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Heat transfer asymptote in laminar flow of non-linear viscoelastic fluidsin straight non-circular tubes

Dennis A. Siginer a,b,⇑, Mario F. Letelier c

a Department of Mechanical Engineering, Petroleum Institute, Abu Dhabi, United Arab Emiratesb Department of Mathematics, Petroleum Institute, Abu Dhabi, United Arab Emiratesc Departamento de Ingenieria Mecanica y Centro de Investigación en Creatividad y Educación Superior, Universidad de Santiago de Chile, Santiago, Chile

a r t i c l e i n f o

Article history:Available online 16 September 2010

Keywords:Heat transferEnhancementNon-linearViscoelasticAsymptoteNon-circularTubes

a b s t r a c t

The fully developed thermal field in constant pressure gradient driven laminar flow of a classof non-linear viscoelastic fluids with instantaneous elasticity in straight pipes of arbitrarycontour @D with constant wall flux is investigated. The non-linear fluids considered are con-stitutively represented by a class of single mode, non-affine constitutive equations. The driv-ing forces can be large. Asymptotic series in terms of the Weissenberg number Wi areemployed to expand the field variables. A continuous one-to-one mapping is used to obtainarbitrary tube contours from a base tube contour @D0. The analytical method presented iscapable of predicting the velocity and temperature fields in tubes with arbitrary cross-sec-tion. Heat transfer enhancement due to shear-thinning is identified together with theenhancement due to the inherent elasticity of the fluid. The latter is to a very large extentthe result of secondary flows in the cross-section but there is a component due to first normalstress differences as well. Increasingly large enhancements are computed with increasingelasticity of the fluid as compared to its Newtonian counterpart. Order of magnitude largerenhancements are possible even with slightly viscoelastic fluids. The coupling between iner-tial and viscoelastic non-linearities is crucial to enhancement. Isotherms for the temperaturefield are discussed for non-circular contours such as the ellipse and the equilateral triangletogether with the behavior of the average Nusselt number Nu, a function of the Reynolds Re,the Prandtl Pr and the Weissenberg Wi numbers. Analytical evidence for the existence of aheat transfer asymptote in laminar flow of viscoelastic fluids in non-circular contours isgiven for the first time. Nu becomes asymptotically independent from elasticity with increas-ing Wi, Nu = f(Pe, Wi) ? Nu = f(Pe). This asymptote is the counterpart in laminar flows in non-circular tubes of the heat transfer asymptote in turbulent flows of viscoelastic fluids in roundpipes. A different asymptote corresponds to different cross-sectional shapes in straighttubes. The change of type of the vorticity equation governs the trends in the behavior ofNu with increasing Wi and Pe. The implications on the heat transfer enhancement is dis-cussed in particular for slight deviations from Newtonian behavior where a rapid rise inenhancement seems to occur as opposed to the behavior for larger values of the Weissenbergnumber where the rate of increase is much slower. The asymptotic independence of Nu fromelasticity with increasing Wi is related to the extent of the supercritical region controlled bythe interaction of the viscoelastic Mach number M and the Elasticity number E, which mit-igates and ultimately cancels the effect of the increasingly strong secondary flows withincreasing Wi to level off the enhancement. The physics of the interaction of the effects ofthe Elasticity E, viscoelastic Mach M, Reynolds Re and Weissenberg Wi numbers on generat-ing the heat transfer enhancement is discussed.

� 2010 Elsevier Ltd. All rights reserved.

0020-7225/$ - see front matter � 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijengsci.2010.07.010

⇑ Corresponding author at: Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates. Tel.: +971 26075946; fax: +971 26075080.E-mail addresses: dsiginer@pi.ac.ae (D.A. Siginer), mletelier@usach.cl (M.F. Letelier).

International Journal of Engineering Science 48 (2010) 1544–1562

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1. Introduction

Experimental findings concerning heat transfer characteristics of aqueous polymer solutions flowing in straight tubespoint at considerable enhancement as compared to its Newtonian counterpart driven by the same conditions and in thesame geometry. Specifically, it is reported that heat transfer results for viscoelastic aqueous polymer solutions are consid-erably higher in flows fully developed both hydrodynamically and thermally, as much as by an order of magnitude depend-ing primarily on the constitutive elasticity of the fluid and to some extent on the boundary conditions, than those found forwater in laminar flow in rectangular ducts, Hartnett and Kostic [1,2]. Heat transfer phenomena in laminar flow of non-linearfluids has not been the subject of many investigations with the exception of round pipes and the case of inelastic shear-thin-ning fluids in tubes of rectangular cross-section in spite of the widespread use of some specific contours in industry such asflattened elliptical tubes. This statement is true for all cross-sectional shapes for both steady and unsteady phenomenaincluding quasi-periodic flows. Heat transfer with viscoelastic fluids has been declared to be a new challenge in heat transferresearch in the early nineties, Hartnett [3], but progress has been limited since that time. The physics of the phenomenon hasnot been entirely clarified.

Highly enhanced heat transfer to aqueous solutions of polyacrylamide and polyethylene of the order of 40–45% as com-pared to the case of pure water in flattened copper tubes was observed by Oliver [4] and later by Oliver and co-workers asearly as 1969. Recent numerical investigations in rectangular cross-sections of Gao and Hartnett [5,6], Naccache and SouzaMendes [7], Payvar [8] and Syrjala [9] establish the connection between the enhanced heat transfer observed and the sec-ondary flows induced by viscoelastic effects. The former researchers as well as Naccache and Souza Mendes predict for in-stance viscoelastic Nusselt numbers as high as three times their Newtonian counterparts. Gao and Hartnett [5] reportnumerical results in rectangular contours which provide evidence that the stronger the secondary flow (as representedby the dimensionless second normal stress coefficient W2) the higher the value of the heat transfer (as represented bythe Nusselt number Nu) regardless the combination of thermal boundary conditions on the four walls. Constant heat fluxis imposed everywhere on the heated walls in their numerical experiments with the remaining walls being adiabatic. Thecombination of boundary conditions plays some role in the enhancement reported with the largest enhancement occurringwhen two opposing walls are heated. Despite these efforts heat transfer characteristics of viscoelastic fluids in steady lam-inar flow in rectangular tubes remains very much an open question.

Although these studies establish a basis for the mechanism of viscoelastic heat transfer the results are at best indicative ofthe prevailing trends for slight deviations from Newtonian behavior. Gao and Hartnett [5] and Naccache and Souza Mendes[7], as well as Payvar [8] and Syrjala [9] use the Reiner-Rivlin and Criminale–Ericksen–Filbey (CEF) constitutive equations,respectively, to characterize the fluid behavior. Both of these constitutive structures are open to an array of criticism andare hardly representative of any viscoelastic fluid in complex flows except in the case of slight deviations from Newtonianbehavior for both equations, and when the flow is viscometric in which case the CEF equation is exact.

The shear rate dependent viscosity of purely viscous fluids (negligible relaxation time) is also responsible for enhanced andreduced heat transfer in the case of shear-thinning and shear–thickening fluids, respectively, as shown by Gingrich et al. [10]and others. But there is evidence in the literature that in the case of viscoelastic fluids the effect of the shear rate dependent vis-cosity on heat transfer enhancement is at least two orders of magnitude smaller when compared to the influence of the second-ary flow, Naccache and Souza Mendes [7]. Thus the latter remains the dominant mechanism for enhanced heat transfer.

In this paper we investigate the heat transfer behavior of a class of non-affine and non-linear viscoelastic fluids in straighttubes of non-circular contour in pressure gradient driven laminar flow and under constant wall flux conditions. The velocityfield is obtained via hierarchical regular perturbation problems derived through the expansion of the field variables intoasymptotic series in terms of the Weissenberg number Wi, Siginer and Letelier [11]. The solution for the thermal field is pre-sented in this paper. The thermal field at the zeroth order represents the temperature distribution in a Newtonian fluid in atube of arbitrary contour with constant wall flux. The thermal field at the first order is null, a consequence of a null first ordervelocity field. At the second order the thermal field is altered separately by shear-thinning and elasticity with additive super-posed contributions. The longitudinal velocity field is further changed at the third order with a corresponding change in thethermal field due to elasticity, but more importantly at this order a secondary flow triggered by unbalanced second normalstresses brings large changes to the temperature distribution and heat dissipation.

The physics governing the asymptotic behavior of Nu and its independence from elasticity with increasing Wi is clarifiedand discussed. The physics is based on the change of type of the vorticity equation. The analysis leading to the criteria gov-erning the change of type for the class of non-affine quasilinear fluids with interpolated Maxwell type convected derivativesand instantaneous elasticity of interest in this paper is similar to the analysis for upper convected Maxwell (UCM) fluids.When the viscoelastic Mach number, the ratio of a characteristic velocity of the fluid to that of the speed of shear waves intorest, exceeds one M2 = ReWi > 1 a supercritical region develops near the centerline of the tube with a compatible elliptic re-gion near the wall. Inertial terms can only be neglected for small values of the Mach number as the analysis in this paper willalso make it clear. When the Mach number M� 1 the spread of the hyperbolic region is small if the dimensionless Elasticitynumber E = Wi/Re is large and vice versa. A complete discussion of the cause and effect relationship of the interaction of theviscoelastic Mach number, the Elasticity number and the Peclet number on the heat transfer enhancement in relation to theextent of the hyperbolic region is given.

For each given non-circular cross-section the enhancement curves Nu = f(Pe, Wi) level off at a higher critical Wic withincreasing Pe or inertia Re. Beyond the boundary Wic = fc(Pe), that is for Wi > Wic at fixed Pe enhancement asymptotically

D.A. Siginer, M.F. Letelier / International Journal of Engineering Science 48 (2010) 1544–1562 1545

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approaches a constant value and is no longer a function of the elasticity, Nu = f(Pe). This asymptotic behavior is quite similarto Virk’s asymptote for drag reduction in turbulent flow in round tubes under isothermal conditions, Virk et al. [12]. Thecounterpart of Virk’s asymptote for heat transfer with viscoelastic fluids in turbulent flow in round straight tubes was shownexperimentally by Hartnett [3]. We show for the first time in this paper the existence of a similar asymptote Nu = f(Re) ormore generally Nu = f(Pe) in laminar flow of non-linear viscoelastic fluids in non-circular straight tubes. A different asymp-tote exists for each cross-sectional shape separating the region where the enhancement is a function of both Wi and Re (atconstant Pr) at low Wi from the region where it is a function of Re alone at higher Wi.

2. Mathematical analysis

2.1. Field equations

The structure of the class of non-linear and non-affine viscoelastic fluids, which includes the Johnson–Segalman andPhan–Thien–Tanner models, investigated in this work has been described in Siginer and Letelier [11]. The family of singlemode constitutive structures which relates the deformation measure D to the viscoelastic contributed stress tensor s isframed in terms of the Gordon–Schowalter convected derivative ð��Þ through a relaxation time k, a molecular contributed vis-cosity gm and a function f related to the elongational properties of the fluid. The balance equations and the constitutive equa-tion for the fluid read as,

quiuj;i ¼ rji;i ui;i ¼ 0; rij ¼ �Pdij þ sij ð1Þ

2gmDij ¼ f ðe�; trsÞsij þ k sij�; sij

�¼ umsij;m �uimsmj � simujm; uij ¼ uj;i � nDij ð2Þ

gm ¼ bgm0l ¼ gm0lo; b ¼ gm0

go; g0 ¼ gN0 þ gm0; lo ¼ 1þ k2nð2� nÞj2 ð3Þ

where u, r and P represent the velocity, the total stress and the pressure, respectively. The zero shear rate viscosity ofthe Newtonian solvent and the molecular contributed zero shear viscosity are represented by gN0 and gm0, respectively.The Newtonian contribution to the extra-stress field s is assumed to be negligible that is the Newtonian contribution tototal viscosity is negligible gN � 0. Setting b equal to one yields a fluid whose total viscosity is contributed by the longchain molecules alone. n is the slippage factor representing the slippage of the polymer strands with respect to the sur-rounding continuum. Field equations (1) and the constitutive formulation (2) are normalized by introducing dimension-less variables,

r� ¼ ra; z� ¼ z

a; u� ¼ u

V0; v� ¼ v

V0; w� ¼ w

V0; P� ¼ aP

gm0V0; s�ij ¼

asij

gm0V0; D�ij ¼

aV0

Dij; u�ij ¼a

V0uij

based on the molecular contributed zero shear viscosity gm0 (assuming gN0 � 0), a characteristic velocity Vo and a character-istic length a. q and P are scalar parameters representing the density and the total pressure field. The field equations are pre-sented in detail in Siginer and Letelier [11] except for the thermal field. The constitutive and balance equations are written indimensionless form in a cylindrical frame with the dimensionless velocity components u* = (u*, v*, w*) in the (r, h, z)directions,

2l0D�ij ¼ f �ðe�; trs�ijÞs�ij þWis�ij�;

2½1þ 2nð2� nÞWi2trD�2ij D�ij ¼ ð1þ e�Wi trs�ijÞs�ij þWi s�ij

�

u�u�;r� þv�r�

u�;h þw�u�;z� �v�2r�¼ F�r � P�;r�

u�v�;r� þv�r�

v�;h þw�v�;z� þu�v�

r�¼ F�h �

1r�

P�;h

u�w�;r� þv�r�

w�;h þw�w�;z� ¼ F�z � P�;z�

where the linearized form of f �ðe; trs�ijÞ is used and Wi denotes the Weissenberg number Wi ¼ kV0a . F�r ; F�h and F�z represent the

viscoelastic force components in the momentum balance,

F�r ¼ ðr� � s�Þr� ¼1r�ðr�s�r�r� Þ;r� þ

1r�

s�r�h;h �s�hh

r�þ s�z�r� ;z�

F�h ¼ ðr� � s�Þh ¼ s�r�h;r� þ1r�

s�hh;h þ2r�

s�r�h þ s�hz� ;z�

F�z ¼ ðr� � s�Þz� ¼1r�ðr�s�r�z� Þ;r� þ

1r�

s�hz� ;h þ s�z�z� ;z�

The heat diffusion equation defines the thermal field,

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DTDt¼ ar2T; uT ;r þ

vr

T ;h þwT ;z ¼ a T ;rr þ1r

T ;r þ1r2 T ;hh þ T ;zz

� �

where a is the thermal diffusivity. The dimensionless temperature field T* is introduced

T�ðr; h; zÞ ¼ Tðr; h; zÞ � TwðzÞTmðzÞ � TwðzÞ

Tw and Tm denote the constant wall temperature and the average temperature, respectively,

Tm ¼1

AVm

ZZswT ds; Vm ¼

1A

ZZswds; A ¼

ZZsds

We consider the case of constant heat flux along the pipe wall, that is

q00w ¼ hðTm � TwÞ ¼ cte

The temperature at any given cross-section is constant in time, but there is a longitudinal temperature gradient. The dimen-sionless heat diffusion equation reads as

T�;r�r� þ1r�

T�;r� þ1

r�2T�;hh ¼ Pr u�T�;r� þ

v�r�

T�;h þw�T�m;z�� �

where Pr stands for the Prandtl number, and the dimensionless average temperature gradient a0 has been introduced,

Pr ¼ gqa

; a0 ¼ T�m;z� ¼1

DT@Tm

@z�; DT ¼ TmðzÞ � TwðzÞ; c0 ¼

@Tm

@z�

The star notation is omitted from here on, and unless noted otherwise the variables and the expressions used aredimensionless.

2.2. Solution of the field equations

The balance and constitutive equations are solved by expanding the field variables in power series

I ¼X1m¼0

WimIhmi; I ¼ ðu; v;w; s;u; P;D;r; TÞ

2.2.1. Velocity and temperature fields at order O(1)

uh0i � ruh0i ¼ r � rh0i; r � uh0i ¼ 0; rh0i ¼ �Ph0iIþ Re�1sh0i; sh0i ¼ 2Dh0i

uh0i ¼ ð0;0;wh0iÞ; sh0i ¼ ðer ez þ ez erÞwh0i;r þ ðeh ez þ ez ehÞ1r

wh0i;h

r2wh0i ¼ 12

Ph0i;z ¼ �4p; wh0i ¼ p½1� r2 þ ern sinðnhÞ ¼ pw0; 0 < e < 1; n > 1 ð4Þ

To satisfy the no-slip condition on the contour @D we must have

wh0ij@D ¼ 0; 1� r2 þ ern sin nh ¼ 0 ð5Þ

Eq. (5)2 is called the shape factor and defines a mapping of the basic circular cross-sectional shape into various non-circularshapes through the parameters e and n. Varying the parameters continuously in the ranges indicated in (4) corresponds to acontinuous deformation of the circular contour. For instance n = 4, e = 0.22 corresponds to a square. An exact sharp right anglecorner is not possible to attain as a radius of curvature however very small is required at the corner for continuity. But from apractical point of view this does not carry any importance as the corner can be made indeed quite close to sharp by varying e.The study of complex cross-sectional shapes is made possible through the superposition of a finite number of solutions of type(5) each one rotated by a certain angle with respect to the others. Thus it is possible to superpose two or more mapped figuressuch as an ellipse (n = 2, ec > e > 0) and a triangle (n = 3, ec > e > 0) either one rotated by a certain angle with respect to the otherto obtain say tear drop shaped contours commonly used in extruders. Note that for n = 2 and n = 3 and different numericalvalues assigned to e one gets ellipses with different aspect ratios, and triangles with varying degrees of curvature of the sidesand in particular varying degrees of sharpness at the corners, respectively. n = 3 and e = 0.385 yield a triangle with straightsides and well rounded corners with very high curvature which look sharp. Shape factor may not yield closed curves forthe tube contour for arbitrarily assigned pair of values for (e, n). For closed curves the value of e cannot exceed an upper limitec = f(n). The explicit expression for ec flows out of the requirement that at a cusp the velocity gradient should be zero,

ec ¼2n

n� 2n

� �ðn�2Þ=2

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In theory both e and n can assume fractional values. In practice admissible closed form shapes are given only by integer val-ues of n and fractional values of e. Assuming that the boundary @D0 of the domain D0 of the flow is a circle, a continuousdeformation of the circle is observed with increasing values of e > 0 for a fixed integer n up to a limiting closed boundarywith n sharp corners or cusps obtained when e is equal to a critical value ec. These ideas are developed in Siginer and Letelier[11]. They were applied recently by Letelier et al. [13] and Siginer and Letelier [14] to the quasi-periodic flow driven by apulsating pressure gradient of an integral viscoelastic fluid of the fading memory type in straight conduits of arbitrary shape.

The thermal field at this order is defined by,

r2Th0i ¼ Pr a0wh0i ¼ Pr a0p½1� r2 þ ern sinðnhÞ

The structure of the RHS allows the solution to be broken down into the sum of two parts:

Th0i ¼ k1ðrÞ þ ek2ðr; hÞ; r2k1ðrÞ ¼ Pr a0pð1� r2Þ; r2k2ðr; hÞ ¼ Pr a0prn sinðnhÞ;

k1ðrÞ ¼Pr16

a0pðr2 � 3Þð1� r2Þ; k2ðr; hÞ ¼ Pr p1

4ðnþ 1Þ a0rnþ2 sinðnhÞ þ C3rn sinðnhÞ� �

The solution is modified by redefining Th0i to make the shape factor appear explicitly in the solution to force the dimension-less temperature at this order to comply with the zero at the wall condition,

Th0i ¼ wh0i½h1ðrÞ þ eh2ðr; hÞ þ Oðe2Þ

The constant C3 is determined by requiring a continuous function, and the representation for the Newtonian temperaturedistribution becomes,

Th0iðr; hÞ ¼ Pr16

a0pw0 r2 � 3þ eðn� 3Þðnþ 1Þ r

n sinðnhÞ þ Oðe2Þ� �

ð6Þ

2.2.2. Temperature field at order O(Wi)A null velocity field is computed at this order, uh1i = vh1i = wh1i = 0, and the corresponding temperature field is also null,

r2Th1i ¼ Pr a0wh1i; Th1i ¼ 0:

2.2.3. Velocity and temperature fields at order O(Wi2)

uh0i � ruh2i þ uh2i � ruh0i þ uh1i � ruh1i ¼ r � rh2i; rh2i ¼ �Ph2iIþ sh2i; r � uh2i ¼ 0;

sh2i þ e� ðtrsh0iÞsh1i þ ðtrsh1iÞsh0i� ��

¼ 2Dh2i þ 4nð2� nÞðtrDh0i2ÞDh0i � sh1i

�

sh1i�¼ uh0i � rsh1i þ uh1i � rsh0i �uh0ish1i � sh0iuh1iT �uh1ish0i � sh1iuh0iT

sh1i�¼ �½er ez þ ez er þ

1rðeh ez þ ez ehÞnðn� 2Þwh0i;r ðwh0i;r Þ

2 þ 1r2 ðw

h0i;h Þ

2� �

At this order the longitudinal component of the linear momentum Eq. (8)1,

uh2iwh0i;r þv h2i

rwh0i;h ¼ �Ph2i;z þ 2r2wh2i þ ðr � sh1i

�Þz þ 4nð2� nÞ r � ðtrDh0i2ÞDh0i

h iz

�ð7Þ

requires the solution for the secondary field. By introducing a streamfunction wh2i a null transversal field is computed,

rRer4wh2i ¼ 0; uh2i ¼ v h2i ¼ 0

The last term in the longitudinal linear momentum balance (7) is generated because lo (refer to Eq. (3)) depends on the shearrate. If it does not that is if lo = 1 (Johnson–Segalman model) the longitudinal field is altered only by terms related to vis-coelastic non-linearity and is defined by,

r2wh2i1 ¼12

nð2� nÞp3½�32r2 þ 8enðnþ 3Þrn sinðnhÞ ð8Þ

If lo – 1 (MPTT model) and is shear rate dependent an additional longitudinal field due to shear-thinning exists. It is definedby

r2wh2i2 ¼12

nð2� nÞp3½�256r2 þ 64enðnþ 2Þrn sinðnhÞ ð9Þ

Thus the contribution to longitudinal velocity at this order is split in two components,

wh2i ¼ wh2i1 þwh2i2

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The details of the computation of these fields are given in [11],

wh2i1 ðr; hÞ ¼ De�2c p3w0 1þ r2 � e

ðn2 þ 2n� 1Þðnþ 1Þ rn sinðnhÞ þ Oðe2Þ

� �

wh2i2 ðr; hÞ ¼ 8De�2c p3w0 1þ r2 � e

ðn2 þ n� 1Þðnþ 1Þ rn sinðnhÞ þ Oðe2Þ

� �

For values of the Deborah number exceeding a critical value De�2c ¼ nð2� nÞ realistic velocity fields do not exist in round

tubes, Letelier and Siginer [15]. The corresponding temperature fields are computed in an analogous manner to the velocitycomputations detailed in [11]. The temperature field T h2i1 which corresponds to wh2i1 is defined by

r2Th2i1 ¼ Pr a0wh2i1 ¼ Prp3a0

De2c

1� r4 þ enðnþ 3Þðnþ 1Þ r2 � ðnþ 2Þðn� 1Þ

ðnþ 1Þ

� �rn sinðnhÞ

�

The solution for Th2i1 reads as,

Th2i1 ðr; hÞ ¼ Prnð2� nÞ

36p3a0½1� r2 þ ern

� sinðnhÞ r4 þ r2 � 8þ e �ð7n2 þ 21n� 4Þ2ðnþ 2Þðnþ 1Þ ð1þ r2Þ þ ð10n2 þ 11n� 17Þ

ðnþ 1Þ2

" #rn sinðnhÞ þ Oðe2Þ

( )ð10Þ

The field which defines Th2i2 corresponding to wh2i2 is the solution of

r2T h2i2 ¼ Pr a0wh2i2 ¼ 8Pr a0p3w0

De2c

1þ r2 � eðn2 þ n� 1Þðnþ 1Þ rn sinðnhÞ þ Oðe2Þ

�

Th2i2 ¼ 8Pr a0p3w0

36De2c

r4 þ r2 � 8þ e �ð7n2 þ 12n� 4Þ2ðnþ 2Þðnþ 1Þ ð1þ r2Þ þ ð10n2 þ 2n� 17Þ

ðnþ 1Þ2

" #rn sinðnhÞ þ Oðe2Þ

( ) ð11Þ

2.2.4. Velocity and temperature fields at order O(Wi3)

uh0i � ruh3i þ uh1i � ruh2i þ uh2i � ruh1i þ uh3i � ruh0i ¼ r � rh3i;r � uh3i ¼ 0; rh3i ¼ �Ph3iIþ sh3i;

fsh3i þ e�½ðtrsh0iÞsh2i þ ðtrsh1iÞsh1i þ ðtrsh2iÞsh0ig ¼

2Dh3i þ 4nð2� nÞ ðtrDh0i2ÞDh1i þ ½trðDh0iDh1i þ Dh1iDh0iÞDh0i

n o� sh2i

�

The defining equation for the longitudinal velocity at this order, the longitudinal component of (11) is:

uh3iwh0i;r þv h3i

rwh0i;h ¼ �Ph0i;z þ 2r2wh3i þ ðr � sh2i

�Þz � eo½r � ðtrsh2iÞsh0iz ð12Þ

It turns out that the longitudinal field is not affected by the constitutive constant eo up to and including the third order in thisanalysis as the contribution of the last term on the RHS of (12) is zero. The secondary field surfaces at this order for the firsttime and is the solution of,

rr4wh3i ¼ 8en2ð2� nÞp4nðnþ 4Þðn� 1Þrnþ1 cosðnhÞ

The solution for the streamfunction wh3i is given by Siginer and Letelier [11],

wh3iðr; hÞ ¼ 14en2ð2� nÞp4½1� r2 þ ern sinðnhÞ2 nðn� 1Þðnþ 4Þ

ðnþ 1Þðnþ 2Þ rn cosðnhÞ

The solution to (12) for wh3i is,

wh3iðr; hÞ ¼ 132

en2ð2� nÞp5w0n2ðnþ 4Þðn� 1Þðnþ 1Þ2ðnþ 2Þ2

½ðnþ 1Þr2 � ðnþ 3Þrn sinðnhÞ þ Oðe2Þ

The thermal field at this order is computed from,

r2Th3i ¼ Pr uh3iTh0i;r þv h3i

rTh0i;h þ a0wh3i

� �

Thus it is the sum of two components, a component stemming from the first normal stress differences Th3i1 and another T h3i2

due to secondary flows,

D.A. Siginer, M.F. Letelier / International Journal of Engineering Science 48 (2010) 1544–1562 1549

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Th3i ¼ Th3i1 þ Th3i2 ; r2Th3i1 ¼ Pr a0wh3i; r2Th3i2 ¼ Pr uh3iTh0i;r þv h3i

rTh0i;h

� �

The solutions read as,

Th3i1 ðr; hÞ ¼ CT Pr2 ðnþ 1Þr4 � 2r2ðnþ 4Þ þ n2 þ 8nþ 19ðnþ 1Þ

� �ð13Þ

Th3i2 ðr; hÞ ¼ CT Pr2 ðnþ 1Þð2nþ 5Þr4 � ð7n2 þ 32nþ 3Þr2 þ 5n3 þ 36n2 þ 87nþ 68ðnþ 1Þ

� �ð14Þ

CTðr; hÞ ¼ en2ð2� nÞp5a0w0n2ðnþ 4Þðn� 1Þ

384ðnþ 1Þ2ðnþ 2Þ2ðnþ 3Þ

" #rn sinðnhÞ

2.3. Computation of the Nusselt number

The Nusselt number Nu is defined as,

Nu ¼ Dhhk¼ Dh

GradðTw � TmÞ

; Dh ¼4AP; Grad ¼ �1

P

ZðrT � nÞdl

where Dh denotes the effective (hydraulic) diameter defined in terms of the area A and the perimeter P of the cross-section. Theaverage temperature gradient, the wall temperature, the average temperature and the outward unit normal vector are repre-sented by ‘‘Grad”, Tw, Tm and n, respectively. The unit normal vector n = cos her + sin heh to the contour G(r, h) is given by

n ¼ rdhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdrÞ2 þ ðrdhÞ2

q er þdrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðdrÞ2 þ ðrdhÞ2q eh

n ¼ G;rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðG;rÞ2 þ G;h

r

�2r er þ

G;h

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðG;rÞ2 þ G;h

r

�2r eh

When the temperature field T(r, h) and the longitudinal velocity W(r, h) are given the average temperature is computed interms of the volume flow rate Q,

Tm ¼RR

Tðr; hÞWðr; hÞr dr dhRRWðr; hÞr dr dh

¼ Q�1ZZ

Tðr; hÞWðr; hÞr dr dh

Some objections may be raised to the use of the hydraulic diameter in the computation of the Nu number. However, thehydraulic diameter has proven to be a very effective tool as it will be also demonstrated in this paper. For example we re-cover the existing values of the average Nu for some non-circular cross-sections in the literature computed by othermethods.

2.3.1. Newtonian Nusselt numberTo compute the Newtonian Nusselt number in the mapped contour only Eqs. (4) and (6) that is the zeroth order O(1)

velocity and temperature fields are needed. Then the Newtonian ‘‘GradN” is expressed as,

GradN ¼ �1P

I@DrTh0i � ðcos uer þ sin uehÞdl

with the temperature gradient and the line integration length dl on the boundary @D given by,

rTh0i ¼ Th0i;r er þ1r

Th0i;h eh; dl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdrÞ2 þ ðrdhÞ2

qThe Newtonian Nusselt number NuN computed at the zeroth order is independent of the Re and Pr numbers as expected,

ðTw � TmÞN ¼ Q�1ZZ

Th0iðr; hÞwh0iðr; hÞr dr dh; Q ¼ pReZ

w0ðr; hÞr dr dh;

NuN ¼ �Dh

P

H@D T h0i;r er þ 1

r Th0i;h eh

�� ðcos uer þ sin uehÞdlRR

Th0iw0r dr dh

ZZw0ðr; hÞr dr dh ð15Þ

2.3.2. Viscoelastic Nusselt numberThe velocity and temperature fields up to and including the third order read

Iðr; hÞ ¼X3

n¼0

WinInðr; hÞ þ OðWi4Þ; I ¼ ðW; TÞ

1550 D.A. Siginer, M.F. Letelier / International Journal of Engineering Science 48 (2010) 1544–1562

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The viscoelastic Nusselt number NuVE is computed through

NuVE ¼Dhh

k¼ Dh

GradVE

ðTw � TmÞVE

The average temperature Tm and wall temperature Tw and the average temperature difference (Tw � Tm)VE are expressed as,

Tm ¼ Q�1ZZ

Th0iw � Th0iðr; hÞ þW2i ðT

h2iw � Th2iðr; hÞÞ

þW3i ðT

h3iw � Th3iðr; hÞÞ

( )Wðr; hÞr dr dhþ OðW4

i Þ

ðTw � TmÞVE ¼ Q�1ZZ

Th0i þW2i Th2i þW3

i Th3in o

Wðr; hÞr dr dhþ OðW4i Þ

where the volume flow rate Q and the wall temperature Tw are given by,

Q ¼ZZ

wh0i þW2i wh2i þW3

i wh3ih i

r dr dh

�þ OðW4

i Þ

Tw ¼ Th0iw þWi2Th2iw þWi3Th3iw þ OðWi4Þ

The GradVE in the case of viscoelastic flow is computed as follows,

GradVE ¼ �1P

I@D

ðTh0i;r þW2i T h2i;r þW3

i Th3i;r Þer

þ 1r ðT

h0i;h þW2

i Th2i;h þW3i Th3i;h Þeh

!� ðcos uer þ sin uehÞdlþ OðWi4Þ

3. Results and discussion

The isotherms of the temperature field for a non-linear viscoelastic fluid are presented in Figs. 1–3 for tubes with ellip-tical, triangular and square cross-sections, respectively. In each Figure the Weissenberg number Wi and the slippage param-eter n is fixed, Wi = 0.3, n = 0.3. For most viscoelastic fluids the Prandtl number assumes values Pr � 50 or higher. The Figuresare drawn for a Peclet number Pe = 104, Pr = 50 and Re = 200. The evolution of the isotherms in a triangular cross-section isshown in Fig. 4 for increasing Wi starting with the Newtonian case Wi = 0 for Pe = 103, Pr = 10 and Re = 100.

As e ? 0 the mapped boundary approaches the circular cross-section and collapses onto the circle for e = 0. Thus,

Th0iNCðr; hÞ ¼ lime!0

Pr16

a0w0 r2 � 3þ eðn� 3Þðnþ 1Þ r

n sinðnhÞ� �

þ Oðe2Þ �

¼ Pr16

a0pð1� r2Þðr2 � 3Þ ð16Þ

101-X

-1

-0.5

0

0.5

1

Y

2.47472.41872.31542.15221.98901.82591.66271.49951.33641.17321.01000.84690.68370.52050.35740.19420.09170.0310-0.0412-0.1093

Fig. 1. Isotherms of the normalized temperature field in an elliptical tube; working fluid is non-affine and non-linearly viscoelastic: Pr = 50, Wi = 0.3, n = 0.3,Re = 200, n = 2, e = 0.4.

D.A. Siginer, M.F. Letelier / International Journal of Engineering Science 48 (2010) 1544–1562 1551

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where the indices N and C stand for Newtonian and circular tube, respectively, represents the temperature distribution in aNewtonian fluid in a circular pipe. Similarly as e ? 0 the limit of the temperature distributions (10) and (11) due to the elas-tic non-linearity and shear-thinning contributed velocities at the second order, respectively, yields the corresponding con-tributions to the temperature distribution in the circular tube when the fluid is non-linear viscoelastic,

Th2i1C ðr; hÞ ¼ lime!0

Th2i1 ¼1

36Pr p3a0

De2c

ð1� r2Þðr4 þ r2 � 8Þ

Th2i2C ðr; hÞ ¼ lime!0

Th2i2 ¼29

Pr p3a0

De2c

ð1� r2Þðr4 þ r2 � 8Þ

-1 0 1X

-1.5

-1

-0.5

0

0.5

1

Y

0.34980.27850.17750.0765-0.0245-0.0757-0.1009-0.1256-0.2266-0.3276-0.4286-0.5296-0.6306-0.7316-0.8326-0.9336-1.0346-1.1356-1.2065-1.2312

Fig. 2. Isotherms of the dimensionless temperature field in a triangular tube; working fluid is non-affine and non-linearly viscoelastic; Pr = 50, Wi = 0.3,n = 0.3, Re = 200, n = 3, e = 0.384.

-1 0 1X

-1

-0.5

0

0.5

1

Y

3.25443.21843.13353.04872.96492.81692.58512.35332.12151.88971.65791.42611.19420.96240.73060.49880.26700.07070.0352-0.0611-0.1966-0.3779

Fig. 3. Isotherms of the normalized temperature field in a pseudo-square tube; working fluid is non-affine and non-linearly viscoelastic; Pr = 50, Wi = 0.3,n = 0.3, Re = 200, n = 4, e = 0.285.

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The major contribution to the deviation of the temperature distribution from that of the Newtonian fluid of the same vis-cosity at the second order of the analysis for a viscoelastic fluid in this class in a circular pipe comes to a large extent fromthe shear-thinning capability of the fluid. The ratio of the latter Th2i2C ðr; hÞ to Th2i1C ðr; hÞ contributed by the velocity increasewh2i1C ðr; hÞ due to elasticity at this order shows that shear-thinning plays a dominant role in determining the deviation fromthe Newtonian temperature distribution in circular pipes, but that non-linear elasticity also plays a significant role,

Th2i1C ðr; hÞTh2i2C ðr; hÞ

¼ 18

ð17Þ

This prediction does not exactly agree with the findings in the literature that the generalized Newtonian fluid model predictsquite well heat transfer in circular tubes, Hartnett [3] and others, unless at least 10% deviation from the correct result isinterpreted as within experimental uncertainty. To elaborate further on the deviation from the Newtonian temperature pro-file the following ratio is computed,

R ¼ Th0iC

Wi2ðTh2i1C þ Th2i2C Þ

�����r¼0

¼ 14

Dec

pWi

� �2

; n ¼ 0:3;0:1 6Wi 6 0:4;Dec

2Wi

� �2

� Oð1Þ; R � O1p2

� �

Thus unless the pressure gradient assumes unrealistically small values p� O(1) in laminar flows of the class of dilute non-linear fluids discussed in this paper the deviation from the Newtonian temperature profile in circular tubes is large, and var-ies quadratically with the normalized pressure gradient. Thus qualitatively heat transfer with viscoelastic fluids which actessentially as inelastic shear-thinning fluids in circular tubes will show substantial deviations from the Newtonian behavior,and the average Nusselt number Nu = f(Pe, Wi) will be larger than its Newtonian counterpart independent of Pe, which agrees

-1 0 1X

-1.5

-1

-0.5

0

0.5

1

Y

1.4441.2881.0610.8690.6680.4360.2410.0770.015

(a)

-1 0 1X

-1.5

-1

-0.5

0

0.5

1

Y

1.4441.2881.0610.8690.6680.4360.2410.0770.015

(b)

-1 0 1X

-1.5

-1

-0.5

0

0.5

1

Y

1.4441.2881.0610.8690.6680.4360.2410.0770.015

(c)

-1 0 1X

-1.5

-1

-0.5

0

0.5

1

Y

1.4441.2881.0610.8690.6680.4360.2410.0770.015

(d)

Fig. 4. Evolution of the isotherms of the normalized temperature field in a triangular tube with increasing Weissenberg number Wi; (a) Wi = 0; (b) Wi = 0.1;(c) Wi = 0.2; (d) Wi = 0.3: Pr = 10, n = 0.3, Re = 100, n = 3, e = 0.384.

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with experimental facts. It is interesting to note that (17) also holds exactly at the center r = 0 of any non-circular cross-sec-tional shape and approximately at any point (r, h) of any cross-section with n axes of symmetry regardless the Reynoldsnumber Re and the Weissenberg number Wi and for any value of the constitutive slippage constant n. This means thatthe shear-thinning initiated average temperature gradient between any point (r, h) of the cross-section and the wall is atmost an order of magnitude larger than the gradient due to the elasticity,

Th2i2 � Tw

Th2i1 � Tw

� Oð10Þ

Thus, there is evidence that heat dissipation as represented by the corresponding average Nusselt number is governed at thisorder to a large extent by shear-thinning in non-circular cross-sections.

To validate the analysis in this paper a comparison of the trends in the values of the Nusselt number for Newtonian fluidscan be derived using (10), (11) and (6) with e = 0, which gives the velocity profile of the Newtonian fluid in the circular cross-section, to show that NuNC > NuNn that is the Newtonian Nusselt number in circular cross-sections NuNC = 4.364 is larger thanthe Nusselt number in all the other cross-sections with n axes of symmetry. In addition NuNn < NuN(n+1), n = 3, 4, 5, . . . that isthe triangle dissipates the least heat. It has the smallest Nusselt number among all cross-sections with n axes of symmetry.The Nusselt number increases as the number of symmetries of the cross-section increase. In that sense the circle has an infi-nite number of symmetries and dissipates the most heat that is has the highest Nusselt number. Ellipse is the exception to thisrule. The smaller the corner angle in the cross-sectional shape the smaller is the Nusselt number. As the corner angle opens upthe Nusselt number also increases to reach a maximum for the circle. This is due to the fact that as the angle becomes sharperflow velocity in the region close to the corner slows down resulting in a less efficient heat transfer process. The values of NuNn

for n = 2, . . . , 7 are computed and presented in Table 1 including the ellipse n = 2 which dissipates the most heat after the cir-cle NuN2 = 4.36. The values of NuNn in Table 1 reproduce those in the literature obtained by other methods.

The enhancement due to shear-thinning is computed by considering the temperature distribution T h2i2 due to shear-thin-ning at order O(Wi2) in the computation of Nun through (11). It turns out that NuNn < Nun < NuC for each and every n. That isthe enhancement due to shear-thinning is the highest in the circular cross-section and it is the lowest in the trianglealthough higher for the triangle than its Newtonian value.

The variation of the Nusselt number NuVE (the subscript VE refers to viscoelastic) with the Weissenberg number Wi for thefully viscoelastic fluid is shown in Figs. 5–7 for Reynolds numbers Re covering a range from very small inertia (Re = 1) to twoorders of magnitude larger (Re = 250). Figs. 5–7 present the case of elliptical, triangular and square cross-sections, respec-tively. Wi is proportional to first normal stress difference N1. Thus the behavior of the total heat transfer enhancement isrepresented in the figures with increasing N1. There is a rapid rise in the value of NuVE in the range 0 < Wi < 0.1, the largerthe Re the steeper the rise. With increasing Wi the rate of enhancement starts leveling off and asymptotically tends to a con-stant value at constant Pe or Re. For almost creeping flow (Re = 1) enhancement is flat for Wi > 0.3 for all three cross-sections.These results are in striking contrast to those reported by Gao and Hartnett [6], but in line with the conclusions of Ultmanand Denn [16] and Delvaux and Crochet [17] and others as will be clarified in the next two sections. The former authors showusing numerical simulations and the Reiner-Rivlin model that in a rectangular cross-section with constant wall temperatureand different type of boundary conditions imposed on the four walls, such as two opposing adiabatic walls and the other twoat constant temperature, and at a constant value of the Peclet number Pe = 25,000 and of the Reynolds number Re = 500, thepredicted enhancement of the Reiner-Rivlin fluid NuVE = f(Pe, Wi) is a concave function of the second normal stress coefficientthat is the rate of enhancement is becoming steeper with increasing second normal stress coefficient with no end in sightwhereas we show that it is a convex function of the second normal stresses that is the rate of enhancement is decreasingwith increasing second normal stresses and tends asymptotically to a constant value that is NuVE = f(Pe, Wi) ? NuVE = f(Pe).Although ultimately experiments are needed to clarify this issue the well known shortcomings of the Reiner-Rivlin model

s ¼ gðIID; IIIDÞDþ a2ðIID; IIIDÞD2; IID ¼ 2D : D; IIID ¼ det D; g ¼ KIID

2

� �ðn�1Þ=2

; a2 ¼ const

in describing the behavior of realistic fluids lie at the foundation of the opposing predicted trends. IID and IIID represent thesecond and third invariants of the rate of deformation tensor D.

Table 1Normalized heat loss per unit length of tubing hf, dimensionless average velocity W, Newtonian Nusselt number NuN, viscoelastic Nusselt number NuVE andcross-section efficiency g = NuVN/hf for elliptical (n = 2), equilateral triangular (n = 3), square (n = 4), pentagonal (n = 5), hexagonal (n = 6) and heptagonal (n = 7)cross-sections.

Re = 200; Wi = 0.3; n = 0.3n e hf W NuN NuVE g (%)

2 0.4 15.8782 472.61 4.36 6.78 42.683 0.38485 17.75 550.64 3.16 10.57 59.554 0.2465 14.122 524.6133 3.62 7.22 51.155 0.17 13.4194 490.3066 3.92 5.71 42.566 0.13 13.1139 487.52 3.99 5.11 38.937 0.1 13.091 486.56 4.08 4.98 38.02

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3.1. Critical phenomena

There is evidence of the convexity of the Nu in the open literature in the case of external flows that is Nu asymptoticallytends to a limit with increasing velocity. The first experimental studies date back to the early seventies, James and Acosta[18] and James and Gupta [19] who reported critical phenomena in the laminar flow of dilute aqueous solutions across smallwires concerning heat transfer and drag, respectively. In their experiments Nu and the drag coefficient increase as the wirediameter is increased and decreased, respectively, at fixed concentration up to a critical speed Uc in the case of Nu and up to a

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5Wi

Nu

Fig. 5. Dependence of the average Nusselt number Nu on the Weissenberg number Wi for the fluid and elliptical tube in Fig. 1: Pr = 50, n = 0.3 (d) Re = 1 (N)Re = 50 (j) Re = 100 (s) Re = 150 (4) Re = 200 (h) Re = 250.

0

2

4

6

8

10

12

14

16

0 0.1 0.2 0.3 0.4 0.5Wi

Nu

Fig. 6. Dependence of the average Nusselt number Nu on the Weissenberg number Wi for the fluid and triangular tube in Fig. 2: Pr = 50, n = 0.3 (d) Re = 1 (N)Re = 50 (j) Re = 100 (s) Re = 150 (D) Re = 200 (h) Re = 250.

0123456789

0 0.1 0.2 0.3 0.4 0.5Wi

Nu

Fig. 7. Dependence of the average Nusselt number Nu on the Weissenberg number Wi for the fluid and square tube in Fig. 3: Pr = 50, n = 0.3 (d) Re = 1 (N)Re = 50 (j) Re = 100 (s) Re = 150 (D) Re = 200 (h) Re = 250.

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critical Rec in the case of the drag coefficient. Nu increases with inertia (velocity in their experiments) as it would in a New-tonian fluid when U < Uc and becomes clearly independent of inertia for U > Uc. This critical speed Uc decreases with increas-ing concentration for all the wire diameters and different dilute solutions used in the experiments. In addition Nu seems tocorrelate with the inverse of the Elasticity number E when U > Uc that is NuVE = f(1/E) (see Section 3.2). Experiments inves-tigating mass transfer from wires were presented by Ambari et al. [20]. Their data shows a large decrease in mass transferfrom the wire in the neighborhood of some critical value Uc followed by a leveling off of mass transfer from the wire withincreasing speed that is an asymptotic approach to a constant value. Most interestingly the critical values Uc for heat transferand mass transfer from wires are roughly the same, although drag reduction data does not correlate as well. But clearly theUc is diameter and flow independent and very much looks like a material parameter. Ultman and Denn [16] introduced theidea of change of type in an attempt to explain the critical phenomena reported by James et al. [18] with Uc equal to the shearwave speed into the fluid at rest. They show theoretically although not rigorously that NuVE = f(1/E) in agreement with exper-iments. Joseph [21] gives an excellent argument that the wave speed is a decreasing function of the concentration the sameas observed in the experiments for Uc. The results of a numerical simulation for flow past a heated circular cylinder of UCMtype viscoelastic fluids are reported by Delvaux and Crochet [17]. They show that with increasing Pe and therefore withincreasing Mach number M (see Section 3.2) the Nusselt number NuVE becomes independent of Re. The asymptotic levelingoff of NuVE is delayed for smaller Pr. Existing studies on external flows therefore clearly suggest an asymptotic independencefrom inertia with increasing velocities at fixed elasticity (concentration). If internal flows are considered the asymptoticindependence from elasticity, as represented by Wi, at fixed inertia is shown in this paper for the first time. These ratherstriking results are to a large extent physically clarified by the concept of the change of type of the vorticity equation dis-cussed in the next section.

3.2. Change of type of vorticity and its effect on the enhancement

Clearly the coupling between viscoelastic and inertial non-linearities is crucial to enhancement. The smallness of the Rey-nolds number is not a sufficient reason for neglecting inertial non-linearities in the flow of viscoelastic fluids with instanta-neous elasticity, which includes the class of fluids without a Newtonian base viscosity studied in this paper. The pivotalparameter is the product of the Reynolds number and the Weissenberg number M2 = ReWi; where M is the viscoelastic Machnumber the ratio of a characteristic velocity of the fluid to that of the speed of shear waves into the fluid at rest, Joseph [21].The neglect of the inertial terms may only be justified for small values of the Mach number. Yoo and Joseph [22] have shownthat in the Poiseuille flow in a wavy channel the vorticity of an UCM (upper convected Maxwell) fluid will change type whenthe velocity in the center of the channel U is larger than a critical value Uc defined by the propagation of the shear waves.Their conclusions are extended by Joseph [21] to Poiseuille flow in pipes among a number of other flow types. The flow do-main is partitioned into two distinct regimes. There is a relatively high speed central region of the channel where vorticityf =r� v is governed by a hyperbolic equation, and a low speed region near the channel walls where it is governed by anelliptic equation. The hyperbolic region is characterized by elastic effects like wave propagation along the characteristicsand undamped oscillations of the perturbed vorticity whereas in the elliptic region the vorticity oscillations are dampedand discontinuities are smoothed. In the hyperbolic region it is possible that the derivatives of the vorticity are discontinuousacross the characteristics if the boundary data is discontinuous. The supercritical and subcritical subdomains are compatibleand are determined by the viscoelastic Mach number M. The Mach number is larger than one in the supercritical hyperbolicregion around the center of the channel M > 1. The vorticity in this region shows an oscillatory behavior along the charac-teristics. The period of the oscillation is governed by M. When the Mach number M� 1 the thickness of the hyperbolic regionis small if the dimensionless Elasticity number E is large and vice versa. The Elasticity number is independent of the velocityand is defined as E = Wi/Re = gk/qa2 where g, k, q and a are the zero shear viscosity, the natural relaxation time of the fluid,the density and the radius of the unperturbed boundary, respectively. For large values of E the oscillations are not damped asthe centerline is approached. The larger the Elasticity number the larger the amplitude of the oscillations when M is fixed.There is a rapid damping of vorticity in the supercritical region away from the boundary when M� 1 if the Weissenbergnumber Wi = M

pE 6 O(1). The rate of damping of vorticity decreases with increasing Wi. Flows with high M appear to be

more elastic in the sense that damping is suppressed as the relaxation time of the fluid is increased. Delvaux and Crochet[17] in their study of the external flow of an UCM fluid around a circular cylinder show numerically that at fixed elasticityrepresented by Wi and increasing inertia Re and therefore for increasing and decreasing Mach M and Elasticity E numbers,respectively, the hyperbolic region gradually expands and the elliptic region around the cylinder and in particular down-stream of the cylinder gradually shrinks. However for the same M but doubled elasticity Wi and halved inertia Re so thatthe Elasticity number E is quadrupled the size of the elliptic region is double of the previous case; of course as E decreasesand M increases the hyperbolic region again expands.

Criteria for the class of non-affine fluids with interpolated Maxwell type of convected derivatives are similar and havebeen worked out by Joseph [21]. Also UCM fluid has instantaneous elasticity with no retardation time. In that sense it is sim-ilar to the family of fluids studied in this paper without base Newtonian viscosity. Joseph remarks in [21] that the conclu-sions of the solution of the Poiseuille flow of UCM fluids in circular pipes with wavy walls are universal to axisymmetricproblems, linear or non-linear, and that when M > 1 there is a region near the wall where the perturbed vorticity equationis elliptic and a region of hyperbolicity near the centerline of the pipe. Axisymmetric problems are relatively easier toformulate and solve when change of type of vorticity is discussed as there is only one component of vorticity. However

1556 D.A. Siginer, M.F. Letelier / International Journal of Engineering Science 48 (2010) 1544–1562

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geometries investigated in this work like the equilateral triangle are not axially symmetric. In general we expect three com-ponents of vorticity in these flows and three different criteria for transcriticality. The hyperbolic regions for different com-ponents of the vorticity may or may not overlap. However we do expect the overall behavior of the change of type and thetrends governed by it to be the same qualitatively with significant quantitative differences.

Each curve in Figs. 5–7 starts from the Newtonian value of Nu on the ordinate. Vorticity is elliptic in the neighborhood ofthat point. Elasticity number gradually grows with increasing Wi on each Re constant curve but it always stays smaller thanone in the range of Wi in the Figures, E = Wi/Re < 1. It is largest on the almost creeping flow curve Re = 1 but the viscoelasticMach number M < 1 on Re = 1 and there is no hyperbolic region on the entire curve; on the other hand M > 1 over an over-whelming stretch of all the remaining curves except very close to the ordinate. It increases along each curve with Wi,M2 = ReWi. We expect the existence of a supercritical region away from the wall where the vorticity is hyperbolic whenM exceeds one. M and E increases and decreases, respectively, on a vertical line Wi constant, which implies that the sizeof the supercritical region grows going up on any vertical line starting from nil on Re = 1. E grows with Wi and at (Re1, Wi1)is smaller than at (Re1, Wi2), Wi1 < Wi2, which would have implied that the size of the subcritical region is larger at (Re1, Wi2)had M stayed constant. However M also grows that is M1 at (Re1, Wi1) is smaller than M2 at (Re1, Wi2) and it grows muchfaster than E, which implies a larger supercritical region at (Re1, Wi2). Thus clearly on each curve Re = constant the size ofthe supercritical region grows with growing Wi.

For larger values of the Elasticity number E the oscillations of the vorticity are not damped as the centerline is approachedwhen M is fixed; for smaller values of E they are severely damped. For fixed E vorticity in the hyperbolic region is rapidlydamped away from the wall, the larger the Wi the smaller the rate of damping. We expect a cause and effect relationshipbetween the leveling off of the enhancement with increasing Wi and the gradual increase of the hyperbolic region togetherwith the reduced damping of the vorticity in this region with increasing Wi and M. Vorticity stays elliptic on all three curvescorresponding to almost creeping flow in Figs. 5–7 as M < 1. However there is a point on all other curves where the super-critical region springs to life as M approaches one from below. This point comes closer and closer to the ordinate on curvescorresponding to larger and larger Re, for instance for Re = 10 and 100 it is approximately located at Wi = 0.1 and 0.01, respec-tively, leading to the conclusion that the sooner the supercritical region sets in the steeper the slope of the curve at thatpoint, thus the larger the rate of enhancement on curves corresponding to larger Re. E is smaller for M � 1 on successivecurves corresponding to growing Re indicating that the size of the hyperbolic region at M � 1 is larger if Re is larger. Thusenhancement will be larger if Re is larger, the nascent hyperbolic region will come closer to the Newtonian point on the ordi-nate and its size will be larger if Re is larger triggering a larger enhancement rate. The fact that enhancement is larger whenRe is larger can not be attributed to stronger secondary flows because on curves with larger Re at M � 1 secondary flows areweaker as Wi is smaller.

Assuming linearity for the sake of a simple scaling argument increasing inertial effects by three decades will increase lon-gitudinal velocity by three decades and secondary flow strength will also increase by three decades at fixed Wi for the ratio ofsecondary flow strength to longitudinal flow to stay of the same order O(Wi)3. Thus inertial effects act to increase the sec-ondary flow strength which in turn acts to enhance average heat transfer that is the average value of NuVE. It is worthwhile tonote that secondary flow strength is quite weak. For instance it is only 0.001 of the longitudinal flow at Wi = 0.1. However theleveling off of the enhancement on each curve is not explicable by the secondary flow argument. For instance on any onecurve the strength of the secondary flow increases 27 times when Wi increases from 0.1 to 0.3. In spite of a much strongersecondary flow the rate of enhancement decreases considerably from Wi = 0.1 to Wi = 0.3. The physics must lie in the inter-action of supercritical vorticity regions for the three components of the vorticity and secondary flows. Evidence suggests thatat constant Re and for very small Wi close to the origin the supercritical region may be responsible for the large rate of in-crease of the enhancement rather than the nascent secondary flows. Further, clearly the effect of the increasing secondaryflow strength with Wi is kept in check by the growing supercritical region as M grows with Wi on Re constant, and theenhancement levels off rather than increase. It is noted that the secondary flow strength will be almost of the same magni-tude as the longitudinal flow when Wi � 1 as

ww¼ O

kUL

� �3

¼ OðWi3Þ

Thus we would expect the enhancement to increase drastically, and continue increasing unrealistically with increasing elas-ticity. However it does not. We suspect that this is due to the undamped vorticity oscillations and to the larger period of theoscillation when M is larger in the growing supercritical region. Vorticity is rapidly damped in the hyperbolic region awayfrom the boundary when M� 1 if the Weissenberg number Wi = 6O(1). However the rate of damping of vorticity decreaseswith increasing Wi on each enhancement curve. The physics will be theoretically clarified in forthcoming publications.

3.3. Enhancement due to shear-thinning

Some of the enhancement observed in Figs. 4–6 is due to increased shear-thinning and elastic effects with increasing lon-gitudinal velocity. To get an estimate of the contribution to total enhancement of these two effects Figs. 8–10 are presentedwhich show the contribution of shear-thinning to total enhancement at the second order, Eq. (11). The elasticity driven con-tribution given by Eq. (10) is of smaller magnitude and follows exactly the same trends. The share of the total enhancement

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of the sum of the two is cross-sectional shape, Wi and Re dependent at constant Pr. The highest contribution occurs for theequilateral triangle for any Re, Pr and Wi.

The maxima of the contributions of the non-linear elastic effects and shear-thinning to enhancement for different Re alloccur for quite small values of the Wi � 0.02. We have not searched for the exact values of the maxima or for the exact loca-tion. But it is clear that there is a sharp rise in contribution to enhancement from these two dominated and driven by shear-thinning when Wi slightly exceeds zero, a maximum is reached for a quite small value of Wi at any Re and thereafter there isa steady decline with increasing Wi culminating in an order of magnitude smaller contribution in all cases for Wi � 0.4 andasymptotically tending to an almost null contribution with increasing Wi. This is due to these two contributions to enhance-ment getting drowned by the increasingly larger enhancement caused by secondary flows. The maximum contributionslightly increases the further the cross-sectional shape deviates from the circular in the range of the Re numbers recordedin this work and reaches a maximum for the triangular shape. For almost creeping flow (Re = 1) the maxima of the shear-thinning driven enhancement at the second order are 2.04%, 2.43%, 4.74% of the total enhancement for the elliptical, squareand triangular tubes, respectively, and for Re = 250 they are 2.6%, 3.06% and 5.97% at Wi � 0.02. The contribution of the elas-tic part of the non-linearity is an order of magnitude smaller than that of the shear rate dependent part of the viscosity. In thetriangular tube about 6% of the enhancement is due to shear-thinning at Wi � 0.02 but it drops to 1.2% for an order of mag-nitude increase in Wi � 0.2. It tends to zero on either side of the divide Wi � 0.02 as Wi goes to zero (Newtonian) and as Wiincreases beyond 0.02 for any Re. These numbers are in ball park agreement with the experimental data of Gingrich et al. [10]for shear-thinning fluids. It is remarkable that 250-fold increase in inertia only increases by 1% the contribution of the shear-thinning effects at the second order to total enhancement for the triangular shape and it is smaller than 1% for the ellipticaland square shapes. The same trend holds for the elasticity driven contribution at this order. The contribution to enhance-ment of the non-linear elastic effects and in particular of the shear-thinning increases more rapidly than that of the second-ary flows generated enhancement as Wi increases from zero reaching a peak around Wi � 0.02 (very dilute solutions), butdecreases thereafter very rapidly with increasing Wi as the contribution of the component due to secondary flows increases

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.1 0.2 0.3 0.4 0.5Wi

(Nu

Nu S)/N

u–

Fig. 8. Contribution to total enhancement of shear-thinning at OðW2i Þ as a function of Wi for the fluid and elliptical tube in Figs. 1 and 4: Pr = 50, n = 0.3 (d)

Re = 1 (N) Re = 50 (j) Re = 100 (s) Re = 150 (D) Re = 200 (h) Re = 250. Nu and NuS represent Nusselt numbers with and without the shear-thinning effect,respectively.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.1 0.2 0.3 0.4 0.5Wi

(Nu

Nu S)/N

u–

Fig. 9. Contribution to total enhancement of shear-thinning at OðW2i Þ as a function of Wi for the fluid and triangular tube in Figs. 2 and 5: Pr = 50, n = 0.3: (d)

Re = 1 (N) Re = 50 (j) Re = 100 (s) Re = 150 (D) Re = 200 (h) Re = 250. Nu and NuS represent Nusselt numbers with and without the shear-thinning effect,respectively.

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at a much faster pace ultimately reducing the contribution of shear-thinning to less than 1% at Wi � 0.4, and as the trendsshow to insignificant levels beyond.

3.4. Effect of the slippage factor

The effect of the slippage factor n on the full enhancement including that of the shear-thinning is presented in Fig. 11 forvalues of n varying from 0.0 to 0.5 as a function of the Re for a fixed value of the Wi.

If n = 0 the MPTT model studied collapses onto the original PTT model and the Gordon–Schowalter convected derivativemerges with the upper convected Maxwell derivative. The original PTT model is identical to the UCM model with constantviscosity (equal to the molecular zero shear viscosity) except for the function f which multiplies the extra-stress in (2)1. Inthe context of the analysis in this paper up to and including O(Wi)3 the constitutive constant e� which appears in f does notinfluence the velocity and consequently the temperature field and the enhancement. Thus the analysis in this paper is iden-tical to that of the UCM model when n = 0 as the family of models in question here collapses onto the UCM model whichbehaves like the linear Newtonian fluid in straight tubes. There is no enhancement and NuVE = NuN is constant and indepen-dent of the Re. Enhancement is almost a linear function of Re in the range studied here for constant n and Wi. Deviation fromlinearity becomes significant with increasing n at larger Re. The ratio of the second and first normal stress differences for theMPTT model is given by

W2ðj2ÞW1ðj2Þ ¼

N2

N1¼ � n

2

A quite realistic value of n to adopt is 0.3 as experiments show that the ratio of N2N1

is around 0.1. We have used n = 0.3 for allthe results presented in this paper.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.1 0.2 0.3 0.4 0.5Wi

(Nu

Nu S)/N

u–

Fig. 10. Contribution to total enhancement of shear-thinning at OðW2i Þ to total enhancement as a function of the Wi for the fluid and square tube in Figs. 3

and 6: Pr = 50, n = 0.3 (d) Re = 1 (N) Re = 50 (j) Re = 100 (s) Re = 150 (D) Re = 200 (h) Re = 250. Nu and NuS represent Nusselt numbers with and without theshear-thinning effect, respectively.

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

0 50 100 150 200 250 300

Nu

Re

Fig. 11. Influence of slippage factor n on total enhancement as a function of Re for a triangular tube: Pr = 50, Wi = 0.3: (j) n = 0 (s) n = 0.1 (D) n = 0.3 (h)n = 0.5.

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3.5. Concept of efficient cross-section

The concept of efficient cross-section is also investigated by searching through all cross-sections corresponding to theinteger values of the mapping parameter n = 2, 3, 4, . . . representing the number of sides as well as the symmetries of thecross-sectional shape. Efficiency g is defined as the ratio of NuVE to hf with the latter symbol referring to the normalized headloss per unit length of tubing for the same volume flow rate and W is the dimensionless average velocity. The results arepresented in Table 1 for six cross-sectional shapes n = 2, 3, . . . , 7 starting with the ellipse n = 2 together with the NewtonianNusselt numbers NuN corresponding to each shape. The first three NuN values in the table for the ellipse, the equilateral tri-angle and the square are a check of the correctness of the computations as they reproduce the existing values from the lit-erature. The remaining values for the pentagon, hexagon and heptagon are to our knowledge new. The triangle is by far themost efficient shape followed by the square and the ellipse. The pentagon is a very close fourth. As the number of sides in-creases the efficiency decreases. In terms of head loss alone the triangle is the most inefficient cross-section followed by theellipse and the square. But if heat transfer capability alone is considered the order is almost reversed with the triangle lead-ing followed by the square and the ellipse.

3.6. Temperature field in arbitrary cross-sections

The mapping used to obtain various shapes with n axes of symmetry is quite versatile and yields a large spectrum ofunconventional shapes with a lesser number of symmetries by superposition. Thus if the superposition of a finite numberof mappings are conceived of, the first order velocity will read as,

wh0i ¼ p 1� r2 þXN

j¼1

ejsjrnjsinðnjhþ hnj

Þcosðnjhþ hnj

Þ

( )" #

Keeping only three terms the first order Newtonian temperature field in the cross-section resulting from the superposition ofthree mappings with ni, si and ei i = 1, 2, 3, si = ±1 together with the second order temperature distribution due to elastic ef-fects is written as,

Th0iðr; hÞ ¼ Pr16

a0p 1� r2 þX2

i¼1

eisirni sinðniðhþ hiÞÞ þ e3s3rn3 cosðn3ðhþ h3ÞÞ" #

� r2 � 3þX2

i¼1

eisiðni � 3Þðni þ 1Þ r

ni sinðniðhþ hiÞÞ þ e3s3ðn3 � 3Þðn3 þ 1Þ r

n3 sinðn3ðhþ h3ÞÞ" #

Th2i1 ðr; hÞ ¼ Prnð2� nÞ

36p3a0 1� r2 þ

X2

i¼1

eisirni sinðniðhþ hiÞÞ þ e3s3rn3 cosðn3ðhþ h3ÞÞ" #

�r4 þ r2 � 8þ

P2i¼1

eisi½�Bið1þ r2Þ þ Dirni sinðniðhþ hiÞÞþ

e3s3½�B3ð1þ r2Þ þ D3rn3 cosðn3ðhþ h3ÞÞ

8><>:

9>=>;

Bi ¼ð7n2

i þ 21ni � 4Þ2ðni þ 2Þðni þ 1Þ ; Di ¼

ð10n2i þ 11ni � 17Þðni þ 1Þ2

; i ¼ 1;2;3

The rules for admissible ei are expounded in detail in Siginer and Letelier [11]. The role of the parameter si = ±1 in theseexpressions is to rotate the corresponding figure in the superposition by an angle p/n in the counterclockwise directionwhereas the phase angle hi allows any one of the superposed figures it may correspond to rotate in either clockwise or coun-terclockwise direction by the angle specified by hi.

3.7. Heat transfer asymptotes in laminar flow

It is well established that in isothermal turbulent flows increasing the elasticity of the fluid decreases drag. However thereis a limit called minimum drag asymptote beyond which drag is no longer a function of elasticity but of Re alone. In turbulentpipe flow the friction factor f is a function of elasticity and inertia f (Re, Wi) when Wi < 10 but it becomes a function of the Reonly f(Re) if Wi > 10, Hartnett [3]. The minimum drag asymptote in turbulent flow in round tubes is determined experimen-tally and is given by

f ¼ 0:20Re�0:48

A similar behavior is observed in heat transfer enhancement. Concentration increase corresponds to a decrease in heattransfer in turbulent flows in round tubes. The Nu is a function of Nu(Re, Wi) when Wi < 100 however it becomes indepen-dent of Wi if Wi > 100, Hartnett [3]. The minimum heat transfer asymptote in turbulent pipe flow determined experimentallyis given by

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Nu ¼ 0:03Re�0:45

We show for the first time the existence of a conceptually similar asymptote in laminar flow of viscoelastic fluids in thispaper. For each given non-circular cross-section the enhancement curve levels off at a higher Wi with increasing inertia Re,Figs. 5–7. Joining the critical points Wic where each curve for Re constant levels off an asymptote is obtained beyond whichenhancement is not a function of the elasticity and depends only on inertia.

4. Conclusions

Heat transfer enhancement in the case of a class of non-affine and non-linear viscoelastic fluids in straight axisymmetrictubes with n sides and n symmetries is studied. The enhancement is generated to a large extent by secondary flows, but thereare also components due to shear-thinning, to non-linear elastic effects and to first normal stress differences. The enhance-ment due to the latter is of order O(e) that is it does not exist in circular tubes with this class of fluids whereas the enhance-ment due to the former two are of order O(1). All components of the enhancement are cross-sectional shape, Weissenbergnumber Wi, Reynolds number Re and Prandtl number Pr dependent Nu = f(Pe, Wi). The approach and the analysis have beenvalidated by computing the Newtonian Nusselt numbers in circular tubes and in cross-sections with n symmetries,n = 2, . . . , 7, demonstrating that the Newtonian Nusselt number is the highest for the circle, and the Nusselt number getssmaller as the number of symmetries gets smaller except for the ellipse. The enhancement due to shear-thinning is com-puted by considering the corresponding temperature distribution at the second order in Wi. The enhancement due to sheardependent viscosity is the largest for the circular cross-section and decreases with decreasing number of symmetries toreach the lowest value for the triangle with three symmetries. However for each cross-section the Nu corresponding tothe inelastic fluid is larger than its Newtonian counterpart. Although there is a component of the enhancement due to elasticeffects at OðW2

i Þ its contribution is overwhelmed by that of the shear-thinning and constitutes at most about 10% of the devi-ation from the Newtonian Nusselt number at OðW2

i Þ depending on the values assumed by Wi, Re and Pr. The full viscoelasticeffects that is the unbalanced first and second normal stresses appear at the third order in Wi. Secondary flows driven bysecond normal stresses contribute a large component to the total enhancement. With growing Wi at fixed inertial effectsand constant Pr the enhancement due to secondary flows grows rapidly. The percentage of the contribution to total enhance-ment of shear-thinning peaks for dilute solutions in all cross-sections studied in this paper at about Wi = 0.02 and thereafterdecreases rapidly with increasing Wi becoming almost insignificant for Wi � 0.4 regardless of Re and Pr and in particular isalmost zero for the triangular cross-section at that value of the Weissenberg number. At fixed inertia and Prandtl number theNusselt number asymptotically approaches a constant value with increasing Wi and becomes independent of Wi that is Nu-VE = f(Pe, Wi) ? NuVE = f(Pe). However the leveling off that is the decrease in the rate of increase of enhancement withincreasing Wi is gradually delayed with increasing Re. The rate of increase of the enhancement is quite high for more dilutesolutions as Wi starts deviating from zero, and it is higher for larger Re. The physics of the enhancement, its rate of increasewith increasing Wi and the asymptotic approach to a constant value independent of elasticity rests with the change of type ofthe vorticity equation in the cross-section and its interaction with the rapidly developing secondary flows in the cross-sec-tion. The structure of the secondary flows, the number of vortices, does not change with increasing Wi but the intensity of thevortices does. The vorticity equation is elliptic at the Newtonian point and stays elliptic for slight deviations from Newtonianbehavior. With further increase in Wi it is also elliptic closer to the wall, but a hyperbolic region in the cross-section springsto life when the viscoelastic Mach number M exceeds one. Vorticity shows an oscillatory behavior with period governed by Mas it propagates along the characteristics in the supercritical region. The larger M the larger is the period of the oscillations.The extent of the supercritical region as well as the rate of damping of the vorticity waves as the centerline is approacheddepends on the value of the Elasticity number. The leveling off of the enhancement with increasing elastic effects is due tothe mitigation of the second normal stress effects by the oscillations of undamped vorticity waves in the supercritical region.The mechanics of the enhancement based on these concepts will be further clarified mathematically in forthcoming publi-cations. Perhaps the most important finding reported in this paper is the heat transfer asymptote in laminar flows in non-circular tubes. The asymptote is different for each cross-section and quite naturally flows out of the above considerations. Fora given cross-section at fixed Re if elasticity is larger than a critical value Wi > Wic heat transfer is no longer a function ofelastic effects and becomes constant. For the same cross-section different Re corresponds to different Wic, joining all theWic an asymptote is obtained for that particular cross-section beyond which heat transfer enhancement is a function ofRe alone.

Acknowledgments

The support of the Chilean Foundation for Research and Development (FONDECYT) is gratefully acknowledged, Grant Nos.1010173 and 7010173.

References

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