International Journal of Heat and Mass Transfer 53 (2010) 4728–4740

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Effect of temperature-dependent viscosity on forced convection heat transferfrom a cylinder in crossflow of power-law fluids

A.A. Soares a,b, J.M. Ferreira a, L. Caramelo a,b, J. Anacleto a,c, R.P. Chhabra d,*

a Departamento de Física, Universidade de Trás-os-Montes e Alto Douro, Apartado 1013, 5001-801 Vila Real, Portugalb CITAB/UTAD, Quinta de Prados, Apartado 1013, 5001-801 Vila Real, Portugalc IFIMUP and IN – Institute of Nanotechnology, Departamento de Fisica da Faculdade de Ciencias da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugald Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India

a r t i c l e i n f o

Article history:Received 4 August 2009Received in revised form 14 January 2010Accepted 10 February 2010Available online 12 July 2010

Keywords:Circular cylinderNusselt numberTemperature-thinningShear-thinningShear-thickening

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.06.019

* Corresponding author. Tel.: +91 512 259 7393; faE-mail address: chhabra@iitk.ac.in (R.P. Chhabra).

a b s t r a c t

The steady, two-dimensional and incompressible flow of power-law fluids across an unconfined isother-mal heated circular cylinder is investigated numerically to ascertain the effect of temperature-dependentviscosity on the flow and forced convection heat transfer phenomena. Extensive numerical results eluci-dating the variation of the heat transfer characteristics and drag coefficient on the severity of tempera-ture dependence of viscosity (0 6 b 6 0.5), power law index (0.6 6 n 6 1.6), Prandtl number(1 6 Pr 6 100) and Reynolds number (1 6 Re 6 30) are presented. The coupled momentum and energyequations are expressed in the stream function/vorticity formulation and solved using a second-orderaccurate finite difference method to determine the local and surface-averaged Nusselt numbers, the dragcoefficient, and to map the flow domain in terms of the temperature and flow fields near the cylinder. Thevariation of viscosity with temperature is shown to have a substantial effect on both the local and sur-face-averaged values of the Nusselt number. As expected, the results also suggest that the rate of heattransfer shows positive dependence on the Reynolds number and Prandtl number. Furthermore, strongerthe dependence of viscosity on the temperature, the greater is the enhancement in the rate of heat trans-fer. Finally, all else being equal, shear-thinning fluid behaviour facilitates heat transfer while the shear-thickening behaviour has deleterious effect on heat transfer.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The laminar convection from a heated cylinder is an importantproblem in the domain of heat transfer. It is used to simulate a widerange of engineering applications as well as to develop better in-sights into more complex systems of flow and heat transfer. Thus,the flow and heat transfer in the forced convection regime from acircular cylinder are well documented in the literature, especiallyfor Newtonian fluids and in power-law fluids e.g. [1–11]. It is, how-ever, fair to say that in contrast to the vast body of information avail-able for Newtonian fluids, the analogous body of knowledge forpower-law fluids in not only meagre, but is also of recent vintage.Furthermore, only a few studies take the variation of viscosity withtemperature into consideration, even though the majority of fluidsused in industrial processes exhibit varying levels of viscosity–temperature dependence. Indeed, the literature is very scant on thisissue even for Newtonian fluids. While the assumption of tempera-ture-independent viscosity is valid as long as the differences in tem-peratures involved remain small in an application, a more accurate

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x: +91 512 259 0007/0104.

prediction of the flow and heat transfer can only be achieved byaccounting for the variation of the viscosity with temperature, asis evidenced by numerous experimental correlations which accountfor this effect by including a factor involving the ratio of viscosityevaluated at the surface temperature and at the mean fluid temper-ature. From a theoretical standpoint, the inclusion of this featureleads to the coupled momentum and thermal energy equationswhich must be solved simultaneously. However, it must be men-tioned here that, for engineering applications, the effects of thevariation of viscosity with temperature are usually taken into ac-count by the application of correction factors with limited empiricalor theoretical justification, thus leading to some ambiguity and awk-wardness [12]. It is also important to recognize that for most liquids(including non-Newtonian polymer melts and solutions), relevantthermo-physical properties such as density, thermal conductivityand specific heat are relatively insensitive to the temperature vari-ation, whereas viscosity decreases markedly with increasingtemperature. For instance, for water, the values of density andviscosity decrease by 2.5% and �65%, respectively over thetemperature range 20–80 �C. Similarly the corresponding changesfor Ethylene glycol are 5.6% and �87%, respectively over thetemperature interval 27 6 T 6 116 �C. Similar trends can also be

Nomenclature

a radius of the cylinder (m)A constant (Pa s)b temperature-thinning index (–)Cd drag coefficient (–)cp heat capacity (J kg�1 K�1)E activation energy (J/mol)F dimensionless function, Eq. (10e)h heat transfer coefficient (W m�2 K�1)I2 dimensionless second invariant of the rate-of-deforma-

tion tensor (–)�I2 dimensionless second invariant of the rate-of-deforma-

tion tensor in real space (–)K power-law consistency index (Pa sn)k thermal conductivity of the fluid (W m�1 K�1)n power-law index (–)Nu average Nusselt number (–)Nu(h) local Nusselt number (–)p dimensionless pressure (–)Pr Prandtl number (–)r cylindrical coordinate (m)R molar gas constant (J mol�1 K�1)Re Reynolds number (=q(2a)n U2�n

1 /K) (–)U1 uniform approach velocity (m/s)T dimensionless temperature (–)�T dimensionless temperature in real space (–)Ts temperature on the surface of the cylinder (K)

T1 far-field fluid temperature (K)vz z-component of velocity (m/s)

Greek lettersc dimensionless function, Eq. (10d) (–)h angle (radians)k dimensionless function, Eq. (10b) (–)b dimensionless function, Eq. (10c) (–)w dimensionless stream function (–)�w dimensionless stream function in real space (–)l viscosity (Pa s)ls viscosity at the temperature on the surface of the cylin-

der (Pa s)l1 viscosity at the far-field fluid temperature (Pa s)g dimensionless viscosity (–)x dimensionless vorticity (–)�x dimensionless vorticity in real space (–)q fluid density (kg/m3)e dimensionless polar coordinate (=ln(r/a)) (–)eij components of rate-of-deformation tensor (s�1)sij dimensionless components of extra stress tensor (–)

Subscriptsh angular componentr radial component

A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740 4729

demonstrated for scores of other Newtonian liquids [13,14], albeit itis neither possible nor justifiable to offer generalisations in terms ofthe severity of viscosity–temperature relationship. Similarly, sincemost polymeric systems (melts and solutions) are modelled asentangled entities of long chain (linear or branched) molecules, theirmass density is independent of chain architecture and dynamicswhereas their viscosity is primarily governed by the degree ofentanglement. Limited available data on polymer solutions andmelts confirm this expectation in so far that their density is littleinfluenced by temperature variation whereas the viscosityconforms to the usual Arrhenius-type temperature dependence[15–17]. Hence, the temperature-induced buoyancy effects aremuch weaker in liquids than in gases. It is therefore fair to neglectthe effects stemming from the temperature-dependence of densityin comparison with those arising from temperature-dependent vis-cosity. In view of this, the variation of viscosity with temperaturehas attracted a great deal of attention due to its effect on flow/heattransfer models with engineering applications and, for the case ofNewtonian fluids, such interest has ranged from heat transport inporous media [18] to forced [19] and mixed [20,21] convection overhorizontal/vertical isothermal plates, as well as in heat transfer toNewtonian fluids from heated bluff bodies whose surfaces are main-tained at constant temperature. For instance, Chang and Finlayson[22] studied numerically the heat transfer for the flow past a cylin-der at Re < 150 for constant fluid properties, and subsequentlyextended this work using experimental data for a fluid with temper-ature-dependent viscosity and determined heat transfer rates forhot and cold cylinders [13]. For a uniform heat flux boundary condi-tion, Perkins and Leppert [23] conducted a study on the overall heattransfer from a circular cylinder to water and ethylene glycol for40 6 Re 6 105 and 1 6 Pr 6 300. Subsequently, they studied thelocal heat transfer behaviour from a cylinder in water over theranges of conditions as 2 � 103

6 Re 6 1.2 � 105 and 1 6 Pr 6 7[24]. They proposed an empirical correlation and reported that thevariation of viscosity with temperature could be corrected using afactor of (ls/l1)0.25. The index of 0.25 here is significantly larger

than the usual value of 0.14 used for internal flow in ducts and pipes.Zhukauskas [25] and Whitaker [26] proposed similar correlationsfor heat transfer coefficient from a circular cylinder in crossflowfor the range 1 6 Re 6 105. Pantokratoras [27] studied the lami-nar-assisted mixed convection heat transfer from a vertical isother-mal cylinder to water with variable physical properties. Molla et al.[28] investigated the natural convection flow from an isothermalcircular cylinder with temperature dependent viscosity. Cheng[29] investigated the effect of temperature-dependent viscosity onthe natural convection heat transfer from a horizontal isothermalcylinder of elliptic cross section. More recently, Ahmad et al. [30]have also reported significant effects of temperature-dependent vis-cosity, buoyancy, and Prandtl number, on the flow and heat transfercharacteristics for an isothermally heated circular horizontal cylin-der. In contrast to the aforementioned studies of the effects of tem-perature on Newtonian flow/heat transfer, much less attention hasbeen devoted in the literature to the non-Newtonian case [3–11],despite its relevance to a wide range of industrial applications whichinvolve non-Newtonian fluids. Such applications include polymerand nanofluids technology, petroleum industry, fluid dropletssprays, purification of crude oil, magneto-hydrodynamic pumpsand power generation, aerodynamic heating, e.g. [31–33]. Many ofthe substances involved in the aforementioned applications displayshear-thinning and/or shear-thickening behaviour e.g. [32,34], andowing to their high viscosity levels they are generally processed inlaminar flow conditions and thus, the corresponding Reynolds num-ber is generally small. Besides, in view of the aforementioned dis-cussion on the weak buoyancy currents in liquids, the main modeof heat transfer here is believed to be forced convection.

For the case of power-law fluid flow past a circular cylinder,during the past 10 years or so, reliable numerical results on thedrag, wake and heat transfer characteristics have accrued in the lit-erature, albeit these predictions are restricted to the so-called two-dimensional steady flow regime e.g. [3–11], with the notableexception of a recent study [35] dealing with the laminar vortexshedding regime. The corresponding scant heat transfer results

4730 A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740

also suggest that heat transfer is facilitated by shear-thinningbehaviour, and it is somewhat impeded in shear-thickening fluids,for both a confined as well as an unconfined cylinder. It needs to beemphasized here that most of the aforementioned studies arereally based on the assumption of the steady flow regime inpower-law fluids up to about Re = 40. Strictly speaking, this isnot so, as the limit of the steady flow is dependent on the valueof the power law index [36]. For instance in a highly shear-thicken-ing fluid with n = 1.8, the steady flow regime ceases to exist atRe = 33. Furthermore, all heat transfer studies have implicit inthem the assumption of constant thermo-physical propertiesthereby severely limiting their applicability to the situationswherein the temperature difference is small. This observation isalso applicable for cylinders of square and elliptical cross sectionsin the two-dimensional steady flow regime [11,37–39], except forthe very recent work on momentum and heat transfer characteris-tics in the laminar vortex shedding regime for a square cylindersubmerged in power-law fluids [40,41].

In summary, it is thus fair to say that, to the best of our knowl-edge, there has been no prior numerical or experimental study toascertain the influence of temperature-dependent viscosity in thepure forced convection heat transfer over a circular cylinder sub-merged in power-law fluids. However, preliminary results on heattransfer in a scraped surface heat exchanger [42] suggest thatwhen this effect is included, the flow characteristics may be sub-stantially changed compared to the constant viscosity case. There-fore, it is useful and instructive to ascertain the role of temperaturedependent viscosity on heat transfer from a cylinder to power-lawfluids. The present study aims to alleviate this situation by investi-gating the forced convection heat transfer in incompressiblepower-law fluids from an isothermal horizontal circular cylinderimmersed in it. The viscosity of the fluid is assumed to be a func-tion of both temperature and shear rate. The surface temperatureTs of the cylinder is higher than that of the ambient fluid tempera-ture T1. It has been assumed that the viscosity of the fluid is inver-sely proportional to an exponential function of temperature whichis in line with the form used by Sun et al. [42].

In summary, the objective of this work is to study numerically theeffect of temperature varying viscosity on the gross engineeringparameters such as Nusselt number and drag coefficient as well ason the detailed flow and thermal fields, i.e., isothermal lines andstream function for the steady power-law flow across a long heatedcircular cylinder over a range of power-law indices (0.6 6 n 6 1.6),temperature-thinning indices 0 6 b 6 0.5, Reynolds numbers 1 6Re 6 30 and Prandtl numbers 1 6 Pr 6 100. Over these ranges ofconditions, the flow is expected to be steady and two-dimensional.

2. Problem statement and basic equations

Consider the two-dimensional, laminar, steady incompressibleflow of a power-law fluid with a uniform velocity (U1) and temper-ature (T1), normal to an infinitely long cylinder. The unconfined

y

r e

xcylinder

Coupled Navier-Stokes and energy equations

upstream downstream

T∞

U∞

far stream

R∞

Fig. 1. The real (x, y) plane and the computational (e, h) plane. Distance (R1) from the extapproach velocity (U1).

flow condition is simulated here by enclosing the isothermal circu-lar cylinder in a cylindrical envelope of the fluid thereby constitut-ing the circular outer boundary (of radius R1), as shown in Fig. 1.The radius of the outer boundary is taken to be sufficiently large(54.6 radii away from the cylinder surface) to minimize the bound-ary effects on the flow and heat transfer at the cylinder surfacemaintained at a constant temperature Ts (>T1). The thermo-physi-cal properties (density q, heat capacity cp, thermal conductivity k,and power law index n) are assumed to be constant and the viscousdissipation term in the thermal energy equation is neglected. Forpower-law fluids, the power-law index is known to be relativelyinsensitive to moderate variation in temperatures [31,32], andthe viscosity is assumed to exhibit an exponential dependence onthe temperature [42]. This assumption leads to the coupling ofthe momentum and the thermal energy equations. It needs to beemphasized here that the viscous dissipation term has been ne-glected in the energy equation used here because for the range ofconditions of 1 6 Re 6 30, the effective viscosity and/or shear rateclose to the cylinder’s surface is not expected to be excessively high.

Since the present study is restricted to a long cylinder and flowconditions of Re 6 30, as noted above, the flow is expected to besteady and two dimensional [36], i.e., all flow variables are inde-pendent of the z-coordinate and are therefore functions of cylindri-cal coordinates r and h alone. The equation of continuity, the r andh components of the equations of motion and thermal energy (inthe absence of viscous dissipation) in cylindrical coordinates, canbe expressed in their dimensionless stream function/vorticity for-mulation in terms of the polar coordinates (e, h) with e = ln(r/a),where a is the radius of cylinder, as follows [4]:

Continuity equation1ee

@

@eee @w@h

� �� @

@hwþ @w

@e

� �¼ 0; ð1Þ

e-momentum component

@w@h

@2w@e@h

� @w@eþ w

� �@2w

@h2 þ@w@eþ w

!

¼ �12@p@e� 2n

Ree�e @

@eeesrrð Þ þ @srh

@e� shh

� �; ð2aÞ

h-momentum component

� @w@h

@2w@e2 þ

@w@e

!þ @w

@eþ w

� �@2w@e@h

¼ �12@p@h� 2n

Ree�2e @

@ee2esrh� �

þ @shh

@h

� �ð2bÞ

and the energy equation

@2T@e2 þ

@2T

@h2 �@T@e

2þ eeRePr2

@w@h

� �þ @T@h

eeRePr2

wþ @w@e

� �� �

þ T 1þ eeRePr2

@w@h

� �¼ 0; ð3Þ

Transformed coupled Navier-Stokes and energy equations

downstream

upstream

cylin

der

0

far stream

ernal boundary to the cylinder surface. Free stream fluid temperature (T1), uniform

A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740 4731

where p, T, Re and Pr are the dimensionless pressure, temperature,Reynolds number and Prandtl number, respectively.

The dimensionless components of the extra stress tensor for anincompressible power-law fluid are written as

sij ¼ �geij; ð4Þ

where g is the dimensionless viscosity, i, j = x, y and eij are thedimensionless components of the rate-of-deformation tensor.

The fluid viscosity is a function of both the temperature andshear rate. In this study, the equation for the dimensionless powerlaw viscosity is generalised for the temperature dependence as fol-lows [42,43]:

g ¼ e�bTe�eI

n�12

2 ; ð5Þ

where n is the power-law index (shear-thinning for n < 1, Newto-nian for n = 1 and shear-thickening for n > 1). By analogy with therole of power-law index (n), b is called the temperature-thinning in-dex and I2 the dimensionless second invariant of the rate-of-defor-mation tensor. Evidently, larger the value of b, stronger is thedependence of viscosity on temperature. The dimensionless tem-perature (T) used in the present study, is related to the previouslyused e.g. [37] dimensionless temperature in real space (�T) asT ¼ ee�T , and I2 is given as:

I2 ¼ e�2e w� @2w@e2 þ

@2w

@h2

!2

þ 4@2w@h@e

!224

35; ð6Þ

The vorticity in its scaled form is given as:

@2w@e2 þ

@2w

@h2 þ 2@w@eþ wþx ¼ 0: ð7Þ

In Eqs. (1), (2a), (2b), (3)–(7) the dimensionless stream function w,vorticity x and pressure p are related to their dimensional counter-parts as eeU1aw, e�e(U1/a)x and ðU2

1q=2Þp, respectively. Thedimensionless components of the extra stress tensor sij and thedimensionless second invariant of the rate of deformation tensorI2 are related to their dimensional counterparts as K(U1/a)nsij and(U1/a)2I2, respectively. In Eq. (3), the dimensionless temperature Tis related to its dimensional counterpart as e�eT(Ts � T1).

The Reynolds number is defined as

Re ¼ qð2aÞnU2�n1

K; ð8Þ

where K denotes the power law consistency index.The Prandtl number is defined as

Pr ¼ cpKk

U12a

� �n�1

: ð9Þ

Elimination of the pressure terms in Eqs. (2) by the usual method ofcross-differentiation and introducing the vorticity x, followed bysome rearrangement, leads to:

g@2x@e2 þ

@2x@h2

!þ 2k

@x@eþ 2b

@x@hþ cx ¼ F; ð10aÞ

where

k ¼ @g@e� g� Re ee

2nþ1

@w@h

; ð10bÞ

b ¼ @g@hþ Re ee

2nþ1

@w@eþ w

� �; ð10cÞ

c ¼ �2@g@eþ gþ Re ee

2n@w@h

; ð10dÞ

F ¼ w� @2w@e2 þ

@2w

@h2

!@2g@h2 �

@2g@e2 þ 2

@g@e

!

� 4@2w@h@e

@g@h� @2g@h@e

!: ð10eÞ

The exponential scaling for the stream function, the vorticity, andthe temperature is appropriate here since the stream function at-tains large values far away from the cylinder; the vorticity is expo-nentially small everywhere except in the region of the wake, andthe temperature decreases rapidly away from the cylinder. Thisscaling procedure, used in previous studies [3,4,44], is also em-ployed in the present study because it suppressed the numericalinstabilities and thus enabled convergent solutions in the range ofconditions of interest here.

The realistic physical boundary conditions for this flow are ex-pressed as follows:

On the cylinder surface, i. e. at e = 0, the usual no-slip conditionis applied:

@w@e¼ @w@h¼ 0; ð11aÞ

which together with Eq. (7) gives

w ¼ 0 and x ¼ � @2w@e2 : ð11bÞ

The thermal boundary conditions at the surface of the solid cylinderis scaled as

T ¼ 1: ð11cÞ

On the plane of symmetry at h = 0, p:

w ¼ x ¼ @T@h¼ 0: ð11dÞ

Far away from the cylinder surface, for e1 = 4, we use the asymp-totic approximation for stream function and vorticity given byChhabra et al. [3],

w � sinðhÞ þ Cd2

e�e hp� erfðQÞ

� �; ð11eÞ

x � �CdReI1�n

22

2nþ1 ffiffiffiffipp Qe�Q2

; ð11fÞ

where Cd is the drag coefficient,

Q ¼ ee2

ffiffiffiffiffiffiRe2n

rI

1�n4

2 sinh2

� �; ð11gÞ

and erf(Q) is the standard error function.The far away stream temperature boundary condition is

T ¼ 0: ð11hÞ

It also needs to be emphasized here that the aforementioned temper-ature boundary condition at the far field (T = 0) does not depend onthe type of fluid and is therefore valid for both Newtonian and non-Newtonian conditions. It is, however, appropriate to mention herethat, although the vorticity and stream function boundary conditionsusually applied in the far-field mimic the well-known asymptoticsolution for the Newtonian case, the non-Newtonian viscosity valuewas incorporated [3,4] in the aforementioned solution with the aimof extending its validity to power-law fluids. In terms of velocityboundary conditions, there is no ambiguity as the free stream velocitycondition is valid for any type of fluid. However, in developing thecorresponding boundary conditions in terms of vorticity and streamfunction, any possible loss of accuracy in such far-field asymptoticsolution for non-Newtonian fluids will become less significant asone approaches the surface of the cylinder (main region of interest)

4732 A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740

where the no-slip boundary condition is clearly valid for bothNewtonian and non-Newtonian conditions. Therefore, the far-fieldboundary conditions for power-law fluids developed in our previousworks [3,4] are believed to be quite adequate for this purpose. The factthat the use of this boundary condition led to the prediction of reliablevalues of drag coefficient and Nusselt number (for the constantviscosity case) inspires confidence in the use of these conditions inthe present case also. Owing to the symmetry of the flow over therange of conditions studied herein, the computations have beencarried out only in the upper half (y P 0 and x2 + y2 P 1) of thecomputational domain. Thus, the corresponding region in the (e, h)plane is defined by e P 0 and 0 6 h 6 p (Fig. 1).

The numerical solution of the system of coupled elliptic govern-ing equations, given by Eqs. (3), (7) and (10) in conjunction withthe above noted boundary conditions (Eqs. (11)) yields the valuesof the stream function w, vorticity x and temperature T variablesin the flow domain, 0 6 e 6 4. The flow and thermal fields, in turn,are used to determine the values of the local and surface-averagedNusselt numbers (Nu(h) and Nu), as well as the drag coefficient Cd.

The local Nusselt number at a point on the surface of the cylin-der is defined by

NuðhÞ ¼ hð2aÞk¼ �2

@T@e� T

� �e¼0: ð12Þ

The surface-averaged Nusselt number is given by

Nu ¼ 1p

Z p

0NuðhÞdh: ð13Þ

The drag coefficient is given by:

Cd ¼ 2nþ1

Re

Z p

0e�b�T�I

n�12

2@ �x@e� �x

� ���

þ �xn� 1

2�I2

@�I2

@e� b

@�T@e

� ���e¼0

sinðhÞdh; ð14Þ

where �x is the dimensionless vorticity in real space ( �x ¼ e�ex). Thedimensionless second invariant of the rate of deformation tensor inreal space (�I2) is obtained through the insertion into Eq. (6) ofw ¼ e�e �w, where �w is the dimensionless stream function in realspace.

Thus, in summary, once the values of the stream function,vorticity and temperature fields are known, these can be post-pro-cessed to obtain the values of drag coefficient, local and surface-averaged Nusselt numbers as functions of the physical (n, b) and

Table 1Comparison of present results of average Nusselt numbers (Nu) for heat transfer in Newliterature.

Source Re = 1

Pr = 1 Pr = 10 Pr = 50

b = 0Zhukauskas [25]b and Whitaker [26]b 0.71 1.33 2.21Chang et al. [13]b – – 2.92Bharti et al. [8,11] 0.8781 1.6639 2.669Present 0.94 1.69 2.67

b = 0.2

Zhukauskas [25]b and Whitaker [26]b 0.73 1.38 2.31Chang et al. [13]b – – 2.95Present 0.95 1.72 2.73

b = 0.5

Zhukauskas [25]b and Whitaker [26]b 0.77 1.47 2.47Chang et al. [13]b – – 3.00Present 0.97 1.77 2.83

a Numerical correlation.b Experimental correlation.

kinematic variables (Re, Pr). These results elucidate the influenceof the temperature-dependent viscosity on the interplay betweenthe fluid rheology and the characteristic conditions of flow.

3. Numerical solution method

The numerical solution procedure used here is an iterativeGauss–Seidel relaxation method, identical to that employed inour earlier studies [4,44,45], and to avoid redundancy, only themain features are included here. The set of non-linear differentialequations, namely the vorticity, stream function and energy Eqs.(3), (7), and (10) which are coupled by the power-law tempera-ture-dependent viscosity (Eq. (5)), have been solved by a finite dif-ference scheme using a second-order upwind differencingtechnique to discretize the convective terms of T and x in the vor-ticity and energy equations, whereas for the diffusion terms, thecentral difference approximation was used. For all other terms inthese equations, central difference approximations have also beenemployed. The resulting system of equations is solved using aGauss–Seidel iterative method with an under relaxation factor of0.8 to the temperature and vorticity variables. In general, the diffi-culty to meet the convergence criterion increases as the value ofthe power-law index deviates increasingly from unity and/or thevalue of temperature-thinning index, b, increases. In both cases,the nonlinearity of the system of equations increases. To obtainconsistent approximations for w, x and T, for each iteration asweep is made through all mesh points and updated values ofthe drag coefficient Cd and Nusselt number Nu are determinedby numerical integration in accordance with Eqs. (13) and (14)on the cylinder surface using Simpson’s rule. Convergence wasdeemed to have been achieved when, for the same iteration, thevariation in both Cd and Nu values in two successive iterationswas less than a preset value of 10�8. To accelerate the convergenceof the numerical solution, the Newtonian steady solution values ofw, x and T at every point of the grid were used as the initialguesses for non-Newtonian flow. The outer boundary was posi-tioned at e1 = 4, corresponding to asymptotic boundary conditionsat a distance of �54.6 radii away from the cylinder. For a(N + 1) � (M + 1) computational grid, the spacings in the radial eand angular h directions are e1/N and p/M, respectively.

For all range of conditions, meshes with 201 � 101 points wereused in radial and angular directions, respectively. All results re-ported herein have been checked for mesh independence by usingat least two different meshes.

tonian fluids from an unconfined circular cylinder with available results from the

Re = 30

Pr = 100 Pr = 1 Pr = 10 Pr = 50 Pr = 100

b = 02.78 3.02 6.74 12.03 15.472.57 – – 12.46 16.683.3471 3.23a 7.08a 12.34a 15.49a

3.28 3.17 6.70 11.61 14.81

b = 0.2

2.91 3.16 7.08 12.63 16.253.63 – – 12.59 16.643.37 3.23 6.92 12.09 15.45

b = 0.5

3.12 3.39 7.61 13.60 17.503.72 – – 12.78 17.043.50 3.31 7.26 12.84 16.54

0 45 90 135 1800.0

0.5

1.0

1.5

2.0

2.5

n =1.6

n =1.0

n =0.6

θ (degrees)

Re =1; Pr =10Su

rfac

e vi

scos

ity,

η s

b = 0b = 0.2b = 0.5

2.5Re =30; Pr =100

b = 0b = 0.2b = 0.5

(a)

A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740 4733

4. Results and discussion

In the present study, the numerical solutions of the coupledmomentum and thermal energy equations have been sought forthe following values of the dimensionless parameters: Reynoldsnumber: 1 6 Re 6 30, Prandtl number: 1 6 Pr 6 100, power-lawindices: 0.6 6 n 6 1.6 for the temperature-thinning indices b = 0,0.2 and 0.5. Extensive results on the local (Nu(h)) and surface-aver-aged (Nu) Nusselt numbers, the drag coefficient (Cd), as well as onthe detailed flow and temperature fields, have been obtained todelineate the effect of the temperature-dependent viscosity onheat and momentum transfer characteristics of a cylinder sub-merged in power-law fluids.

4.1. Validation of results

In view of the fact that extensive validation of the presentnumerical method has already been reported in our previous stud-ies corresponding to b = 0 for Newtonian and power-law fluids[3,4,44], the comparison presented herein is restricted to the caseof temperature-dependent viscosity using the limited informationavailable in the literature for Newtonian fluids. Table 1 comparesthe surface-averaged values of the Nusselt number (Nu) for heattransfer in Newtonian fluids (n = 1) from a circular cylinder atRe = 1 and 30, and Pr = 1, 10, 50 and 100 with literature values.For b = 0, an excellent correspondence can be seen to exist betweenthe present numerical results and those of Bharti et al. [8,11] forRe = 1 and 30, respectively (Table 1); the maximum differencebeing of the order of �7%. Deviations of this order are not at alluncommon in such numerical studies and these arise due to differ-ences in the numerical methods (for instance finite volume methodused by Bharti et al. [8,11] versus finite difference method usedhere), problem formulations, flow schematics, grid and/or domainsizes, discretization schemes, convergence criterion, etc.

For b = 0, 0.2 and 0.5 the present numerical results were com-pared with experimental correlations [25,26] and [13], respectivelygiven by

Nu ¼ 0:25þ 0:4Re12 þ 0:006Re

23

Pr0:37 l1

ls

� �14

ð15aÞ

with 1 6 Re 6 105, and

Nu ¼ ð0:36þ 0:58Re0:48ÞPrð0:29þ0:028log10ReÞ l1ls

� �ð�0:12þ0:10log10PrÞ

ð15bÞ

with 0.01 6 Re 6 50, 19 < Pr < 1948 and 0:009 < l1ls

< 110.

Table 2Viscosity l1 (Pa s) and temperature T1 (K) far-field values for benzene and 100 wt%glycerine, obtained from the corresponding values at the cylinder surface (ls, Ts) fortemperature-thinning indices b = 0.2 (l1 = 1.22ls) and b = 0.5 (l1 = 1.65ls). Theviscosity–temperature curves of benzene (A = (9.20 ± 2.0)10�6Pa s, E = (103.67 ±0.58)102 J/mol) and 100 wt% glycerine (A = (4.4 ± 1.4)10�12 Pa s, E = (646.5 ± 7.6)102

J/mol) both satisfy the relation l ¼ A exp ERH

� �.

Fluid b ls (�10�3 Pa s) Ts (K) l1 (�10�3 Pa s) T1 (K)

Benzene 0.2 0.350 343 0.427 3250.389 333 0.475 3160.436 323 0.532 307

0.5 0.350 343 0.577 3010.389 333 0.641 2940.436 323 0.719 286

100% Glycerine 0.2 624 303 762 300764 300.5 933 298945 298 1154 296

0.5 624 303 1029 297764 300.5 1260 295945 298 1558 292

The comparison showed discrepancies between the presentnumerical results and experimental correlation (15a) of <33% and<6% for Re = 1 and 30, respectively, while similar comparison withexperimental correlation (15b) showed discrepancies of less than10% for both Re = 1 and 30 (Table 1). Unfortunately, as far as knownto us, there are no such experimental results available for power-law fluids. In spite of this, in view of the aforementioned limitedcomparisons together with our past experience, the new results re-ported herein are believed to be reliable to ±4–5%.

At this juncture, it is also important to validate the viscosity’stemperature-dependent component e�b�T appearing in Eq. (5). Sincethis component is used for both Newtonian [13] and non-Newto-nian [42] fluids, for the sake of simplicity only the Newtonian fluidbehaviour is considered here. The validity of the temperature-dependent component e�b�T requires simultaneous satisfaction ofthe following two conditions:

(i) The variation of the fluid viscosity l with temperature mustbe of the form

0

0

1

1

2

Surf

ace

visc

osity

, ηs

Fig. 2.(gs) ove

l ¼ AeE

RH; ð16Þ

where H(K) is the temperature, R the molar gas constant, and acti-vation energy E needs to be independent of viscosity. In practice,

0 45 90 135 180.0

.5

.0

.5

.0

n =1.6

n =1.0

n =0.6

θ (degrees)

(b)Effects of temperature-thinning index b on the distribution of the viscosityr the surface of the cylinder at n = 0.6, 1.0 and 1.6. (a) Re = 1 and (b) Re = 30.

-3.0

-2.4

-1.8

-1.2

-0.6

0.00 45 90 135 180

-3.0

-2.4

-1.8

-1.2

-0.6

0.00 45 90 135 180

Re = 1

n =1.6

n =1.0

n =0.6

θ (degrees)

Pr =10

Surf

ace

vort

icity

, ωs

b = 0b = 0.2b = 0.5b = 0b = 0.2b = 0.5b = 0b = 0.2b = 0.5

(a)

θ (degrees)

Pr =100

Surf

ace

vort

icity

, ωs

-10

-8

-6

-4

-2

0

0 45 90 135 180

-10

-8

-6

-4

-2

0

0 45 90 135 180

n =1.6

n =1.0

n =0.8

θ (degrees)

Pr =10

Surf

ace

vort

icity

, ωs

b = 0b = 0.2b = 0.5b = 0b = 0.2b = 0.5b = 0b = 0.2b = 0.5

Re = 30

(b)

θ (degrees)

Pr =100

Surf

ace

vort

icity

, ωs

Fig. 3. Effects of temperature-thinning index b on the distribution of the vorticity (xs) over the surface of the cylinder at n = 0.6–0.8, 1.0 and 1.6. (a) Re = 1, Pr = 10 and 100, (b)Re = 30, Pr = 10 and 100.

4734 A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740

most fluids do obey this form of temperature dependence, albeitover varying ranges of temperature intervals.

(ii) The following approximation must be valid for the dimen-sionless viscosity g:

g ¼ ll1¼ e

ERH

eE

RT1� e�b�T ; ð17Þ

where �T ¼ e�eT is related to H (K), to the cylinder’s surface temper-ature Ts (K), and to the far field temperature T1 (K) as:

�T ¼ H� T1Ts � T1

: ð18Þ

The approximation used in Eq. (17) requires that (e.g. [13]):

Ts � T1 � T1: ð19Þ

Since the cylinder is heated Ts > T1, and using Eq. (19), it can beshown that the temperature-thinning index in Eq. (17) is given bythe positive quantity:

0 45 90 135 180

0.8

1.6

2.4

3.2

4.0

4.8

5.6

0 45 90 135 180

0.8

1.6

2.4

3.2

4.0

4.8

5.6

0 45 90 135 180

0.8

1.6

2.4

3.2

4.0

4.8

5.6

Pr =1

Pr =50

Pr =100

Pr =10

n = 0.6

Nu(

θ)

θ (degrees)

b = 0b = 0.2b = 0.5

Re = 1

n =1.0

Nu(

θ)

θ (degrees)

(a)

n = 1.6

Nu(

θ)

θ (degrees)

0 45 90 135 180

5

10

15

20

25

30

35

0 45 90 135 180

5

10

15

20

25

30

35

0 45 90 135 180

5

10

15

20

25

30

35

Pr =50

Pr =1

Pr =100

Pr =10

n = 0.8

Nu(

θ)

θ (degrees)

b = 0b = 0.2b = 0.5

Re = 30

(b)

n = 1.0

Nu (

θ)

θ (degrees)

n = 1.6

Nu(

θ)

θ (degrees)

Fig. 4. Effects of temperature-thinning index b on the distribution of the local Nusselt number (Nu(h)) over the surface of the cylinder at n = 0.6–0.8, 1.0 and 1.6. (a) Re = 1,Pr = 1–100 and (b) Re = 30, Pr = 1–100.

A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740 4735

b ¼ ERT1

Ts � T1T1

¼ lnl1ls

; ð20Þ

giving

l1 ¼ ebls: ð21Þ

Thus, for instance, for b = 0.2 the far-field viscosity l1 = 1.22ls, i.e.the variation in fluid viscosity from the cylinder surface (l = ls) to

the far field (l = l1) is 22% whereas, for b = 0.5, l1 = 1.65ls andthe corresponding variation in viscosity is 65%. Such relations forl1 = l1(b, ls), in conjunction with calibration curves l = l(T) foreach fluid, enable the determination of T1 for each fluid given thevalues of temperature-thinning index b and cylinder surface tem-perature Ts. Typical examples for benzene and 100 wt% glycerine,given in Table 2, clearly show the condition Ts � T1� T1 weakensslightly with a decrease in cylinder surface temperature and/or

0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

0

2

4

6

8

10

12

14

16

18

0

2

4

6

8

10

12

14

16

18n =0.8

Pr = 100

Pr = 50

Pr = 1

Nu

Re

Pr = 10

n =1.0

Nu

Re

b = 0b = 0.2b = 0.5

n =1.6

Nu

Re

Fig. 5. Variation of surface-averaged Nusselt number (Nu) with Re and b, for n = 0.8, 1.0 and 1.6.

4736 A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740

increase in temperature thinning index from b = 0.2 to b = 0.5,although it still holds good in both cases by a comfortable margin.

4.2. Effect of Reynolds number, power-law index and temperature-thinning index on the viscosity on the surface of cylinder

Representative results depicting the effects of temperature-thinning index b on dimensionless surface viscosity are shown inFig. 2 for a range of values of power-law index (n = 0.6, 1, 1.6), Rey-nolds number (Re = 1, 30) and Prandtl number (Pr = 10, 100). Sincethe effect of the Prandtl number on the surface viscosity variationwere found to be rather insignificant, no further figures are pre-sented for the other values of the Prandtl number, Pr.

As expected, the results showed the surface viscosity to be adecreasing function of the temperature-thinning index (Fig. 2),and that the dependence of the local viscosity on temperature-thinning index was seen to be less pronounced in the regions ofthe cylinder surface where the viscosity was lower, i.e., the regionsof relatively high shear rate. It was found that for Re = 1, the valueof surface viscosity for shear-thinning fluids (n = 0.6, 0.8) wasgreatest towards the front (h = 180�) and rear (h = 0�) ends of thecylinder with a minimum in-between, whereas for shear-thicken-ing fluids (n = 1.6), the reverse behaviour was observed (Fig. 2a).At high Reynolds number (Re = 30), although the trends were qual-itatively similar, a more complex behaviour was observed, with thepresence of two minima for shear-thinning fluids and two maximafor shear-thickening fluids (Fig. 2b). This is so presumably due tothe flow separation at Re = 30 while it remains attached to the sur-face at Re = 1. For Newtonian fluids (n = 1), the surface viscositywas independent of h (Fig. 2). The variation of viscosity aroundthe cylinder surface, which was significant at both n = 0.6 andn = 1.6, became less pronounced towards n = 1 where no variationin surface viscosity occurred (Fig. 2). The aforementioned featureswere present for all values of the temperature-thinning indexinvestigated in this study (b = 0, 0.2, 0.5) and the variation of sur-face viscosity with h was, for each n, qualitatively similar for all val-ues of b, and was also consistent with the similar profiles reportedby Bharti et al. [7] at Re = 5, 10, 20 and 40 using the temperature-independent viscosity approximation.

4.3. Surface vorticity distribution

Typical results showing the effects of temperature-thinning in-dex (b) and power-law index (n) on surface vorticity are presentedfor Pr = 10 and 100 in Fig. 3, for Re = 1 (Fig. 3a) and Re = 30 (Fig. 3b).The results showed that an increase in temperature-thinning indexand/or a decrease in power-law index resulted in an increase insurface vorticity (Fig. 3), and that the dependence of surface vortic-ity on temperature-thinning index became more pronounced atlower power-law indices. This is so in part due to the enhancedmobility of the fluid on both counts, namely, small value of nand large value of b. For instance at Re = 1 with Pr = 10, forn = 1.6, the maximum value of the surface vorticity increased from0.62 at b = 0 to 0.80 at b = 0.5, whereas for n = 0.6 it increased from1.89 at b = 0 to 3.03 at b = 0.5. For b – 0, i.e. temperature-depen-dent viscosity case, it was found that an increase in the value ofthe Prandtl number resulted in a slight increase in the magnitudeof the maximum surface vorticity for shear-thinning (n = 0.6, 0.8)and shear-thickening (n = 1.6) fluids (Fig. 3). Overall, the vorticityprofiles shown in the present study were qualitatively similar tothose previously obtained [3,7] for power-law fluids with the tem-perature-independent viscosity approximation.

4.4. Heat transfer characteristics

This study showed that for Pr = 1–100, Re = 1, 30 and n = 0.6,0.8–1.6, for all values of temperature-thinning parameter investi-gated, i.e. b = 0, 0.2 and 0.5, local Nusselt number Nu(h) was anincreasing function of Re and/or Pr and a decreasing function ofpower-law index n, except for Re = 30 at the rear of the cylinder(h = 0�) where the dependence of Nu(h) on n was reversed(Fig. 4). Thus, for the aforementioned conditions, non-zero valuesof temperature-dependent viscosity parameter b resulted in Nu(h)profiles which are qualitatively similar to those observed in theabsence of temperature-dependent viscosity (b = 0) by Soareset al. [4]. However, quantitatively, an increase in the value oftemperature-thinning index b always resulted in an increase invalue of Nu(h), and such an increase was more pronounced forshear-thinning fluids than that for shear-thickening fluids

Tabl

e3

Nor

mal

ized

Nus

selt

num

ber

valu

esN

u(b

=0.

2)/N

u(b

=0)

and

Nu(

b=

0.5)

/Nu(

b=

0).P

ower

-law

inde

x(n

),Re

ynol

dsnu

mbe

r(R

e),P

rand

tlnu

mbe

r(P

r).

Re

bn

=0.

8n

=1.

0n

=1.

6

Pr=

1Pr

=10

Pr=

50Pr

=10

0Pr

=1

Pr=

10Pr

=50

Pr=

100

Pr=

1Pr

=10

Pr=

50Pr

=10

0

10.

21.

0104

1.02

061.

0274

1.03

031.

0089

1.01

811.

0238

1.02

611.

0065

1.01

251.

0161

1.01

760.

51.

0259

1.05

171.

0689

1.07

661.

0229

1.04

551.

0596

1.06

561.

0163

1.03

141.

0404

1.04

41

50.

21.

0151

1.02

571.

0337

1.03

831.

0137

1.02

251.

0286

1.03

131.

0100

1.01

541.

0193

1.02

110.

51.

0374

1.06

401.

0849

1.09

601.

0340

1.05

621.

0723

1.07

961.

0250

1.03

881.

0487

1.05

33

100.

21.

0177

1.02

801.

0383

1.04

311.

0152

1.02

491.

0323

1.03

581.

0113

1.01

771.

0232

1.02

540.

51.

0413

1.07

001.

0959

1.10

811.

0376

1.06

141.

0814

1.09

101.

0282

1.04

451.

0593

1.06

48

200.

21.

0177

1.03

291.

0456

1.05

161.

0164

1.02

861.

0390

1.04

251.

0127

1.02

151.

0256

1.02

880.

51.

0432

1.08

171.

1178

1.13

481.

0405

1.07

341.

1004

1.10

981.

0317

1.05

571.

0653

1.06

96

300.

21.

0180

1.03

891.

0495

1.05

631.

0175

1.03

271.

0411

1.04

351.

0136

1.02

281.

0267

1.03

110.

51.

0441

1.09

311.

1230

1.13

921.

0427

1.08

311.

1058

1.11

721.

0340

1.05

911.

0678

1.07

26

A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740 4737

(Fig. 4). This is clearly due to the lowering of the effective viscos-ity in shear-thinning fluids near the cylinder due to its high tem-perature. Moreover, the increase in Nu(h) with b was greatestclose to the maximum of Nu(h) which was located at the frontof the cylinder near h = 180�, i.e., the location of maximum shearrate and temperature gradient (Fig. 4). At Re = 30, an increase intemperature-thinning index from b = 0 to b = 0.5 resulted, forn = 0.8, in an increase in the maximum of Nu(h) of 10% atPr = 10 and 14.2% at Pr = 100, the corresponding increases forn = 1 being 8% and 12.4% and, for n = 1.6, 6.2% and 7.7%. ForRe = 1, under the same conditions (b, Pr), the corresponding in-creases for n = 0.6 were 7.8% and 10.6%, for n = 1 were 6.5% and8%, and for n = 1.6 were 4.9% and 5.7%. This clearly suggests that,at least in part, this enhancement is also due to the progressivethinning of the thermal boundary layer with the increasing valueof the Prandtl number. Hence, in other words, at least for the re-gion near the maximum of Nu(h), the increase in heat transferwith b becomes more pronounced for higher Prandtl numbers.It was also found that the increase in Nu(h) with b was least closeto the minimum value of Nu(h) which was located downstream ofthe cylinder (Fig. 4). It is clear from the above discussion that theextent of the increase of Nusselt number with b is determined bya complex interplay between the kinematic (Re, Pr) and physical(b, n) characteristics of the system at each point on the cylindersurface, and to better understand the overall effect of these char-acteristics on heat transfer, it is both useful and convenient to usethe mean value of the Nusselt number (averaged over the cylin-der surface), thus eliminating one variable from the functionaldependence. The results showed that the surface-averaged Nus-selt number Nu was an increasing function of temperature-thin-ning index b (Fig. 5). For b = 0, 0.2 and 0.5, the surface-averagedNusselt number showed positive dependence on the Reynoldsnumber Re and on Prandtl number Pr, and was a decreasing func-tion of the power-law index n (Fig. 5), thus extending to the tem-perature-dependent viscosity case (b – 0) a result which hadpreviously only been reported for power-law fluids [4] usingthe temperature-independent viscosity approximation (b = 0).Furthermore, the dependence on the Reynolds number was foundto be stronger than that on the Prandtl number. Such an increasein Nu with Re and/or Pr can be explained in terms of the in-creased contribution of convection to heat transfer which resultsfrom an increase in Peclet number Pe = RePr. Likewise, an increasein the value of b and/or a decrease in the value of power-law in-dex n, cause a decrease in the effective viscosity and, therefore,increased convection, again resulting in larger values of the Nus-selt number, Nu.

Further examination of the results revealed that the increase inthe value of the average Nusselt number Nu with b was more pro-nounced for lower values of power-law index (Fig. 5), in line withthe Nu(h) profiles (Fig. 4). The increase in the value of the averageNusselt number Nu with b was also more pronounced for largervalues of Re and/or Pr (Fig. 5), again due to the gradual thinningof the thermal boundary layer. For instance, at Pr = 1, an increasein the temperature-thinning index from b = 0 to b = 0.5 resulted,for n = 0.8, in an increase in Nu of 2.6% at Re = 1 and 4.4% atRe = 30; the corresponding increases for n = 1 were 2.3% and4.3%, and for n = 1.6 were 1.6% and 3.4%. For Pr = 100 under thesame conditions (b, Re), the corresponding increases for n = 0.8were 7.7% and 13.9%, for n = 1 were 6.6% and 11.7%, and forn = 1.6 were 4.4% and 7.3%. Thus, ignoring the temperature-dependent viscosity leads to an underestimate in Nu, and aban-doning such restriction can lead to a correction of as much as14% in the value of Nu. Further estimates of these correctionscan be computed using Table 3.

The aforementioned results clearly show that the increase invalue of the average Nusselt number Nu with increase in b and/

0

1

2

3

-3 -1 1 3 5-3

-2

-1

0

b = 0.2

0

1

2

3

-3 -1 1 3 5-3

-2

-1

0

b = 0.5

0

1

2

3

-3 -1 1 3 5-3

-2

-1

0

Re = 1, Pr = 100b = 0

0

1

2

3

-3

-2

-1

0 0

1

2

3

-3

-2

-1

00

1

2

3

-3 -1 1 3 5 -3 -1 1 3 5-3 -1 1 3 5-3

-2

-1

0

n =

0.6

n =

1.0

0

1

2

3

-3

-2

-1

0 0

1

2

3

-3

-2

-1

00

1

2

3

-3 -1 1 3 5 -3 -1 1 3 5-3 -1 1 3 5-3

-2

-1

0

n =

1.6

Fig. 6. Effects of b and n on the streamlines (top) and isothermals (bottom) for Re = 1 and Pr = 100. Flow direction left to right.

4738 A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740

or decrease in n became more pronounced for larger values ofPeclet number, Pe. Since heat transfer by conduction is indepen-dent of fluid viscosity, this trend is consistent with the decreasein the contribution of conduction to heat transfer which resultsfrom an increase in Peclet number.

Representative plots of the streamlines and isotherm contoursfor Re = 1 with Pr = 100 (Fig. 6), and Re = 30 with Pr = 10 (Fig. 7),showed that for a fixed value of the temperature-thinning indexb, an increase in power-law index resulted in a slight increasein distance between the streamlines, and between the isothermallines, in the downstream region. Qualitatively similar trends havealso been reported in previous studies, based on the approxima-tion of temperature-independent viscosity for power-law fluids[3,4]. Moreover, for a fixed value of power-law index n, an in-crease in temperature-thinning index b did not seem to have pro-duced any significant change in the qualitative nature of thestreamline and isotherm patterns (Figs. 6 and 7). Both the afore-mentioned observations indicate that the effect of temperature-thinning index on the streamline and isotherm patterns is quitelimited. Moreover, further observations at Re = 30 with Pr = 10(Fig. 7) showed similar increase in wake length with power-lawindex for all values of temperature-thinning index. For Re = 30and Pr = 10, the wake length was found to be almost independentof temperature-thinning index for shear-thickening (n = 1.6) andNewtonian (n = 1) fluids, although for shear-thinning fluids(n = 0.8) there was a slight increase in wake length with the tem-perature-thinning index, b.

4.5. Drag coefficient

The dependence of the drag coefficient on the dimensionlessgroups Re and Pr as well as on the physical characteristics (b, n)of the system is shown in Fig. 8. The results showed that forn = 0.8, 1.0 and 1.6, an increase in the temperature-thinning indexfrom b = 0 to 0.5 caused a decrease in the drag coefficient Cd(Fig. 8). Such effect can be attributed to a reduction in the fric-tional drag component due to the overall decrease in the effectiveviscosity which resulted from larger temperature-thinning index.Moreover, such dependence of drag coefficient on temperature-thinning index became more pronounced as the Prandtl numberdecreased from Pr = 100 to Pr = 1. The latter trend can qualita-tively be explained by the decrease in compactness of the iso-therms with the thickening of the boundary layer, resulting inlower temperature gradients close to the cylinder surface, andtherefore increased the effect of cylinder temperature on fluidtemperature. In turn the more pronounced effect on fluid temper-ature, and thus on effective viscosity, resulted in a more pro-nounced decrease in the value of Cd with b at lower Pr values.It was also found that the dependence of drag coefficient on thetemperature-thinning index became more pronounced at lowerRe values (Fig. 8). Finally, the dependence of drag coefficient onthe temperature-thinning index became more pronounced at low-er values of the power-law index (Fig. 8), consistent with thegreater dependence of the surface vorticity on the temperature-thinning index shown in Fig. 3.

0

1

2

3

-3

-2

-1

0 0

1

2

3

-3

-2

-1

0 0

1

2

3

-3

-2

-1

0

Re = 30, Pr = 10b = 0

0

1

2

3

-3

-2

-1

0 0

1

2

3

-3

-2

-1

0 0

1

2

3

-3

-2

-1

0

b = 0.2

0

1

2

3

-3 -1 1 3 5 7-3

-2

-1

0 0

1

2

3

-3 -1 1 3 5 7-3

-2

-1

0 0

1

2

3

-3 -1 1 3 5 7

-3 -1 1 3 5 7 -3 -1 1 3 5 7 -3 -1 1 3 5 7

-3 -1 1 3 5 7 -3 -1 1 3 5 7 -3 -1 1 3 5 7

-3

-2

-1

0

b = 0.5

n =

0.8

n =

1.0

n =

1.6

Fig. 7. Effects of b and n on the streamlines (top) and isothermals (bottom) for Re = 30 and Pr = 10. Flow direction left to right.

0 5 10 15 20 25 300

2

4

6

8

10

12

0 5 10 15 20 25 300

2

4

6

8

10

12

0 5 10 15 20 25 300

2

4

6

8

10

12n = 0.8

Cd

Re

n = 1.0

Cd

Re

n = 1.6

b = 0b = 0.5, Pr = 100b = 0.5, Pr = 1

Cd

Re

Fig. 8. Variation of drag coefficient (Cd) with Re and b, for n = 0.8, 1.0 and 1.6.

A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740 4739

Before closing this discussion, it needs to be emphasized herethat while the assumption of forced convection affords a great sim-plification, it also restricts the applicability of the present results tosituations where the difference in densities between the cylinderand the surrounding fluid is sufficiently small for the imposedvelocity to be large in comparison to that induced by buoyancy,and/or the density of the fluid is much less sensitive than viscosityto temperature variation over the range of interest. This assump-

tion is much less serious for liquids (both Newtonian and power-law) than for gases. Similarly, the flow has been assumed to besteady up to Re = 30 for the highest value of n used here, i.e.,n = 1.6. This is probably justified as the flow is known to be steadyup to Re = 36–37 for this value for b = 0. Lastly, it should be borne inmind that these are the very first set of results elucidating the ef-fects of temperature-dependent viscosity for a circular cylinder inpower-law fluids. Naturally, these will get superseded by future

4740 A.A. Soares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4728–4740

studies incorporating the effect of temperature-dependent densityand/or delineating the limits of the steady flow regime under theseconditions.

5. Conclusions

The steady crossflow of power-law fluids over an isothermallyheated circular cylinder has been investigated numerically todetermine the effect of the temperature-dependent viscosity onthe flow and heat transfer characteristics using temperature-thin-ning indices 0 6 b 6 0.5, for a wide range of power-law indices(0.6 6 n 6 1.6), Reynolds numbers (1 6 Re 6 30) and Prandtl num-bers (1 6 Pr 6 100). This study showed that the surface viscositydecreased with the temperature-thinning index and that suchdependence was more pronounced in regions of higher surface vis-cosity or of low shearing. An increase in the temperature-thinningindex b resulted in a decrease in drag coefficient and increased sur-face vorticity, but both the drag coefficient and surface vorticityprofiles were qualitatively similar to those observed for the caseof the temperature-independent viscosity approximation (i.e.b = 0). A decrease in the power-law index was found to increasethe magnitudes of both drag coefficient and surface vorticity, andalso increased their dependence on temperature-thinning index.The dependence of drag coefficient on temperature-thinning indexwas also more pronounced at lower Reynolds and Prandtl num-bers. Both the local Nusselt number Nu(h) and the surface-aver-aged Nusselt number Nu were increasing functions of thetemperature-thinning index b, and their dependence on b becamemore pronounced for lower values of n. The dependence of Nu on bwas also more pronounced for larger Reynolds and Prandtl or Pec-let numbers. For b – 0, the profiles of both Nu(h) and Nu werefound to be qualitatively similar to those for b = 0. The study ofstreamline and isotherm patterns showed that, for each power-law index, an increase in temperature-thinning index did not pro-duce any significant variation in either of these patterns. For eachtemperature-thinning index value, an increase in power-law indexresulted in slight increase in distance between adjacent stream-lines, as well as between the adjacent isotherms, in the down-stream region.

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