International Journal of Heat and Mass Transfer 54 (2011) 4069–4077

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International Journal of Heat and Mass Transfer

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Heat transfer due to electroosmotic flow of viscoelastic fluids in a slit microchannel

Arman Sadeghi, Mohammad Hassan Saidi ⇑, Ali Asghar MozafariCenter of Excellence in Energy Conversion (CEEC), School of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran

a r t i c l e i n f o

Article history:Received 7 May 2010Received in revised form 17 March 2011Accepted 21 March 2011Available online 4 May 2011

Keywords:Electroosmotic flowMicrochannelJoule heatingViscous dissipationPTT modelFENE-P model

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.04.004

⇑ Corresponding author. Tel.: +98 21 66165522; faxE-mail addresses: armansadeghi@mech.sharif.edu

edu (M.H. Saidi), mozafari@sharif.edu (A.A. Mozafari)

a b s t r a c t

The bio-microfluidic systems are usually encountered with non-Newtonian behaviors of working fluids.The rheological behavior of some bio-fluids can be described by differential viscoelastic constitutiveequations that are related to PTT and FENE-P models. In the present work, thermal transport character-istics of the steady fully developed electroosmotic flow of these fluids in a slit microchannel with con-stant wall heat fluxes have been investigated. The Debye–Huckel linearization is adopted and theeffects of viscous dissipation and Joule heating are taken into account. Analytical solutions are obtainedfor the transverse distributions of velocity and temperature and finally for Nusselt number. Two differentbehaviors are observed for the Nusselt number variations due to increasing �geWe2 which are an increas-ing trend for positive wall heat flux and a decreasing one for negative wall heat flux. However, the influ-ence of �geWe2 on Nusselt number vanishes at higher values of the dimensionless Debye–Huckelparameter. It is also realized that the effect of viscous heating is more important at small values of both�geWe2 and the dimensionless Debye–Huckel parameter. Furthermore, the results show a singularity inNusselt number at higher negative values of the dimensionless Joule heating parameter.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In the recent decades, micronsize mechanical and biochemicaldevices have become more prevalent both in commercial applica-tions and in scientific investigations. Transport phenomena atmicroscale reveal many features that are not observed in macro-scale devices. These features are quite different for gas and liquidflows. In gas microflows we encounter four important effects: com-pressibility, viscous heating, thermal creep and rarefaction [1]. Rar-efaction effects are treated using slip velocity and temperaturejump boundary conditions at solid surfaces. For liquid flow throughmicrochannels, the classical boundary conditions of no slip velocityand no temperature jump are quite accurate [1]. However, liquidflows are encountered with other microscale features such as sur-face tension and electroosmotic effects. Electroosmosis refers to li-quid flow induced by an electric field along electrostaticallycharged surfaces. The electric field may be the result of externalor flow induced potentials. The electrokinetic effect due to the flowinduced potential is unfavorable, as it causes moving the chargesand molecules in the opposite direction of the flow, creating extraimpedance to the flow motion. Nevertheless, electroosmosis hasmany applications in sample collection, detection, mixing and sep-aration of various biological and chemical species. Another andprobably the most important application of electroosmosis is the

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: +98 21 66000021.(A. Sadeghi), saman@sharif.

.

fluid delivery in microscale at which the electroosmotic micro-pump has many advantages over other types of micropumps. Elec-troosmotic pumps are bi directional, can generate constant andpulse free flows with flow rates well suited to microsystems andcan be readily integrated with lab-on-a-chip devices. The magni-tude and the direction of flow of an electroosmotic pump can bechanged instantly [2]. In addition, electroosmotic pumps have nomoving parts and have much simpler design and easier fabrication.

Hydrodynamic aspects of electroosmotic flow in ultrafine capil-lary slits were analyzed by Burgreen and Nakache [3]. Rice andWhitehead [4] investigated fully developed electroosmotic flowin a narrow cylindrical capillary for low zeta potentials, using theDebye–Huckel linearization. Levine et al. [5] extended the Riceand Whitehead’s work to high zeta potentials by means of anapproximation method originally proposed by Philip and Wooding[6]. More recently, an analytical solution for electroosmotic flow ina cylindrical capillary was derived by Kang et al. [7] by solving thecomplete Poisson–Boltzmann equation for arbitrary zeta-poten-tials. Hydrodynamic aspects of fully developed electroosmotic flowin a semicircular microchannel were studied by Wang et al. [8].Analytical series solutions were found for two basic cases whichcan be superposed to yield solutions for any combination of con-stant zeta potentials on the flat or curved wall boundaries. Xuanand Li [9] developed general solutions for electrokinetic flow inmicrochannels with arbitrary geometry and arbitrary distributionof wall charge. Electroosmotic flow in parallel plate microchannelsfor the cases in which electric double layers interact witheach other was analyzed by Talapatra and Chakraborty [10].

Nomenclature

cp specific heat at constant pressure (kJ kg�1 K�1)Dh hydraulic diameter of channel (=4H)e proton charge (C)Ex electric field in the axial direction (V m�1)F body force vector (N m�3)G dimensionless electric field (Eq. (12))h heat transfer coefficient (W m�2 K�1)H half channel height (m)k thermal conductivity (W m�1 K�1)kB Boltzmann constant (J K�1)n0 ion density (m�3)Nu Nusselt number (=hDh/k)p pressure (Pa)q dumbbell extensibility (FENE-P model)q00 wall heat flux (W m�2)s volumetric heat generation due to Joule heating

(W m�3)S dimensionless Joule heating term (¼ E2

x H=q00r0)Sv dimensionless viscous heating parameter

(¼ gger0U2=E2x k2

D)t time (s)T temperature (K)u axial velocity (m s�1)u⁄ dimensionless axial velocity (=u/U)u velocity vector (m s�1)U mean velocity (m s�1)We Weissenberg number (¼ kgeU=H)x axial coordinate (m)

y transverse coordinate (m)y⁄ dimensionless transverse coordinate (=y/H)z valence number of ions in solution

Greek symbols� PTT parameterf wall zeta potential (V)f⁄ dimensionless wall zeta potentialg viscosity coefficient (kg m�1 s�1)h dimensionless temperature (Eq. (28))K dimensionless Debye–Huckel parameter (¼ H=kD)k relaxation time (s)kD Debye length (m)q density (kg m�3)qe net electric charge density (C m�3)r liquid electrical resistivity (X m)s�xy dimensionless shear stress (=Hsxy/ggeU)s stress tensor (Pa)w EDL potential (V)w⁄ dimensionless EDL potential (=ezw/kBTav)

Subscriptsav averageb bulkc criticalge generalizedw wall0 neutral liquid

4070 A. Sadeghi et al. / International Journal of Heat and Mass Transfer 54 (2011) 4069–4077

Hydrodynamically developing flow between two parallel plates forelectroosmotically generated flow has been reported in a numeri-cal study by Yang et al. [11]. Also several researches have been per-formed to study heat transfer characteristics of electroosmoticflow. Maynes and Webb [12] analytically have studied fully devel-oped electroosmotically generated convective transport for a par-allel plate microchannel and circular microtube under imposedconstant wall heat flux and constant wall temperature boundaryconditions. Yang et al. [13] investigated forced convection in rect-angular ducts with electrokinetic effects for both hydrodynami-cally and thermally fully developed flow. They investigated theeffects of streaming potential on flow and heat transfer.

All the foregoing studies are related to Newtonian fluids. Never-theless, the bio-microfluidic systems are usually encountered withnon-Newtonian behaviors of working fluids for which other consti-tutive equations rather than Newton’s law of viscosity are needed.In the literature, various models have been proposed to analyzenon-Newtonian fluid flow behavior such as power-law model[14], Moldflow first order model [15], Bingham model [16], Eyringmodel [17] and PTT and FENE-P models [18]. The study of electro-osmotic flow of non-Newtonian fluids is new and the open litera-ture shows a limited number of relevant papers. Berli andOlivares [17] theoretically studied the electrokinetic flow of differ-ent non-Newtonian fluids through slit and cylindrical microchan-nels, using three constitutive equations comprising power-lawmodel, Bingham model and Eyring model. The resulting equationsallow one to predict the flow rate and electric current as functionsof the simultaneously applied electric potential and pressure gradi-ents. Electroviscous effects in steady, fully developed, pressure-driven flow of power-law liquids through a cylindrical microchan-nel have numerically been investigated by Bharti et al. [19], using afinite difference method. With the implementation of an approxi-mate scheme for the hyperbolic sine function initially introducedby Philip and Wooding [6], an approximate analytical solution for

velocity distribution in electroosmotic flow of power-law fluidsin slit microchannels has been presented by Zhao et al. [20]. Anumerical study of electroosmotic flow in parallel plate micro-channels considering the power-law non-Newtonian fluid has beencarried out by Tang et al. [21]. The simulation results showed thatthe fluid rheological behavior is capable of significantly changingthe electroosmotic flow pattern and the flow behavior index playsan important role. In a recent study, Afonso et al. [18] developedclosed from solutions for hydrodynamic characteristics of com-bined electroosmotically and pressure driven flow of two visco-elastic fluids, namely, the PTT and FENE-P models. To theauthors’ best knowledge, the only research work considering thethermal transport features of non-Newtonian fluids electroosmoticflow has been undertaken by Das and Chakraborty [22] whichstudied electrokinetic effects in fully developed flow of power-law fluids in parallel plate microchannels. They derived solutionsfor the transverse distributions of velocity, temperature and soluteconcentration. However, the Nusselt number which is an impor-tant parameter in design and active control of microdevices wasnot considered.

The rheological behavior of some bio-fluids can be described bydifferential viscoelastic constitutive equations that are related tothe PTT and FENE-P models, as in the case of blood [23], saliva[24], synovial fluid [25] or other biofluids containing long chainmolecules. This motivated us to analyze the electroosmotic flowof these models in the present work. In most lab-on-a-chip sys-tems, the cross section of microchannels made by modernmicromachining technology is close to a rectangular shape[26,27] and it may be effectively represented by a two dimensionalslit when the width is much larger than the height. Therefore, forconvenience of analysis, a slit microchannel having constant heatfluxes on its walls is considered and the effects of Joule heatingand viscous dissipation are taking into account. Although accord-ing to Tang et al. [28], in general, a conjugate heat transfer problem

A. Sadeghi et al. / International Journal of Heat and Mass Transfer 54 (2011) 4069–4077 4071

has to be solved to simultaneously account for heat transfer in boththe liquid and the channel wall; however, as shown by Bejan [29], aconjugate heat transfer problem may be reduced to the classicalboundary condition of constant heat flux when the ratio of theexternal to internal heat transfer coefficients is very small. For a slitmicrochannel surrounded by stagnant air, the external heat trans-fer coefficient due to free convection is much smaller than theinternal one which takes high values at microscale. Therefore, aconstant heat flux boundary condition is consistent with heattransfer physics of the practical applications. Surprisingly, theanalysis presented here can also cover the other classical boundarycondition of constant temperature, as is a special case of constantheat flux boundary condition in the presence of internal heating.The governing equations for fully developed conditions are firstmade dimensionless and then closed form expressions are ob-tained for the transverse distributions of velocity and temperatureand also for Nusselt number. The interactive effects of flow param-eters on the temperature field and Nusselt number are shown ingraphical form and also discussed in detail.

2. Problem formulation

2.1. Problem assumptions

Consider the situation where both hydrodynamically and ther-mally fully developed electroosmotic flow of a viscoelastic fluidtakes place through a slit microchannel with channel half heightof H. An illustration of the problem is depicted in Fig. 1. In the anal-ysis the following assumptions are considered:

� Thermophysical properties are constant in the whole domainincluding the EDL.� Constant values of the heat flux and zeta potential are consid-

ered at the walls.� Liquid contains an ideal solution of fully dissociated symmetric

salt.� The charge in the EDL follows Boltzmann distribution.� Wall potentials are considered low enough for Debye–Huckel

linearization to be valid.� In calculating the charge density, it is assumed that the temper-

ature variation over the channel cross section is negligible com-pared with the absolute temperature. Therefore, the chargedensity field is calculated on the basis of an averagetemperature.

Fig. 1. Geometry of the physical problem, coo

� The Fourier law for heat conduction is valid for viscoelastic flu-ids being considered here.

2.2. Velocity distribution

The constitutive equations to be considered here are the simpli-fied PTT model derived by Phan-Thien and Tanner [30] from net-work theory arguments and the FENE-P model based on thekinetic theory for finitely extensible dumbbells with a Peterlinapproximation for the average spring force [31]. The presentationof the constitutive equation for both of these models has been gi-ven in detail by Afonso et al. [18] and it is omitted here to savespace. They showed that at fully developed conditions, there isan exact equivalence of the both models solutions in the sense ofa parameter to parameter match. Therefore, for convenience ofanalysis, it is useful to define generalized parameters and performan analysis based on these generalized parameters instead of twoanalyses for both models. The following generalized parametersare introduced for FENE-P model [18]

kge;FENE-P ¼ kqþ 2qþ 5

ð1Þ

�ge;FENE-P ¼1

qþ 5ð2Þ

gge;FENE-P ¼ g ð3Þ

in which k is the relaxation time of the fluid, q is a parameter thatmeasures the extensibility of the dumbbell, g is the viscosity coef-ficient, and � is a parameter that imposes an upper limit to the elon-gational viscosity. For PTT model, the generalized parameters arethe same as those belonging to the model, i.e.,

kge;PTT ¼ k; �ge;PTT ¼ �; gge;PTT ¼ g ð4Þ

At fully developed conditions, the transverse velocity vanishes;therefore, the velocity vector may be written as u = {u(y), 0}. Basedon the findings of Afonso et al. [18], the axial velocity gradient canbe related to the shear stress, sxy, as

dudy¼ 1

gge1þ 2

�gek2ge

g2ge

s2xy

!sxy ð5Þ

rdinate system and electric double layer.

4072 A. Sadeghi et al. / International Journal of Heat and Mass Transfer 54 (2011) 4069–4077

and in dimensionless form

du�

dy�¼ s�xy þ 2�geWe2s�3

xy ð6Þ

where y� ¼ y=H; u� ¼ u=U; s�xy ¼ Hsxy=ggeU, and We is the Weiss-enberg number, a measure of the level of elasticity in the fluid,which is given by

We ¼ kgeUH

ð7Þ

with U being the mean velocity.The momentum exchange through the flow field is governed by

the Cauchy equation

qDuDt¼ �rpþr � sþ F ð8Þ

in which q denotes the density, p represents the pressure, s is thestress tensor, and F is the body force vector. Here, the body forceacts in the x direction and equals qeEx with Ex denoting the electricfield and qe representing the net electric charge density. Based onthe Debye–Huckel linearization, the net electric charge densitymay be written as [32]

qe ¼ �2n0e2z2

kBTavw ð9Þ

where n0 is the ion density, e is the proton charge, z is the valencenumber of ions in solution, w is the EDL potential, kB is the Boltz-mann constant, and Tav is the average absolute temperature overthe channel cross section. It should be pointed out that, assuminga univalent solution at 25 �C, the Debye–Huckel linearization is va-lid for zeta potentials below 25 mV which is usually an upper limitfor bio-applications [33–38]. At fully developed conditions Du/Dt = 0 and the pressure gradient is absent in purely electroosmoticflow. Therefore, the momentum equation in the axial direction is re-duced to

dsxy

dy¼ �qeEx ¼

2n0e2z2Ex

kBTavw ð10Þ

Using dimensionless parameters, the momentum equation (10)may be written as

ds�xy

dy�¼ Gw� ð11Þ

in which w� ¼ ezw=kBTav and dimensionless electric field, G, is givenby

G ¼ 2n0ezH2Ex

ggeUð12Þ

The dimensionless EDL potential w⁄ from our previous work dealingwith electroosmotic flow of Newtonian fluids [32] may be writtenas

w� ¼ f�coshðKy�Þ

cosh Kð13Þ

in which f⁄ is the dimensionless wall zeta potential, i.e., f⁄ = ezf/kBTav and K represents the dimensionless Debye–Huckel parametergiven by K ¼ H=kD with kD being the Debye length. By substitutingw⁄ from the above equation and noting that the shear stress at thecenterline is zero, Eq. (11) can be integrated to yield the followingdistribution for dimensionless shear stress

s�xy ¼Gf�

K cosh KsinhðKy�Þ ð14Þ

Using the above expression for the dimensionless shear stress, thedimensionless velocity gradient from Eq. (6) becomes

du�

dy�¼ Gf�

K cosh KsinhðKy�Þ þ 2�geWe2 Gf�

K cosh KsinhðKy�Þ

� �3

ð15Þ

Eq. (15) can be integrated subject to the no slip boundary conditionat the wall ðu�ð1Þ ¼ 0Þ and the resulting velocity profile is

u� ¼ aþ b coshðKy�Þ þ c coshðKy�Þsinh2ðKy�Þ ð16Þ

where

a ¼ �Gf�

K2 þ2�geWe2

3KGf�

K cosh K

� �3

ð2 cosh K � cosh Ksinh2KÞ ð17Þ

b ¼ 1K

Gf�

K cosh K

� �� 4�geWe2

3KGf�

K cosh K

� �3

ð18Þ

c ¼ 2�geWe2

3KGf�

K cosh K

� �3

ð19Þ

Since in the definitions of u⁄, G and We we have used the meanvelocity, the two latest parameters are not independent. Theirdependency can be obtained invoking the fact that average dimen-sionless velocity over the cross section of the channel is equal tounity. The dependency of G and We then may be written in compactform as

a0Gf�

K cosh K

� �3

þ b0Gf�

K cosh K

� �þ 1 ¼ 0 ð20Þ

in which

a0 ¼ 2�geWe2

3Kcosh Ksinh2K � 2 cosh K � sinh3K

3Kþ 2

sinh KK

!

ð21Þ

b0 ¼ cosh KK

� sinh K

K2 ð22Þ

From the general formulas for the roots of algebraic cubic equa-tions, it can be readily shown that the real solution of Eq. (20) is

Gf�

K cosh K¼ 1

3a0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð�27a02 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi729a04 þ 108a03b03

qÞ

3

r"

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð�27a02 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi729a04 þ 108a03b03

qÞ

3

r #ð23Þ

Therefore, the dimensionless velocity distribution depends only ontwo parameters which are K and �geWe2.

2.3. Temperature distribution

The conservation of energy including the effects of viscous dis-sipation and Joule heating provides

qcpDTDt¼ r � ðkrTÞ þ ru : sþ s ð24Þ

In the above equation, s and ru:s denote the rate of volumetricheat generation due to Joule heating and viscous dissipation,respectively. The Joule heating term equals s ¼ E2

x=r with r beingthe liquid electrical resistivity given by [5]

r ¼ r0

cosh w�ð25Þ

in which r0 is the electrical resistivity of the neutral liquid. Thehyperbolic term in the above equation accounts for the fact thatthe resistivity within the EDL is lower than that of the neutral li-quid, due to an excess of ions close to the surface. For low wall zetapotentials, which is the case in this study, cosh w⁄? 1 and the Joule

A. Sadeghi et al. / International Journal of Heat and Mass Transfer 54 (2011) 4069–4077 4073

heating term may be considered as the constant value of s ¼ E2x=r0

[39]. For steady fully developed flow DT/Dt = u(oT/ox) andru:s = sxy(du/dy). Therefore, the energy equation (24) becomes

qcpu@T@x¼ k

@2T@x2 þ

@2T@y2

!þ sxy

dudyþ E2

x

r0ð26Þ

The relevant boundary conditions for the energy equation are asfollows

@T@y

� �ðx;0Þ¼ 0 Tðx;HÞ ¼ TwðxÞ and k

@T@y

� �ðx;HÞ¼ q00 ð27Þ

The dimensionless temperature h is introduced in the following,which depends only on y for fully developed flow

hðyÞ ¼ T � Twq00H

k

ð28Þ

Taking differentiation of Eq. (28) with respect to x gives

@T@x¼ dTw

dx¼ dTb

dxð29Þ

in which Tb is the bulk temperature. From an energy balance on alength of duct dx, the following expression is obtained for dTb/dx

dTb

dx¼ q00

qcpUH1þ Sþ b

SSv

K2

� �ð30Þ

where Sv ¼ gger0U2=E2x k

2D is the dimensionless viscous heating

parameter, S ¼ E2x H=q00r0 is the dimensionless volumetric heat gen-

eration due to Joule heating, and

b ¼Z 1

0s�xy

du�

dy�dy�

¼ Gf�

K cosh K

� �c8� b

2

� �K þ ðb� cÞ sinhð2KÞ

4þ 3c

32sinhð4KÞ

� �ð31Þ

Since oT/ox is constant, the axial conduction term in the energyequation vanishes. Therefore, the energy equation in dimensionlessform may be written as

d2hdy�2

¼ 1þ Sþ bSSv

K2

� �u� � S� SSv

K2 s�xydu�

dy�ð32Þ

By substituting s�xy from Eq. (14) and using Eq. (16), we come upwith the following dimensionless energy equation

d2hdy�2

¼ A1 þ A2 coshðKy�Þ þ A3 coshð2Ky�Þ þ A4

� coshðKy�Þsinh2ðKy�Þ þ A5sinh4ðKy�Þ ð33Þ

where the coefficients A1 to A5 are given by

A1 ¼a 1þ Sþ bSSv

K2

� �� Sþ ðbþ 2cÞ SSv

2KGf�

K cosh K

� �; A2

¼b 1þ Sþ bSSv

K2

� �; A3 ¼ �ðbþ 2cÞ SSv

2KGf�

K cosh K

� �;

A4 ¼c 1þ Sþ bSSv

K2

� �; A5 ¼ �3c

SSv

KGf�

K cosh K

� �ð34Þ

The thermal boundary conditions in the dimensionless form arewritten as

dhdy�

� �ð0Þ¼ 0; hð1Þ ¼ 0 ð35Þ

After integrating Eq. (33) twice and applying the above boundaryconditions, the following dimensionless temperature distributionis obtained

h ¼ B1y�2 þ B2 coshðKy�Þ þ B3 coshð2Ky�Þ þ B4

� coshðKy�Þsinh2ðKy�Þ þ B5sinh4ðKy�Þ þ B6 ð36Þ

in which

B1 ¼A1

2þ 3

16A5; B2 ¼

A2

K2 �2A4

9K2 ; B3 ¼A3

4K2 �3A5

32K2 ;

B4 ¼A4

9K2 ; B5 ¼A5

16K2 ; B6 ¼ �B1 � B2 cosh K � B3 coshð2KÞ

� B4 cosh Ksinh2K � B5sinh4K ð37Þ

Once the temperature distribution is obtained, the quantities ofphysical interest, including the bulk temperature of the fluid andthe heat transfer rate can be obtained. The dimensionless bulk tem-perature is given by

hb ¼R 1

0 u�hdy�R 10 u�dy�

¼Z 1

0u�hdy� ¼ aC1 þ bC2 þ cC3 þ B6 ð38Þ

where

C1 ¼B1

3þ 3

8B5 þ B2

sinh KKþ B3

2K� 3B5

16K

� �sinhð2KÞ

þ B4

3Ksinh3K þ B5

4Kcosh Ksinh3K

C2 ¼B2

2� B4

8þ 1

Kþ 2

K3

� �B1 þ

B3

3Kþ B5

5K

� �sinh K � 2B1

K2 cosh K

þ B2

4K� B4

16K

� �sinhð2KÞ þ 2

B3

3K� B5

5K

� �sinh Kcosh2K

þ B4

4Ksinh Kcosh3K þ B5

5Ksinh Kcosh4K

C3 ¼ �B2

8þ B4

16� 1

3Kþ 14

27K3

� �B1 �

B3

15Kþ B5

7K

� �sinh K

þ B4

32K� B2

16K

� �sinhð2KÞ þ 2B1

3K2 cosh K � 2B1

9K2 cosh3K

þ 13Kþ 2

27K3

� �B1 �

7B3

15Kþ 3B5

7K

� �sinh Kcosh2K

þ B2

4K� 7B4

24K

� �sinh Kcosh3K þ 2B3

5K� 3B5

7K

� �sinh Kcosh4K

þ B4

6Ksinh Kcosh5K þ B5

7Ksinh Kcosh6K

ð39Þ

The heat transfer rate can be expressed in terms of Nusselt numberas

Nu ¼ hDh

k¼ q00Dh

kðTw � TbÞ¼ � 4

hbð40Þ

with Dh = 4H. It is noteworthy that for Newtonian behavior, i.e.,�geWe2 = 0, the Nusselt number obtained in the present study andthe one given in our previous work [32] are exactly identical.

It is common in bio-applications that the wall heat flux takesnegative values. This occurs in cases that a fraction of the energygenerated by internal heating is dissipated through the walls.When all the internal heating is dissipated through the wall, theaxial variation of temperature vanishes, i.e., oT/ox = 0. This is tosay that, the classical boundary condition of constant wall temper-ature has been recovered from the constant wall heat flux bound-ary condition. It is worth mentioning that although any negativevalue of the wall heat flux is plausible, there is just a particularvalue of the wall heat flux which corresponds to a constant wall

Fig. 3. Transverse distribution of dimensionless temperature at different values ofSv (a) S = 5 and (b) S = �5.

4074 A. Sadeghi et al. / International Journal of Heat and Mass Transfer 54 (2011) 4069–4077

temperature boundary condition and it is a function of the totalJoule heating and viscous dissipation as

q00 ¼ �E2x H=r0 � bggeU2=H ð41Þ

Also for this case, according to Eq. (30), the values of S, Sv, and K arenot independent.

3. Results and discussion

It has been shown that the main parameters governing heat andfluid flow in fully developed electroosmotic flow of viscoelastic flu-ids being considered here in a slit microchannel are K, �geWe2, S andSv. Here, their interactive effects on the transverse distribution oftemperature and Nusselt number are analyzed. Although a nega-tive S is more encountered in practice, however, for the sake ofgenerality, both negative and positive values of the dimensionlessJoule heating parameter are considered. Based on the practicalranges of the electroosmotic velocity and channel height reportedby Karniadakis et al. [1], and also the reported values of the relax-ation time for human blood [40], a wide range of 0–1000 is consid-ered for �geWe2. It is also worth mentioning that the chosen valuesof Sv are those used by Maynes and Webb [41].

The transverse distribution of dimensionless temperature at dif-ferent values of S in the absence of viscous heating is presented inFig. 2. Both positive and negative values of the wall heat flux areconsidered. Positive values of dimensionless Joule heating termcorrespond to the wall cooling case where heat is transferred fromthe wall to the fluid, while the opposite is true for negative valuesof S. In the absence of Joule heating, the temperature distribution isindependent of whether the wall is heated or cooled. As observed,increasing values of S lead to lower values of dimensionless tem-perature which implies that Joule heating increases the wall tem-perature rather than the bulk temperature. The reason is thatalthough the distribution of energy generated by Joule heating isuniform throughout the channel cross section, but the energytransferred by the flow decreases near the wall and it equals zeroat the wall.

Fig. 3 demonstrates the transverse distribution of dimensionlesstemperature at different values of Sv. Different trends are observedfor wall cooling and heating cases. To increase viscous heating ef-fects is to decrease dimensionless temperature for wall cooling,while the opposite is true for wall heating. Viscous dissipation be-haves like an energy source increasing the temperature of the fluidespecially near the wall, since the highest shear rates occur at this

Fig. 2. Transverse distribution of dimensionless temperature at different values of Sin the absence of viscous heating.

region, while it is zero at centerline. Therefore, the maximum tem-perature rise occurs at the wall. For wall cooling case, the maxi-mum temperature occurs at the wall. So, increasing the walltemperature results in increasing the difference between tempera-tures of the wall and the fluid particles, while the opposite is truefor wall heating case, since for this case the temperature of the wallis the minimum in temperature field. This is why the trends aredifferent for two cases. It can be seen that for S = �5, as Sv in-creases, the sign of the dimensionless bulk temperature is changedfrom negative to positive. So, for a value of the dimensionless vis-cous heating parameter called Sv,c, which depends on flow param-eters, the value of the dimensionless bulk temperature will be zero,which this, according to Eq. (40) causes a singularity in Nusseltnumber values.

Fig. 4 exhibits the transverse distribution of dimensionless tem-perature at different values of �geWe2 in the absence of viscousheating. As �geWe2 increases, the dimensionless temperature in-creases for wall cooling, while the contrary is right for wall heating.As �geWe2 increases, the velocity profile becomes more plug-likeand consequently the energy transferred by convection increasesnear the wall, which this, in the following, leads to decreasingthe wall temperature. Therefore, for wall cooling, increasing �geWe2

leads to decreasing the temperature difference between the walland the fluid particles, while for wall heating it is vice versa. Itshould be pointed out here that although for wall heating heat istransferred from the flow to the wall, however, because of Joule

Fig. 4. Transverse distribution of dimensionless temperature at different values of�geWe2 in the absence of viscous heating (a) S = 5 and (b) S = �5.

Fig. 5. Nusselt number versus 1/K at different values of S in the absence of viscousheating.

Fig. 6. Nusselt number as a function of dimensionless viscous heating parameter atdifferent values of S.

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heating, the net energy carried by the flow is positive for this spe-cial case.

Fig. 5 shows the Nusselt number values versus 1/K at differentvalues of S in the absence of viscous heating. Generally speaking,to increase S is to decrease Nusselt number. This behavior may

be explained by Fig. 2. As seen, increasing values of S lead to higherdimensionless bulk temperatures with negative sign, which this,according to Eq. (40) leads to lower values of Nusselt number.For sufficiently high values of S with negative sign such asS = �10, the behavior is quite different. For these cases, a singular-ity occurs in Nusselt number values. At the singularity point thewall and the bulk temperatures are the same, so heat transfer can-not be expressed in terms of Nusselt number. Note that after singu-larity point the Nusselt number takes negative values (not shownin the figure). This phenomenon takes place as a result of the bulktemperature being lower than the wall temperature and it does notmean that heat transfer takes place in the opposite direction. Ex-cept for S = �10, a higher value of K causes a higher Nusselt num-ber. As K goes to infinity, for all values of S, the Nusselt numberapproaches 12 which is the classical solution for slug flow [42].

Fig. 6 illustrates the Nusselt number as a function of the dimen-sionless viscous heating parameter at different values of S forK = 10 and �geWe2 = 1. As seen, increasing values of Sv lead to smal-ler values of the Nusselt number. This is due to increasing the dif-ference between the wall and bulk temperatures due to increasingviscous heating effects.

Fig. 7 demonstrates the Nusselt number values versus 1/K atdifferent values of Sv. For a given Sv, the effect of viscous heatingis decreased with increasing K and it is actually zero at the limitK ?1. This is due to the fact that, the velocity gradient over themajority of the channel cross section is decreased with increasingK. The exception is a region near the wall in which sharp gradientsexist. At higher values of K, the extent of this region is of the orderkD. Increasing K while keeping the dimensionless viscous heatingparameter constant means that the Debye length remains un-changed, while the channel height increases. Therefore, at the limitK ?1 the extent of this region compared with the channel heightis actually zero and, as a result, the effect of viscous heatingvanishes.

Fig. 8 exhibits the Nusselt number as a function of �geWe2 at dif-ferent values of K in the absence of viscous dissipation. As �geWe2

increases the velocity increases near the wall resulting in lower va-lue of the wall temperature. Consequently, the temperature differ-ence between the wall and the bulk flow decreases for wall coolingresulting in a higher Nusselt number, as seen in Fig. 8a. Since forwall heating, the wall possesses the minimum temperature inthe flow field, therefore, decreasing its temperature results in high-er dimensionless bulk temperature with negative sign and

Fig. 7. Nusselt number versus 1/K at different values of Sv.

Fig. 8. Nusselt number as a function of �geWe2 at different values of K in the absenceof viscous dissipation (a) S = 5 and (b) S = �5.

Fig. 9. Nusselt number as a function of �geWe2 at different values of Sv.

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consequently lower Nusselt number for wall heating, as observedin Fig. 8b. At higher values of K, the effect of �geWe2 on Nusseltnumber becomes insignificant. This is due to the fact that at higher

values of K the velocity profile is very similar to that of slug flowand the value of �geWe2 does not notably affect the velocity distri-bution. The Nusselt number as a function of �geWe2 at different val-ues of Sv for wall cooling case is presented in Fig. 9. The effect ofincreasing �geWe2 is found to be decreasing the viscous heating ef-fects. This is an expected behavior, because the effects of increasingvalues of �geWe2 and K on the velocity profile are the same.

4. Conclusions

The thermal transport features of the steady fully developedelectroosmotic flow of two viscoelastic fluids, namely PTT andFENE-P models, in a slit microchannel have been considered. Theclassical boundary condition of constant wall heat flux was consid-ered at the walls and the influences of viscous dissipation and Jouleheating were taken into consideration. Based on appropriate gen-eralized parameters, closed form expressions were obtained forthe transverse distributions of velocity and temperature and finallyfor Nusselt number. Some results of this study are summarizedbelow:

� As �geWe2 increases, the Nusselt number increases for wall cool-ing case, whereas the contrary is right for wall heating case.Nevertheless, the influence of �geWe2 on Nusselt numberbecomes insignificant at higher values of the dimensionlessDebye–Huckel parameter.� For a given value of the dimensionless Joule heating parameter,

the effect of viscous heating is more important at smaller valuesof �geWe2 and K and it vanishes at extreme limits of theseparameters.� Depending on the value of flow parameters, a singularity may

occur in Nusselt number values, especially at higher values ofthe dimensionless Joule heating term.� Generally speaking, an increase in the value of the dimension-

less Joule heating term decreases the Nusselt number.

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