bird et al., 1987).

Alternatively, ; can be estimated from measurements of the ratio ofthe second to the first normal stress difference in shear flow, since for aMPTT/WM fluid

N2/N] = -;/2. (4.8)

Tabulated values for N2/N1 for different concentrations of polyacrylamide in water can be found in Tanner (1985). However, the concentration of the fluid under consideration (0.05% by weight) is outsidethe range of concentrations listed. Extrapolation suggests that; shouldbe approximately 0.01. On the other hand, the difficulty of accuratelymeasuring the second normal stress difference N2 it is well known,especially for low values. As shown in Bird et al. (1987), considerablescatter can exist in the data of N 2/N1 against the shear rate. Therefore,at this point, it is difficult to specify with confidence anything more thanan order of magnitude estimate for the parameter ;, ; = 0(0.01).

07:13:19

TABLE VIII. Flow resistance fRen for MPTT/WM with the PCFD method (Nx = 16,Np= 100, uniform), Parameters: a=0.3, N=0.1592, n=O.54, m= I Psn- I ,n' = 1.14, m' = 0.26 P sn' - I, E = 0.0,

Reynolds number, Re, ; = 0,0") ;=0.01 ;=0.1 ;=0.05

1.528 9,0094 9,0096 9.0112 9,016412.484 9,8403 9,8349 9.8398 9,848421.581 10,4099 10,4154 10.4404 10.470336.912 11.0995 11.1130 1l.l573 11,177850.430 11.5281 11.5538 11.607462.906 11.8299 11.8633 11.9447

"The results for the WM fluid are different from those reported in Table VII because aslightly different viscosity function has been used here,

The values of the fRen calculated with the interpolated value of~ = 0.01 are, basically, indistinguishable from the ones of the WM fluid.The maximum increase observed, in the range of Re, where calculationswere performed, was of the order of 0.3%. This slight, yet consistent,increase in the flow resistance prompted us to perform a parametricstudy on the effect of ~ on the fRen.

Results obtained with values of ~ up to 0.5 have been tabulated inTable VIII, Also, the effect of ~ on the flow resistance has been graphically demonstrated in Fig. 18 for two values of Re., It is clear that, byincreasing the value of ~, or, equivalently, having a fluid with highersecond to the first normal stress difference ratio and/or higher stressovershoot in start up of shear flow, the fRe" definitely increases but anincrease larger than 1% was never observed for the geometry of interest,for the range of flow rates under investigation. Furthermore, the flowresistance levels off at higher values of ~, as it can be deducted from Fig.18 (a and b). Thus, any dramatic increase of the fRe", even at unrealistically high values of t: is highly unlikely. The results have beensummarized in Fig. 19.

It is also important to mention here that computational difficultieswere encountered in obtaining and validating with mesh refinementsome of the numerical predictions for the MPTI/WM fluid with finite~. In particular, the numerical predictions for ~ = 0.1 were validated upto Re, = 50.430. The fine mesh UN6 (Nx = 32, Np = 141) has beenused for this purpose. For ~ = 0.5, a solution with the mesh UN3(Nx = 16, Np = 100) was obtained up to a maximum Re, = 36.912,while with the fine mesh UN6 the calculations were carried up toRe, = 21.581. After those points, it was very difficult for the Newton's

07:13:19

1046 l- •c

Ree=21.581

10400.0 02 04 06 0.8 1.0

11.16 t •

af 11.14o:'+-

::::L Ree=36.912

, , ! I, ! " ! , ! I ,

0.0 0.1 0.2 03 04 0.5 0.6

FIG. 18. Influence of ~ on the flow resistance fRe. for an MPTTIWM fluid. Parametersare: a = 0.3, N = 0.1592. The rheological parameters of the fluid are shown in Table VIII.(a) Re, = 21.581. mesh UN3. (b) Re, = 36.912. mesh UN3.

iteration to converge with zero order continuation with respect to ~

(starting from the WM solution, i.e., ~ = 0) or with respect to Re,(starting from a converged solution at lower Ren) . In any case, the

07:13:19

"15.0 ~

0:::~ " till!>'+-

".. til..<l>

10.0 f- "ti" ..

6 6 §Mo,l::.t::.

5J0.1 1 10 100

FIG. 19. Comparison between numerical predictions for the flow resistance IRen as afunction of the Reynolds number Re, obtained using the PCFD method, and the experimental data of Deiber and Schowalter (1981). The symbols correspond to: (\l) experimental data, (+) MPTIIWM fluid, ~=O.I, mesh UN3, (0) WM fluid, mesh UN3.Parameters are: a = 0.3, N = 0.1592. The rheological parameters of both fluids are shownin Table VIII.

results obtained with the mesh UN6 were never more than 0.03% different from the values listed in Table VIII.

It is probable that the problems of convergence are due to theGordon-Schowalter derivative which is used when ~ is finite. Thesedifficulties in the calculations reinforce the cautionary statement madebefore (see Sec. IV B), about the possibility of loss of evolution and,therefore, numerical instabilities at certain flow parameters when; isnonzero. The work with the Giesekus constitutive model, mentionedabove, is expected to give some answers on this subject.

Finally, the possibility that the presence of inertia and/or shear thinning dampens the effect of; in the calculated flow resistance was examined. The inertialess flow of a non-shear-thinning MPTI fluid wasinvestigated. The geometrical parameters were: a = 0.1 and N = 0.5.The ratio of the solvent to the polymeric viscosity {3 was set equal to5.666. The value of; was set equal to 0.1. Zheng et al. (1990) have

07:13:19

PILITSIS, SOUVALIOTIS, AND BERIS

Q)18.50::

18.0 ~

'" ""x

~ *n 9.4 )(.f;!. x o ~ +b.X " X " X "

17.5 Ell I I I ! ! I ! ! ! I !

0.0 1.0 20 3.0 4.0 5.0 6.0

FIG. 20. Flow resistance IRe as a function of the Weissenberg number We obtained withthe MPTT model, using the PCFD method and the DEM method. The symbols correspond to: (\7) PCFD predictions, ~ = 0.1, f3 = 5.66ti (Nx = 16, Np = 100), (X) PCFDpredictions, ~ = 0.1, (3= 5.666 (Nx = 32, Np = 141), ( + ) PCFD predictions (Pilitsisand Beris, 1989), ~ = 0.0, (3= 5.666, (0) DEM predictions (Zheng et al., 1990), ~ = 0.1,(3= 5.666. Geometrical parameters are: a = 0.1, N = 0.5.

presented results for the same combination of rheological and geometrical parameters. Pilitsis and Beris (1989) performed calculations withthe PCFD method for an Oldroyd-B fluid, with the same retardationtime, in the same geometry. As it was shown in that work, gradedtowards the wall meshes can enhance the accuracy of calculations withnon-shear-thinning viscoelastic fluids. This suggestion was followedhere. Two meshes have been used in our calculations (Nx = 16,Np = 100 and Nx = 32, Np = 141).

Results from these calculations have been plotted in Fig. 20, alongwith results for the Oldroyd-B fluid (Pilitsis and Beris, 1989). On thesame plot, the findings of Zheng et al. (1990) are presented. The calculations with the MPTT fluid were interrupted at relatively low Weissenberg number since the focus on this work was the detection of anyeffects of ; in the flow resistance. No numerical difficulties have beenencountered with either one of the two meshes. A very small increase in

07:13:19

the flow resistance, above the Oldroyd-B value, was found, in agreementwith the findings for the shear-thinning fluid. Our results agree withthose of Zheng et al. (1990) up to We -::::; 1. However, we were unable toverify the substantial increase reported by them for higher Weissenbergnumbers. Since our results are converged with mesh refinement andgiven the very fine size of our discretization, we attribute the discrepancy to an inadequate mesh size used in their calculations.

v. CONCLUSIONS

In this work, the flow of a shear-thinning viscoelastic fluid (SeparanAP-30) through a peT has been numerically simulated. A variety ofconstitutive models has been used and the results have been comparedagainst the experimental data of Deiber and Schowalter (1981). Inaddition, some results for a non-shear-thinning MPTT fluid have beenpresented.

For the shear-thinning fluid, it was found that the presence of theelasticity in the mathematical modeling caused an increase in the flowresistance over the value calculated for the viscous fluid. It was alsofound that the use of more complex constitutive equations which canrepresent the dynamic (transient) properties of the fluid can shift thecalculated data towards the correct direction. However, in all cases, thenumerical results seriously underpredicted the experimentally measuredflow resistance. [A more substantial increase in the flow resistance withthe MPTT /WM fluid previously reported in a Society of Rheologymeeting (Santa Fe, 1990) and in Figs. 6.24, 6.25 and Table 6.10 inPilitsis (1990) was found to be caused by a programming error.] Thesefindings were verified by calculations with a non-shear-thinning fluidunder inertialess flow conditions. Note that, in the non-shear-thinningcase, there is also an experimentally measured increase in the flow resistance (Huzarewicz et al., 1991) which has not been predicted theoretically as yet (Pilitsis and Beris, 1989; Burdette et al., 1989).

A major result of this work is the consistency seen among the predictions obtained by a variety of models. All of them represent constitutive equations which exactly fit the same steady shear data. Thisindicates a robustness among the various models. However, the predictions do not match the experimental data in this simple, yet not viscometric flow. The remaining issue is what is the particular rheologicalfeature which is not captured by anyone of the used models and whichis responsible for this discrepancy. At this point, the possibility that theexperimentally observed phenomena correspond to an instability, result-

07:13:19

ing in a three dimensional or time periodic or non-axially periodic flow,cannot be excluded as an explanation. These effects, which have notbeen examined in the present work, would be of interest for furtherinvestigation, both theoretical and experimental.

ACKNOWLEDGMENTS

The financial support from the National Science Foundation throughan Engineering Research Center grant to the Center for CompositeMaterials at the University of Delaware is gratefully acknowledged. Wewould also like to acknowledge the computational support provided bythe Academic Computing Services at the University of Delaware and bythe Cornell National Supercomputer Facility (CNSF).

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