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

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

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Heat transfer due to electroosmotic flow of viscoelastic fluids in a slit microchannel Arman Sadeghi, Mohammad Hassan Saidi ⇑, Ali Asghar Mozafari Center 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 2010 Received in revised form 17 March 2011 Accepted 21 March 2011 Available online 4 May 2011 Keywords: Electroosmotic flow Microchannel Joule heating Viscous dissipation PTT model FENE-P model

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 constitutive equations that are related to PTT and FENE-P models. In the present work, thermal transport characteristics of the steady fully developed electroosmotic flow of these fluids in a slit microchannel with constant wall heat fluxes have been investigated. The Debye–Huckel linearization is adopted and the effects of viscous dissipation and Joule heating are taken into account. Analytical solutions are obtained for the transverse distributions of velocity and temperature and finally for Nusselt number. Two different behaviors are observed for the Nusselt number variations due to increasing geWe2 which are an increasing trend for positive wall heat flux and a decreasing one for negative wall heat flux. However, the influence of geWe2 on Nusselt number vanishes at higher values of the dimensionless Debye–Huckel parameter. 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 in Nusselt number at higher negative values of the dimensionless Joule heating parameter. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction In the recent decades, micronsize mechanical and biochemical devices have become more prevalent both in commercial applications and in scientific investigations. Transport phenomena at microscale reveal many features that are not observed in macroscale devices. These features are quite different for gas and liquid flows. In gas microflows we encounter four important effects: compressibility, viscous heating, thermal creep and rarefaction [1]. Rarefaction effects are treated using slip velocity and temperature jump boundary conditions at solid surfaces. For liquid flow through microchannels, the classical boundary conditions of no slip velocity and no temperature jump are quite accurate [1]. However, liquid flows are encountered with other microscale features such as surface tension and electroosmotic effects. Electroosmosis refers to liquid flow induced by an electric field along electrostatically charged surfaces. The electric field may be the result of external or flow induced potentials. The electrokinetic effect due to the flow induced potential is unfavorable, as it causes moving the charges and molecules in the opposite direction of the flow, creating extra impedance to the flow motion. Nevertheless, electroosmosis has many applications in sample collection, detection, mixing and separation of various biological and chemical species. Another and probably the most important application of electroosmosis is the ⇑ Corresponding author. Tel.: +98 21 66165522; fax: +98 21 66000021. E-mail addresses: [email protected] (A. Sadeghi), saman@sharif. edu (M.H. Saidi), [email protected] (A.A. Mozafari). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.04.004

fluid delivery in microscale at which the electroosmotic micropump has many advantages over other types of micropumps. Electroosmotic pumps are bi directional, can generate constant and pulse free flows with flow rates well suited to microsystems and can be readily integrated with lab-on-a-chip devices. The magnitude and the direction of flow of an electroosmotic pump can be changed instantly [2]. In addition, electroosmotic pumps have no moving parts and have much simpler design and easier fabrication. Hydrodynamic aspects of electroosmotic flow in ultrafine capillary slits were analyzed by Burgreen and Nakache [3]. Rice and Whitehead [4] investigated fully developed electroosmotic flow in a narrow cylindrical capillary for low zeta potentials, using the Debye–Huckel linearization. Levine et al. [5] extended the Rice and Whitehead’s work to high zeta potentials by means of an approximation method originally proposed by Philip and Wooding [6]. More recently, an analytical solution for electroosmotic flow in a cylindrical capillary was derived by Kang et al. [7] by solving the complete Poisson–Boltzmann equation for arbitrary zeta-potentials. Hydrodynamic aspects of fully developed electroosmotic flow in a semicircular microchannel were studied by Wang et al. [8]. Analytical series solutions were found for two basic cases which can be superposed to yield solutions for any combination of constant zeta potentials on the flat or curved wall boundaries. Xuan and Li [9] developed general solutions for electrokinetic flow in microchannels with arbitrary geometry and arbitrary distribution of wall charge. Electroosmotic flow in parallel plate microchannels for the cases in which electric double layers interact with each other was analyzed by Talapatra and Chakraborty [10].

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Nomenclature cp Dh e Ex F G h H k kB n0 Nu p q q00 s S Sv t T u u⁄ u U We x

specific heat at constant pressure (kJ kg1 K1) hydraulic diameter of channel (=4H) proton charge (C) electric field in the axial direction (V m1) body force vector (N m3) dimensionless electric field (Eq. (12)) heat transfer coefficient (W m2 K1) half channel height (m) thermal conductivity (W m1 K1) Boltzmann constant (J K1) ion density (m3) Nusselt number (=hDh/k) pressure (Pa) dumbbell extensibility (FENE-P model) wall heat flux (W m2) volumetric heat generation due to Joule heating (W m3) dimensionless Joule heating term (¼ E2x H=q00 r0 ) dimensionless viscous heating parameter (¼ gge r0 U 2 =E2x k2D ) time (s) temperature (K) axial velocity (m s1) dimensionless axial velocity (=u/U) velocity vector (m s1) mean velocity (m s1) Weissenberg number (¼ kge U=H) axial coordinate (m)

Hydrodynamically developing flow between two parallel plates for electroosmotically generated flow has been reported in a numerical study by Yang et al. [11]. Also several researches have been performed to study heat transfer characteristics of electroosmotic flow. Maynes and Webb [12] analytically have studied fully developed electroosmotically generated convective transport for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Yang et al. [13] investigated forced convection in rectangular ducts with electrokinetic effects for both hydrodynamically and thermally fully developed flow. They investigated the effects of streaming potential on flow and heat transfer. All the foregoing studies are related to Newtonian fluids. Nevertheless, the bio-microfluidic systems are usually encountered with non-Newtonian behaviors of working fluids for which other constitutive equations rather than Newton’s law of viscosity are needed. In the literature, various models have been proposed to analyze non-Newtonian fluid flow behavior such as power-law model [14], Moldflow first order model [15], Bingham model [16], Eyring model [17] and PTT and FENE-P models [18]. The study of electroosmotic flow of non-Newtonian fluids is new and the open literature shows a limited number of relevant papers. Berli and Olivares [17] theoretically studied the electrokinetic flow of different non-Newtonian fluids through slit and cylindrical microchannels, using three constitutive equations comprising power-law model, Bingham model and Eyring model. The resulting equations allow one to predict the flow rate and electric current as functions of the simultaneously applied electric potential and pressure gradients. Electroviscous effects in steady, fully developed, pressuredriven flow of power-law liquids through a cylindrical microchannel have numerically been investigated by Bharti et al. [19], using a finite difference method. With the implementation of an approximate scheme for the hyperbolic sine function initially introduced by Philip and Wooding [6], an approximate analytical solution for

y y⁄ z

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

Greek symbols PTT parameter f wall zeta potential (V) f⁄ dimensionless wall zeta potential g viscosity coefficient (kg m1 s1) h dimensionless temperature (Eq. (28)) K dimensionless Debye–Huckel parameter (¼ H=kD ) k relaxation time (s) kD Debye length (m) q density (kg m3) qe net electric charge density (C m3) r liquid electrical resistivity (X m) sxy dimensionless shear stress (=Hsxy/ggeU) s stress tensor (Pa) w EDL potential (V) w⁄ dimensionless EDL potential (=ezw/kBTav)



Subscripts av average b bulk c critical ge generalized w wall 0 neutral liquid

velocity distribution in electroosmotic flow of power-law fluids in slit microchannels has been presented by Zhao et al. [20]. A numerical study of electroosmotic flow in parallel plate microchannels considering the power-law non-Newtonian fluid has been carried out by Tang et al. [21]. The simulation results showed that the fluid rheological behavior is capable of significantly changing the electroosmotic flow pattern and the flow behavior index plays an important role. In a recent study, Afonso et al. [18] developed closed from solutions for hydrodynamic characteristics of combined electroosmotically and pressure driven flow of two viscoelastic fluids, namely, the PTT and FENE-P models. To the authors’ best knowledge, the only research work considering the thermal transport features of non-Newtonian fluids electroosmotic flow has been undertaken by Das and Chakraborty [22] which studied electrokinetic effects in fully developed flow of powerlaw fluids in parallel plate microchannels. They derived solutions for the transverse distributions of velocity, temperature and solute concentration. However, the Nusselt number which is an important parameter in design and active control of microdevices was not considered. The rheological behavior of some bio-fluids can be described by differential viscoelastic constitutive equations that are related to the PTT and FENE-P models, as in the case of blood [23], saliva [24], synovial fluid [25] or other biofluids containing long chain molecules. This motivated us to analyze the electroosmotic flow of these models in the present work. In most lab-on-a-chip systems, the cross section of microchannels made by modern micromachining technology is close to a rectangular shape [26,27] and it may be effectively represented by a two dimensional slit when the width is much larger than the height. Therefore, for convenience of analysis, a slit microchannel having constant heat fluxes on its walls is considered and the effects of Joule heating and viscous dissipation are taking into account. Although according to Tang et al. [28], in general, a conjugate heat transfer problem

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has to be solved to simultaneously account for heat transfer in both the liquid and the channel wall; however, as shown by Bejan [29], a conjugate heat transfer problem may be reduced to the classical boundary condition of constant heat flux when the ratio of the external to internal heat transfer coefficients is very small. For a slit microchannel surrounded by stagnant air, the external heat transfer coefficient due to free convection is much smaller than the internal one which takes high values at microscale. Therefore, a constant heat flux boundary condition is consistent with heat transfer physics of the practical applications. Surprisingly, the analysis presented here can also cover the other classical boundary condition of constant temperature, as is a special case of constant heat flux boundary condition in the presence of internal heating. The governing equations for fully developed conditions are first made dimensionless and then closed form expressions are obtained for the transverse distributions of velocity and temperature and also for Nusselt number. The interactive effects of flow parameters on the temperature field and Nusselt number are shown in graphical form and also discussed in detail. 2. Problem formulation

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 The Fourier law for heat conduction is valid for viscoelastic fluids being considered here. 2.2. Velocity distribution The constitutive equations to be considered here are the simplified PTT model derived by Phan-Thien and Tanner [30] from network theory arguments and the FENE-P model based on the kinetic theory for finitely extensible dumbbells with a Peterlin approximation for the average spring force [31]. The presentation of the constitutive equation for both of these models has been given in detail by Afonso et al. [18] and it is omitted here to save space. They showed that at fully developed conditions, there is an exact equivalence of the both models solutions in the sense of a parameter to parameter match. Therefore, for convenience of analysis, it is useful to define generalized parameters and perform an analysis based on these generalized parameters instead of two analyses for both models. The following generalized parameters are introduced for FENE-P model [18]

kge;FENE-P ¼ k

qþ2 qþ5

ð1Þ

2.1. Problem assumptions Consider the situation where both hydrodynamically and thermally fully developed electroosmotic flow of a viscoelastic fluid takes place through a slit microchannel with channel half height of H. An illustration of the problem is depicted in Fig. 1. In the analysis the following assumptions are considered:  Thermophysical properties are constant in the whole domain including the EDL.  Constant values of the heat flux and zeta potential are considered 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 temperature variation over the channel cross section is negligible compared with the absolute temperature. Therefore, the charge density field is calculated on the basis of an average temperature.

ge;FENE-P ¼

1 qþ5

ð2Þ

gge;FENE-P ¼ g

ð3Þ

in which k is the relaxation time of the fluid, q is a parameter that measures the extensibility of the dumbbell, g is the viscosity coefficient, and  is a parameter that imposes an upper limit to the elongational viscosity. For PTT model, the generalized parameters are the 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}. Based on the findings of Afonso et al. [18], the axial velocity gradient can be related to the shear stress, sxy, as 2

ge kge du 1 1 þ 2 2 s2xy ¼ dy gge gge

!

sxy

Fig. 1. Geometry of the physical problem, coordinate system and electric double layer.

ð5Þ

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 3 du Gf Gf ¼ sinhðKy Þ þ 2ge We2 sinhðKy Þ  dy K cosh K K cosh K

and in dimensionless form

du 3 ¼ sxy þ 2ge We2 sxy dy

ð6Þ

where y ¼ y=H; u ¼ u=U; sxy ¼ Hsxy =gge U, and We is the Weissenberg number, a measure of the level of elasticity in the fluid, which is given by

We ¼

Du q ¼ rp þ r  s þ F Dt

2n0 e2 z2 w kB T av

ð9Þ

where n0 is the ion density, e is the proton charge, z is the valence number of ions in solution, w is the EDL potential, kB is the Boltzmann constant, and Tav is the average absolute temperature over the channel cross section. It should be pointed out that, assuming a univalent solution at 25 °C, the Debye–Huckel linearization is valid for zeta potentials below 25 mV which is usually an upper limit for bio-applications [33–38]. At fully developed conditions Du/ Dt = 0 and the pressure gradient is absent in purely electroosmotic flow. Therefore, the momentum equation in the axial direction is reduced to

dsxy 2n0 e2 z2 Ex w ¼ qe Ex ¼ dy k B T av

ð10Þ

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

dsxy ¼ Gw dy

ð11Þ

in which w ¼ ezw=kB T av and dimensionless electric field, G, is given by

2n0 ezH2 Ex gge U

ð12Þ

The dimensionless EDL potential w⁄ from our previous work dealing with electroosmotic flow of Newtonian fluids [32] may be written as

coshðKy Þ w ¼ f cosh K ⁄

ð13Þ ⁄

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

Gf sinhðKy Þ K cosh K

a¼

ð16Þ

ð14Þ

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

Gf K2

þ

 3 2ge We2 Gf 2 ð2 cosh K  cosh Ksinh KÞ 3K K cosh K

ð17Þ

b¼

   3 1 Gf 4ge We2 Gf  K K cosh K 3K K cosh K

ð18Þ

c¼

 3 2ge We2 Gf 3K K cosh K

ð19Þ

ð8Þ

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

sxy ¼

2

u ¼ a þ b coshðKy Þ þ c coshðKy Þsinh ðKy Þ

ð7Þ

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

G¼

Eq. (15) can be integrated subject to the no slip boundary condition at the wall ðuð1Þ ¼ 0Þ and the resulting velocity profile is

where

kge U H

qe ¼ 

ð15Þ

Since in the definitions of u⁄, G and We we have used the mean velocity, the two latest parameters are not independent. Their dependency can be obtained invoking the fact that average dimensionless velocity over the cross section of the channel is equal to unity. The dependency of G and We then may be written in compact form as

a0



Gf K cosh K

3

0

þb



Gf K cosh K



þ1¼0

ð20Þ

in which 3

2ge We2 sinh K sinh K 2 a ¼ þ2 cosh Ksinh K  2 cosh K  3K K 3K

!

0

ð21Þ 0

b ¼

cosh K sinh K  K K2

ð22Þ

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

"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gf 1 3 1 03 02 þ ð27a 729a04 þ 108a03 b Þ ¼ 3a0 2 K cosh K rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3 1 03 þ ð27a02  729a04 þ 108a03 b Þ 2

ð23Þ

Therefore, the dimensionless velocity distribution depends only on two parameters which are K and geWe2. 2.3. Temperature distribution The conservation of energy including the effects of viscous dissipation and Joule heating provides

qcp

DT ¼ r  ðkrTÞ þ ru : s þ s Dt

ð24Þ

In the above equation, s and ru:s denote the rate of volumetric heat generation due to Joule heating and viscous dissipation, respectively. The Joule heating term equals s ¼ E2x =r with r being the liquid electrical resistivity given by [5]

r¼

r0 cosh w

ð25Þ

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

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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) and ru:s = sxy(du/dy). Therefore, the energy equation (24) becomes

qcp u

@T @2T @2T ¼k þ @x @x2 @y2

! þ sxy

ð26Þ

ð27Þ

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

T  Tw

ð28Þ

q00 H k

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

A1 3 A2 2A4 A3 3A5  ; A5 ; B2 ¼ 2  2 ; B3 ¼ þ 2 16 K 9K 4K 2 32K 2 A4 A5 B4 ¼ ; B5 ¼ ; B6 ¼ B1  B2 cosh K  B3 coshð2KÞ 2 9K 16K 2 2 4  B4 cosh Ksinh K  B5 sinh K ð37Þ

Once the temperature distribution is obtained, the quantities of physical interest, including the bulk temperature of the fluid and the heat transfer rate can be obtained. The dimensionless bulk temperature is given by

R1

u h dy hb ¼ R01 ¼ u dy 0

ð29Þ

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

  dT b q00 SSv ¼ 1þSþb 2 dx qcp UH K

ð30Þ

1

du  dy dy 0      Gf c b sinhð2KÞ 3c K þ ðb  cÞ ¼  þ sinhð4KÞ 8 2 4 32 K cosh K

sxy

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

  2 d h SSv  SSv du u  S  2 sxy  ¼ 1 þ S þ b 2 2 dy dy K K

ð32Þ

By substituting sxy from Eq. (14) and using Eq. (16), we come up with the following dimensionless energy equation

C1 ¼

u h dy ¼ aC 1 þ bC 2 þ cC 3 þ B6

ð38Þ

0

B4 B5 3 4 sinh Kcosh K þ sinh Kcosh K 4K 5K    B2 B4 1 14 B3 B5 C3 ¼  þ B þ   þ sinh K 1 3K 27K 3 8 16 15K 7K   B4 B2 2B1 2B1 3  sinhð2KÞ þ 2 cosh K  2 cosh K þ 32K 16K 3K 9K    1 2 7B3 3B5 2 B þ  þ þ sinh Kcosh K 1 3K 27K 3 15K 7K     B2 7B4 2B3 3B5 3 4  sinh Kcosh K þ  sinh Kcosh K þ 4K 24K 5K 7K þ

2



2



4



 coshðKy Þsinh ðKy Þ þ A5 sinh ðKy Þ     SSv SSv Gf A1 ¼a 1 þ S þ b 2  S þ ðb þ 2cÞ ; A2 2K K cosh K K      SSv SSv Gf ¼b 1 þ S þ b 2 ; A3 ¼ ðb þ 2cÞ ; 2K K cosh K K     SSv SSv Gf A4 ¼c 1 þ S þ b 2 ; A5 ¼ 3c K K cosh K K

B4 B5 5 6 sinh Kcosh K þ sinh Kcosh K 6K 7K ð39Þ

ð33Þ

where the coefficients A1 to A5 are given by

The heat transfer rate can be expressed in terms of Nusselt number as

Nu ¼

ð34Þ

The thermal boundary conditions in the dimensionless form are written as

 hð1Þ ¼ 0

  B1 3 sinh K B3 3B5 sinhð2KÞ þ þ B5 þ B2  K 3 8 2K 16K

B4 B5 3 3 sinh K þ cosh Ksinh K 3K 4K    B2 B4 1 2 B3 B5 2B1 þ 3 B1 þ þ sinh K  2 cosh K C2 ¼  þ K K 2 8 3K 5K K     B2 B4 B3 B5 2  sinhð2KÞ þ 2  sinh Kcosh K þ 4K 16K 3K 5K

d h ¼ A1 þ A2 coshðKy Þ þ A3 coshð2Ky Þ þ A4 dy2

¼ 0;

1

þ

ð31Þ

dh dy

Z

þ

where Sv ¼ gge r0 U 2 =E2x k2D is the dimensionless viscous heating parameter, S ¼ E2x H=q00 r0 is the dimensionless volumetric heat generation due to Joule heating, and



ð36Þ

where

@T dT w dT b ¼ ¼ @x dx dx

Z

4

B1 ¼

    @T @T ¼ 0 T ðx;HÞ ¼ T w ðxÞ and k ¼ q00 @y ðx;0Þ @y ðx;HÞ

b¼

2

 coshðKy Þsinh ðKy Þ þ B5 sinh ðKy Þ þ B6 in which

du E2x þ dy r0

The relevant boundary conditions for the energy equation are as follows

hðyÞ ¼

h ¼ B1 y2 þ B2 coshðKy Þ þ B3 coshð2Ky Þ þ B4

ð35Þ

ð0Þ

After integrating Eq. (33) twice and applying the above boundary conditions, the following dimensionless temperature distribution is obtained

hDh q00 Dh 4 ¼ ¼ hb k kðT w  T b Þ

ð40Þ

with Dh = 4H. It is noteworthy that for Newtonian behavior, i.e., geWe2 = 0, the Nusselt number obtained in the present study and the one given in our previous work [32] are exactly identical. It is common in bio-applications that the wall heat flux takes negative values. This occurs in cases that a fraction of the energy generated by internal heating is dissipated through the walls. When all the internal heating is dissipated through the wall, the axial variation of temperature vanishes, i.e., oT/ox = 0. This is to say that, the classical boundary condition of constant wall temperature has been recovered from the constant wall heat flux boundary condition. It is worth mentioning that although any negative value of the wall heat flux is plausible, there is just a particular value of the wall heat flux which corresponds to a constant wall

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

q00 ¼ E2x H=r0  bgge U 2 =H

ð41Þ

Also for this case, according to Eq. (30), the values of S, Sv, and K are not independent. 3. Results and discussion It has been shown that the main parameters governing heat and fluid flow in fully developed electroosmotic flow of viscoelastic fluids being considered here in a slit microchannel are K, geWe2, S and Sv. Here, their interactive effects on the transverse distribution of temperature and Nusselt number are analyzed. Although a negative S is more encountered in practice, however, for the sake of generality, both negative and positive values of the dimensionless Joule heating parameter are considered. Based on the practical ranges of the electroosmotic velocity and channel height reported by Karniadakis et al. [1], and also the reported values of the relaxation time for human blood [40], a wide range of 0–1000 is considered for geWe2. It is also worth mentioning that the chosen values of Sv are those used by Maynes and Webb [41]. The transverse distribution of dimensionless temperature at different values of S in the absence of viscous heating is presented in Fig. 2. Both positive and negative values of the wall heat flux are considered. Positive values of dimensionless Joule heating term correspond to the wall cooling case where heat is transferred from the wall to the fluid, while the opposite is true for negative values of S. In the absence of Joule heating, the temperature distribution is independent of whether the wall is heated or cooled. As observed, increasing values of S lead to lower values of dimensionless temperature which implies that Joule heating increases the wall temperature rather than the bulk temperature. The reason is that although the distribution of energy generated by Joule heating is uniform throughout the channel cross section, but the energy transferred by the flow decreases near the wall and it equals zero at the wall. Fig. 3 demonstrates the transverse distribution of dimensionless temperature at different values of Sv. Different trends are observed for wall cooling and heating cases. To increase viscous heating effects is to decrease dimensionless temperature for wall cooling, while the opposite is true for wall heating. Viscous dissipation behaves like an energy source increasing the temperature of the fluid especially near the wall, since the highest shear rates occur at this

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

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

region, while it is zero at centerline. Therefore, the maximum temperature rise occurs at the wall. For wall cooling case, the maximum temperature occurs at the wall. So, increasing the wall temperature results in increasing the difference between temperatures of the wall and the fluid particles, while the opposite is true for wall heating case, since for this case the temperature of the wall is the minimum in temperature field. This is why the trends are different for two cases. It can be seen that for S = 5, as Sv increases, the sign of the dimensionless bulk temperature is changed from negative to positive. So, for a value of the dimensionless viscous heating parameter called Sv,c, which depends on flow parameters, the value of the dimensionless bulk temperature will be zero, which this, according to Eq. (40) causes a singularity in Nusselt number values. Fig. 4 exhibits the transverse distribution of dimensionless temperature at different values of geWe2 in the absence of viscous heating. As geWe2 increases, the dimensionless temperature increases for wall cooling, while the contrary is right for wall heating. As geWe2 increases, the velocity profile becomes more plug-like and consequently the energy transferred by convection increases near the wall, which this, in the following, leads to decreasing the wall temperature. Therefore, for wall cooling, increasing geWe2 leads to decreasing the temperature difference between the wall and the fluid particles, while for wall heating it is vice versa. It should be pointed out here that although for wall heating heat is transferred from the flow to the wall, however, because of Joule

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

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.

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be explained by Fig. 2. As seen, increasing values of S lead to higher dimensionless 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 as S = 10, the behavior is quite different. For these cases, a singularity occurs in Nusselt number values. At the singularity point the wall and the bulk temperatures are the same, so heat transfer cannot be expressed in terms of Nusselt number. Note that after singularity point the Nusselt number takes negative values (not shown in the figure). This phenomenon takes place as a result of the bulk temperature being lower than the wall temperature and it does not mean that heat transfer takes place in the opposite direction. Except for S = 10, a higher value of K causes a higher Nusselt number. As K goes to infinity, for all values of S, the Nusselt number approaches 12 which is the classical solution for slug flow [42]. Fig. 6 illustrates the Nusselt number as a function of the dimensionless viscous heating parameter at different values of S for K = 10 and geWe2 = 1. As seen, increasing values of Sv lead to smaller values of the Nusselt number. This is due to increasing the difference between the wall and bulk temperatures due to increasing viscous heating effects. Fig. 7 demonstrates the Nusselt number values versus 1/K at different values of Sv. For a given Sv, the effect of viscous heating is decreased with increasing K and it is actually zero at the limit K ? 1. This is due to the fact that, the velocity gradient over the majority of the channel cross section is decreased with increasing K. The exception is a region near the wall in which sharp gradients exist. At higher values of K, the extent of this region is of the order kD . Increasing K while keeping the dimensionless viscous heating parameter constant means that the Debye length remains unchanged, while the channel height increases. Therefore, at the limit K ? 1 the extent of this region compared with the channel height is actually zero and, as a result, the effect of viscous heating vanishes. Fig. 8 exhibits the Nusselt number as a function of geWe2 at different values of K in the absence of viscous dissipation. As geWe2 increases the velocity increases near the wall resulting in lower value of the wall temperature. Consequently, the temperature difference between the wall and the bulk flow decreases for wall cooling resulting in a higher Nusselt number, as seen in Fig. 8a. Since for wall heating, the wall possesses the minimum temperature in the flow field, therefore, decreasing its temperature results in higher dimensionless bulk temperature with negative sign and

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

heating, the net energy carried by the flow is positive for this special case. Fig. 5 shows the Nusselt number values versus 1/K at different values of S in the absence of viscous heating. Generally speaking, to increase S is to decrease Nusselt number. This behavior may

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

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

Fig. 9. Nusselt number as a function of

geWe2 at different values of Sv.

values of K the velocity profile is very similar to that of slug flow and the value of geWe2 does not notably affect the velocity distribution. The Nusselt number as a function of geWe2 at different values of Sv for wall cooling case is presented in Fig. 9. The effect of increasing geWe2 is found to be decreasing the viscous heating effects. This is an expected behavior, because the effects of increasing values of geWe2 and K on the velocity profile are the same.

4. Conclusions The thermal transport features of the steady fully developed electroosmotic flow of two viscoelastic fluids, namely PTT and FENE-P models, in a slit microchannel have been considered. The classical boundary condition of constant wall heat flux was considered at the walls and the influences of viscous dissipation and Joule heating were taken into consideration. Based on appropriate generalized parameters, closed form expressions were obtained for the transverse distributions of velocity and temperature and finally for Nusselt number. Some results of this study are summarized below:  As geWe2 increases, the Nusselt number increases for wall cooling case, whereas the contrary is right for wall heating case. Nevertheless, the influence of geWe2 on Nusselt number becomes insignificant at higher values of the dimensionless Debye–Huckel parameter.  For a given value of the dimensionless Joule heating parameter, the effect of viscous heating is more important at smaller values of geWe2 and K and it vanishes at extreme limits of these parameters.  Depending on the value of flow parameters, a singularity may occur in Nusselt number values, especially at higher values of the dimensionless Joule heating term.  Generally speaking, an increase in the value of the dimensionless Joule heating term decreases the Nusselt number.

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

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

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