International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

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International Journal of Heat and Mass Transfer

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

Viscous dissipation effects on thermal transport characteristics of combinedpressure and electroosmotically driven flow in microchannels

Arman Sadeghi, Mohammad Hassan Saidi *

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 a b s t r a c t

Article history:Received 2 September 2009Received in revised form 15 March 2010Accepted 26 March 2010Available online 8 May 2010

Keywords:Electroosmotic flowLaminarConvectionMicrochannelViscous dissipationJoule heating

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.04.028

* Corresponding author. Tel.: +98 21 66165522; faxE-mail address: Saman@sharif.edu (M.H. Saidi).

This study investigates the influence of viscous dissipation on thermal transport characteristics of thefully developed combined pressure and electroosmotically driven flow in parallel plate microchannelssubject to uniform wall heat flux. Closed form expressions are obtained for the transverse distributionsof electrical potential, velocity and temperature and also for Nusselt number. From the results it is real-ized that the Brinkman number has a significant effect on Nusselt number. Generally speaking, toincrease Brinkman number is to decrease Nusselt number. Although the magnitude of Joule heatingcan affect Brinkman number dependency of Nusselt number, however the general trend remainsunchanged. Depending on the value of flow parameters, a singularity may occur in Nusselt number valueseven in the absence of viscous heating, especially at great values of dimensionless Joule heating term. Fora given value of Brinkman number, as dimensionless Debye–Huckel parameter increases, the effect of vis-cous heating increases. In this condition, as dimensionless Debye–Huckel parameter goes to infinity, theNusselt number approaches zero, regardless of the magnitude of Joule heating. Furthermore, it is realizedthat the effect of Brinkman number on Nusselt number for pressure opposed flow is more notable thanpurely electroosmotic flow, while the opposite is true for pressure assisted flow.

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1. Introduction Also, precise flow control can be easily achieved by controlling

Due to the rapid development of microdevices such as MEMSsensors, micropumps and microvalves, research efforts in fluidflow and heat transfer in these devices have become attractive re-search fields. Transport phenomena at the microscale reveal manyfeatures that are not observed in the macroscale devices. Conse-quently, fundamental issues related to fluid flow and heat transferin microchannels need to be resolved for efficient design of micro-fluidic devices.

Fluid delivery is crucial in the microfluidic systems since theoperating pressure is substantially high. Although micropumpswhich are capable of delivering such pressures exist [1], howevertheir moving components are complicated to design and fabricateand they are prone to mechanical failure due to fatigue and fabri-cation defects which consequently make them unsuitable formicrofluidic applications. To meet the pumping requirements ofmicrodevices, various techniques have been proposed for fluidpumping in which the electroosmotic micropump has been favoreddue to its many advantages over other types of micropumps. Elec-troosmotic pumps need no moving parts and have much simplerdesign and easier fabrication. It is applicable to a wide range offluid conductivity, which is essential for biomedical applications.

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: +98 21 66000021.

the external electric field.The study of liquid flow in microchannels with consideration of

electrokinetic effects can be traced to 1960s. The early analyticalworks on electroosmotic flow report the electrokinetically drivenfully developed hydrodynamics of microchannels [2–4]. Hydrody-namically developing flow between two parallel plates for elect-roosmotically generated flow has been reported in a numericalstudy by Yang et al. [5]. Also several researches have been per-formed to study heat transfer characteristics of electroosmotic flowin microchannels. Maynes and Webb [6] analytically have studiedfully developed electroosmotically generated convective transportfor a parallel plate microchannel and circular microtube under im-posed constant wall heat flux and constant wall temperatureboundary conditions. Yang et al. [7] investigated forced convectionin rectangular ducts with electrokinetic effects for both hydrody-namically and thermally fully developed flow. They investigatedthe effects of streaming potential on flow and heat transfer.

The flow rate induced by electroosmotic force is usually smalland therefore even a small pressure gradient applied along amicrochannel may cause velocity distributions and correspondingflow rates that deviate from the purely electroosmotic flow. Thepressure gradient may arise from several reasons such as the pres-ence of alternative pumping mechanism, placement of mechanicalvalve in the flow path and the existence of variations in the wallzeta potential [8]. When electroosmotic and traditional pressure

Nomenclature

Br Brinkman number [=lU2/qH]cp specific heat at constant pressure [kJ kg�1 K�1]Dh hydraulic diameter of channel [=4H]e electron charge [C]Ex electric field in the axial direction [V m�1]G1 dimensionless pressure gradient [Eq. (14)]G2 dimensionless electrical potential gradient [Eq. (14)]h heat transfer coefficient [W m�2 K�1]H half channel height [m]ie current density [A m�2]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 wall heat flux [W m�2]s volumetric heat generation due to Joule heating

[W m�3]S dimensionless form of s [=sH/q]T temperature [K]u axial velocity [m s�1]u* dimensionless axial velocity [=u/U]U mean velocity [m s�1]x axial coordinate [m]y transverse coordinate [m]

y* dimensionless transverse coordinate [=y/H]z valence number of ions in solution

Greek symbolse fluid permittivity [C V�1 m�1]f wall zeta potential [V]f* dimensionless wall zeta potentialh dimensionless temperature [Eq. (21)]j Debye–Huckel parameter [m�1]K dimensionless Debye–Huckel parameter [=jH]kD Debye length [m]l dynamic viscosity [kg m�1 s�1]q density [kg m�3]qe net electric charge density [C m�3]r liquid electrical resistivity [Xm]u electrostatic potential [V]U externally imposed electrostatic potential [V]w EDL potential [V]w* dimensionless EDL potential [=ezw/kBTav]

Subscriptsav averageb bulkc criticalw wall

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791 3783

forces are present simultaneously, the resulting velocity profile is asuperposition of the electroosmotic and pressure driven flows [9].Fully developed thermal transport of combined pressure and elect-roosmotically driven flow in circular microtubes has been analyzedby Maynes and Webb [10]. The two classical thermal boundaryconditions of constant wall heat flux and constant wall tempera-ture were considered. Chakraborty [11] and Zade et al. [12] devel-oped closed form solutions for hydrodynamically and thermallyfully developed heat transfer in circular ducts and parallel platechannels, respectively, considering isoflux boundary conditions atthe walls for combined pressure and electroosmotically drivenflow. However, Zade et al. [12] represented the effect of the EDLby a slip velocity at the wall (Helmholtz–Smoluchowsky velocity),which limits their analysis to thin EDL limit only. Jain and Jensen[13] considered fully developed isoflux heat transfer in microchan-nels formed by parallel plates, analyzing the flow and heat transferwithin the EDL but did not consider the effect of Joule heating.Recently, Chen [14] investigated the thermal transport characteris-tics of fully developed mixed pressure and electroosmotically dri-ven flow in parallel plate micro- and nanochannels subject touniform wall heat flux considering Joule heating effects. Analyticalsolutions were obtained for constant fluid properties, whilenumerical solutions were presented for variable fluid properties.Analytical solutions for thermally developing combined pressureand electroosmotically driven flow in microchannels have been ob-tained by Dutta et al. [15–17] for a variety of wall boundaryconditions.

Viscous dissipation effects are typically only significant for highviscous flows or in presence of high gradients in velocity distribu-tion. In macroscale, such high gradients occur in high velocityflows. However, in microscale devices such as microchannels, be-cause of small dimensions, such high gradients may occur evenfor low velocity flows. So, for microchannels the viscous dissipationshould be taken into consideration. Koo and Kleinstreuer [18,19]investigated the effects of viscous dissipation on the temperaturefield and ultimately on the friction factor using dimensional anal-ysis and experimentally validated computer simulations. It was

found that ignoring viscous dissipation could affect accurate flowsimulations and measurements in microconduits. Although thereare numerous works considering viscous dissipation effects inpressure driven flows, unfortunately the open literature shows alimited number of papers that deal with viscous heating effectsin electroosmotic flow through microchannels. The effect of vis-cous dissipation in fully developed electroosmotic heat transferfor a parallel plate microchannel and circular microtube under im-posed constant wall heat flux and constant wall temperatureboundary conditions has been analyzed by Maynes and Webb[20].They concluded that the influence of viscous dissipation isonly important at low values of the relative duct radius. Sharmaand Chakraborty [21] have obtained semi analytical solutions forthe temperature and Nusselt number distribution in the thermalentrance region of parallel plate microchannels under the com-bined action of pressure driven and electroosmotic transportmechanisms, by taking into account the effects of viscous dissipa-tion in the framework of an extended Graetz problem. They consid-ered the constant wall temperature case and represented the effectof the EDL by Helmholtz–Smoluchowsky velocity.

The main theme of the present work is to analytically investi-gate viscous dissipation effects on thermal transport characteris-tics of both hydrodynamically and thermally fully developedcombined pressure and electroosmotically driven flow in parallelplate microchannels subject to uniform wall heat flux. Closed formexpressions are obtained for the transverse distributions of electri-cal potential, velocity and temperature and also for Nusselt num-ber. The results of this investigation will give valuable insight tothe effect of viscous dissipation on the flow and heat transfer char-acteristics that can be useful for active control of electroosmotical-ly driven flow and heat transfer.

2. Problem formulation

We consider flow through a microchannel formed between twoparallel plates with channel half width of H. Geometry of the

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

3784 A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

physical problem is depicted in Fig. 1. The flow is driven by bothpressure gradient and external voltage gradient. In the analysisthe following assumptions are considered:

� The flow is laminar and both thermally and hydrodynamicallyfully developed.� Thermophysical properties are constant. This assumption,

which has successfully been used by Maynes and Webb [20],is valid for temperature variations less than 10 K.� The channel walls are subjected to a constant heat flux.� The liquid contains an ideal solution of fully dissociated sym-

metric salt.� The charge in the EDL follows Boltzmann distribution.� In calculating the charge density, it is assumed that the temper-

ature variation over the channel cross section is negligiblecompared to the absolute temperature. Therefore, the chargedensity field is calculated on the basis of an averagetemperature.� Wall potentials are considered low enough for Debye–Huckel

linearization to be valid.� The external voltage is significantly higher than the flow

induced streaming potential.

2.1. Electrical potential distribution

The electrical potential distribution is obtained from solution ofthe Poisson equation:

r2u ¼ �qe

e; ð1Þ

where e is the fluid permittivity, and qe is the net electric chargedensity. The potential u is due to combination of externally im-posed field U and EDL potential w, namely:

u ¼ Uþ w: ð2Þ

For an ideal solution of fully dissociated symmetric salt, the electriccharge density is given by [9]

qe ¼ �2n0ez sinhezw

kBTav

� �; ð3Þ

where n0 is the ion density, e is the electron charge, z is the valencenumber of ions in solution, kB is the Boltzmann constant, and Tav isthe average absolute temperature over the channel cross section.For fully developed flow, w = w(y) and the external potential gradi-ent is in the axial direction only, i.e., U = U(x). For a constant voltagegradient in the x-direction, Eq. (1) becomes

d2wdy2 ¼

2n0eze

sinhezw

kBTav

� �: ð4Þ

The above non-linear second order one dimensional equation isknown as the Poisson–Boltzmann equation. Eq. (4) in the dimen-sionless form becomes

d2w�

dy�2¼ 2n0e2z2

ekBTavH2 sinh w� ð5Þ

in which w* = ezw/kBTav and y* = y/H. The quantity (2n0e2z2/ekBTav)�1/2 is known as Debye length kD. Defining Debye–Huckelparameter as k = 1/kD, we come up with

d2w�

dy�2� k2H2 sinh w� ¼ 0: ð6Þ

If w* is small enough, namely w*6 1, the term sinh w* can be

approximated by w*. This linearization is known as Debye–Huckellinearization. It is noted that for typical values of e = 1.6 � 10�19 Cand Tav = 298 K and using the values of 1 and 1.38 � 10�23 J/K forvalence number and Boltzmann constant, respectively, this approx-imation is valid for w 6 25.7 mV. Defining dimensionless Debye–Huckel parameter K = kH and invoking Debye–Huckel linearization,Eq. (6) becomes

d2w�

dy�2� K2w� ¼ 0: ð7Þ

The boundary conditions for the above equation are

w�ð1Þ ¼ f�;dw�

dy�

� �ð0Þ¼ 0 ð8Þ

in which f* is the dimensionless wall zeta potential, i.e., f* = ezf/kBTav. Using Eq. (7) and applying boundary conditions (8), thedimensionless potential distribution is obtained as follows:

w� ¼ f�coshðKy�Þ

cosh K: ð9Þ

2.2. Velocity distribution

Using coordinates shown in Fig. 1, the momentum equation inx-direction under combined action of pressure and electric poten-tial gradient is as follows:

ld2udy2 ¼

dpdx� qeEx; ð10Þ

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791 3785

where the electric field in the x-direction Ex is given by

Ex ¼ �dudx¼ �dU

dx: ð11Þ

Substitution of Ex from the above equation and using Eq. (1), themomentum equation becomes

ld2udy2 ¼

dpdx� e

d2wdy2

dUdx

: ð12Þ

The momentum equation in dimensionless form becomes

d2u�

dy�2¼ G1 � G2w

� ð13Þ

in which u* = u/U, where U is the mean velocity and G1 and G2 whichrefer to dimensionless gradients for pressure and external electricalpotential, respectively, are given below:

G1 ¼H2

lUdpdx; G2 ¼

2n0ezH2

lUdUdx

: ð14Þ

With zero velocity at each wall, the dimensionless velocity profilebecomes

u� ¼ �G1

2ð1� y�2Þ þ G2

K2 f� 1� coshðKy�Þcosh K

� �: ð15Þ

It should be noted that since in the definition of u*, G1 and G2 wehave used the mean velocity, the two latest parameters are notindependent. Their dependency can be obtained invoking the factthat average dimensionless velocity over the cross section of thechannel is equal to unity. The dependency of G1 and G2 then willbe as:

G2

K2 f� ¼1þ G1

3

1� tanh KK

: ð16Þ

Using the above correlation, the velocity distribution becomes

u� ¼ �G1

2ð1� y�2Þ þ

1þ G13

1� tanh KK

1� coshðKy�Þcosh K

� �: ð17Þ

Note that G1 = 0 corresponds to purely electroosmotic flow, G1 = �3corresponds to purely pressure driven or Poiseuille flow, �3 < G1<0correspond to pressure assisted flow and other values of G1 corre-spond to pressure opposed flow.

2.3. Temperature distribution

The conservation of energy including the effects of Joule heatingand viscous dissipation requires

qcpu@T@x¼ k

@2T@x2 þ

@2T@y2

!þ sþ l du

dy

� �2

: ð18Þ

In the above equation, s and l(du/dy)2 denote the rate of volumetricheat generation due to Joule heating and viscous dissipation,respectively. Joule heating is due to the electrical current and theresistivity of the fluid and equals to s ¼ i2

er. The term ie is the cur-rent density established by the applied potential and r is the liquidelectrical resistivity. The current density is related to the charge po-tential by the relation [22]

ie ¼Ex

rcosh w�: ð19Þ

For low wall zeta potentials, which is the case in this study,cosh w* ? 1 and the Joule heating term may be considered as theconstant value of s ¼ E2

x=r [22]. The relevant boundary conditionsfor the energy equation are as follows:

@T@y

� �ðx;0Þ¼ 0

Tðx;HÞ ¼ TW ðxÞ and k@T@y

� �ðx;HÞ¼ q

ð20Þ

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

hðyÞ ¼ T � TwqHk

: ð21Þ

Taking differentiation of Eq. (21) with respect to x gives

@T@x¼ dTw

dx¼ dTb

dxð22Þ

in which Tb is the bulk temperature. The energy balance for an ele-ment of duct with the dimensions of H and dx may be written as

qcpUHdTb ¼ qþ sH þ lZ H

0

dudy

� �2

dy

" #dx ð23Þ

After required manipulations, the following expression is obtainedfor dTb/dx

dTb

dx¼ 1

qcpUHqþ sH þ lU2

Hb

!; ð24Þ

where b is given by

b ¼ G21

3� 2G1

1þ G13

1� tanh KK

!� K2

2cosh2K

1þ G13

1� tanh KK

!2

þ K2

1þ G13

1� tanh KK

!2

þ 2K

G11þ G1

3

1� tanh KK

!24

35 tanh K: ð25Þ

Since oT/ox is constant, the axial conduction term in the energyequation will be zero. So the energy equation in dimensionless formwill be as follows:

d2hdy�2

¼ a� by�2 � c coshðKy�Þ � dsinh2ðKy�Þ � ey� sinhðKy�Þ; ð26Þ

where a, b, c, d and e are given by

a ¼ �G1

2þ

1þ G13

1� tanh KK

!ð1þ Sþ bBrÞ � S; ð27Þ

b ¼ �G1

2

� �ð1þ Sþ bBrÞ þ G2

1Br; ð28Þ

c ¼1þ G1

3

1� tanh KK

!ð1þ Sþ bBrÞ

cosh K; ð29Þ

d ¼ K2

cosh2K

1þ G13

1� tanh KK

!2

Br; ð30Þ

e ¼ 4K

cosh K�G1

2

� �1þ G1

3

1� tanh KK

!Br; ð31Þ

in which Br = lU2/qH is the Brinkman number and S = sH/q is thedimensionless volumetric heat generation due to Joule heating.The thermal boundary conditions in the dimensionless form arewritten as

dhdy�

� �ð0Þ¼ 0; hð1Þ ¼ 0: ð32Þ

Using Eq. (26) and applying boundary conditions (32), the dimen-sionless temperature distribution is obtained as:

Fig. 2. Transverse distribution of dimensionless velocity at different values of G1.

3786 A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

h ¼ a2

y�2 � b12

y�4 � c

K2 coshðKy�Þ � dcoshð2Ky�Þ

8K2 � y�2

4

� �

� ey�

K2 sinhðKy�Þ � 2K3 coshðKy�Þ

� �þ f ; ð33Þ

where

f ¼ � a2þ b

12þ cosh K

K2 c � 14� coshð2KÞ

8K2

� �d

þ sinh K

K2 � 2K3 cosh K

� �e: ð34Þ

To obtain the Nusselt number, first the dimensionless bulk temper-ature hb must be calculated, which is given by

hb ¼R 1

0 u�hdy�R 10 u�dy�

¼Z 1

0u�hdy�

¼ �G1

2þ

1þ G13

1� tanh KK

!g1 þ

G1

2g2 �

1þ G13

1� tanh KK

!g3

cosh Kþ f ; ð35Þ

where g1, g2 and g3 are as follows:

g1 ¼a6� b

60� sinh K

K3 c þ 112� sinhð2KÞ

16K3

� �d

� cosh K

K3 � 3sinh K

K4

� �e; ð36Þ

g2 ¼a

10� b

84� sinh K

K3 � 2cosh K

K4 þ 2sinh K

K5

� �c

þ 1160

8� 10sinhð2KÞ

K3 þ 10coshð2KÞ

K4 � 5sinhð2KÞ

K5

� �d

� cosh K

K3 � 5sinh K

K4 þ 10cosh K

K5 � 10sinh K

K6

� �e; ð37Þ

g3 ¼sinh K

2K� cosh K

K2 þ sinh K

K3

� �a

� sinh K12K

� cosh K

3K2 þsinh K

K3 � 2cosh K

K4 þ 2sinh K

K5

� �b

� 12K2 þ

sinhð2KÞ4K3

� �c

þ 148

12sinh K

K� 24

cosh K

K2 þ 21sinh K

K3 � sinhð3KÞK3

� �d

� 18� 8

K3 þ 2coshð2KÞ

K3 � 5sinhð2KÞ

K4

� �e: ð38Þ

Therefore, the Nusselt number which is based on the definitiongiven by

Nu ¼ hDh

k¼ qDh

kðTw � TbÞ¼ � 4

hbð39Þ

can be obtained. It is noteworthy that for purely electroosmoticflow, the Nusselt number obtained in the present study and theone given by Chen [14] are in quite agreement. Unfortunately, inthe presence of pressure gradient the use of different referencevelocities prevents any comparison. For the case that the flow ispurely driven by pressure gradient the present results are in excel-lent agreement with those given by Jeong and Jeong [23].

Although the analysis presented in this section assumed a uni-form heat flux at the walls, however, the asymptotic solution of aconstant wall temperature boundary condition for which oT/ox = 0 may also be explored. This corresponds to the conditionwhere all volumetric heating in the fluid is dissipated at the walls,yielding q < 0. Thus, the fully developed condition for imposed con-stant wall temperature in presence of internal heating is one of

constant wall heat flux as well, where q gives a particular valuewhich is a function of the total Joule heating and viscous dissipa-tion and may be obtained from q + sH + (lU2/H)b = 0. The asymp-totic solution without internal heating requires an independentprocedure and cannot be obtained from the solution presentedhere.

3. Results and discussion

The dimensionless pressure gradient, dimensionless Debye–Huckel parameter, dimensionless Joule heating term and Brinkmannumber are the main parameters governing heat and fluid flow infully developed combined pressure and electroosmotically drivenflow in parallel plate microchannels. Here, their interactive effectson the transverse distributions of velocity and temperature and fi-nally on Nusselt number are analyzed.

The transverse distribution of dimensionless velocity at differ-ent values of dimensionless pressure gradient for K = 50 is pre-sented in Fig. 2. The value of Debye–Huckel parameter, equal to50, implies that EDL is limited to a small region close to the wallsand a significant portion of the channel width is outside the EDL,so, the velocity distribution is nearly a slug flow profile for purelyelectroosmotic flow (G1 = 0), while as expected, it is a Poiseuilleflow for purely pressure driven flow (G1 = �3). The velocity profilefor other values of dimensionless pressure gradient is the superpo-sition of both purely electroosmotic and Poiseuille flows. So, thevelocity distribution for pressure assisted flow (G1 = �1) showsboth a maximum value at the centerline which is related to Poiseu-ille flow and a sharp gradient at the wall which is inherited fromelectroosmotic flow. For pressure opposed flow which correspondsto G1 = 1 and G1 = 3, the velocity distribution attains its maximumvalue at a point close to the wall and reaches a local minimum atthe centerline as a result of opposed pressure. Note that for suffi-ciently large values of G1, reverse flow may occur at the centerline.

Fig. 3 exhibits transverse distribution of dimensionless temper-ature at different values of S in the absence of viscous heating andK = 10. As seen, increasing values of S lead to smaller values ofdimensionless temperature for purely electroosmotic flow whichimplies that Joule heating increases the wall temperature ratherthan the bulk temperature. The reason is that although the distri-bution of energy generated by Joule heating is uniform throughoutthe channel cross section but the energy transferred by convectiondecreases near the wall and it equals zero at the wall. For S = �10, his approximately constant over much of the duct cross section ris-ing to a maximum at the wall. In this region at which the velocity

Fig. 3. Transverse distribution of dimensionless temperature at different values of Sin the absence of viscous heating: (a) purely electroosmotic flow and (b) pressureopposed flow.

Fig. 4. Transverse distribution of dimensionless temperature at different values ofBr for wall cooling case: (a) purely electroosmotic flow and (b) pressure opposedflow.

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791 3787

profile is uniform, all the energy generated by Joule heating istransferred by the flow. For pressure opposed flow, the tempera-ture distribution shows a quite different behavior. Although a lar-ger S leads to smaller h in the region adjacent to the wall, it leads togreater values of dimensionless temperature in the region close tothe centerline. The interpretation can be expressed is that since asa result of opposed pressure the centerline velocity is a local min-imum, so the energy transferred by the flow in this region is smal-ler compared to the near wall region which attains large velocities.This fact consequently leads to greater dimensionless tempera-tures for the core region.

The transverse distribution of dimensionless temperature at dif-ferent values of Br for wall cooling case, at which heat is trans-ferred from the walls to the fluid, is shown in Fig. 4. Note thatsince the wall heat flux is positive, the Brinkman number cannottake negative values. The viscous dissipation behaves like an en-ergy source increasing the temperature of the fluid especially nearthe wall, since the highest shear rate occurs at this region while itis zero at the centerline. So, for both cases, increasing values of Brlead to smaller dimensionless temperatures. The effect of viscousheating is more notable for pressure opposed flow, as a result ofgreater velocity gradient existence at the wall. The transverse dis-tribution of dimensionless temperature at different values of Br forwall heating case is presented in Fig. 5. For this case, as Brinkmannumber increases with negative sign, the dimensionless tempera-ture increases which is an expected behavior. For sufficiently great

values of Br with negative sign, the maximum dimensionless tem-perature occurs at the duct centerline rather than the wall. Fromthe figure, one can see that as Br increases with negative sign,the sign of dimensionless bulk temperature is changed from nega-tive to positive. So, for a value of Brinkman number called Brc,which its value depends on flow parameters, the value of dimen-sionless bulk temperature will be zero, which this, according toEq. (39) causes a singularity in Nusselt number values.

Fig. 6 depicts Nusselt number versus 1/K at different values of Sin the absence of viscous heating for purely electroosmotic flow.Generally speaking, to increase S is to decrease the fully developedNusselt number. This behavior may be explained by means ofFig. 3a. As seen, increasing values of S lead to greater dimensionlessbulk temperatures with negative sign, which this according to Eq.(39) leads to smaller values of Nusselt number. For sufficientlygreat values of S with negative sign such as S = �10, the behavioris quite different. For these cases, a singularity occurs in Nusseltnumber values. At the singularity point the wall and the bulk tem-peratures are the same, so heat transfer cannot be expressed interms of Nusselt number. Note that after singularity point the Nus-selt number takes negative values (not shown in the figure). Thisphenomenon takes place as a result of the bulk temperature beingsmaller than the wall temperature and it does not mean that theheat transfer takes place from the wall to the fluid (note that thisis a wall heating case). Except for S = �10, a greater value of Kcauses a greater Nusselt number. By increasing K, EDL will be

Fig. 5. Transverse distribution of dimensionless temperature at different values ofBr for wall heating case: (a) purely electroosmotic flow and (b) pressure opposedflow.

Fig. 6. Nusselt number vs. 1/K at different values of S in the absence of viscousheating for purely electroosmotic flow.

Fig. 7. Brinkman number dependency of the Nusselt number at different values ofS: (a) purely electroosmotic flow, (b) pressure assisted flow and (c) pressureopposed flow.

3788 A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

limited to smaller regions close to the walls, resulting in a moreplug-like velocity profile. Therefore, as K goes to infinity, for all val-ues of S, the Nusselt number approaches 12 which is the classicalsolution for slug flow [24].

The Brinkman number dependency of the Nusselt number atdifferent values of S for K = 10 is shown in Fig. 7. As seen, increasingvalues of Brinkman number lead to smaller values of Nusselt num-ber, regardless of the magnitude of dimensionless Joule heatingterm. The effect of Brinkman number on Nusselt number for pres-sure opposed flow is more notable than purely electroosmotic flow,as a result of existing greater velocity gradients especially at the

Fig. 9. Nusselt number vs. G1 at different values of Br in the absence of Jouleheating.

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791 3789

solid surface, while the opposite is true for pressure assisted flow.Although increasing values of S lead to smaller Nusselt numbers,but for greater values of Br, the effect of S on Nusselt numberbecomes slighter. This is due to the fact that for great Brinkmannumbers, viscous dissipation term dominates heat transfer charac-teristics. The effect of S on Nusselt number for pressure assistedflow is more pronounced than purely electroosmotic flow, whileits effect on Nusselt number for pressure opposed flow is negligi-ble. Note that this is a special case that although the magnitudeof dimensionless Joule heating term affects dimensionless temper-ature distribution, but the dimensionless bulk temperature re-mains unchanged.

Fig. 8 presents Nusselt number values versus G1 at different val-ues of S in the absence of viscous heating. Generally speaking, agreater G1 corresponds to a greater Nu, except for S = �10 whichshows a different trend including a singularity point atG1 = �1.46. At G1 = 0.32 the Nusselt number of all values of S coin-cide with each other. Before this point, the effect of increasing val-ues of S is to decrease Nusselt number, while for greater values ofdimensionless pressure gradient it is vice versa. The effect of vis-cous dissipation on Nusselt number is shown in Fig. 9 which de-picts Nusselt number values versus G1 in the absence of Jouleheating. As seen, the Brinkman number has a significant effect onNusselt number especially for Great values of dimensionless pres-sure gradient. So for the channels having small values of wall heatflux and small height and also for high velocity or high viscousflows, the viscous heating term cannot be discarded from energyequation. Although this phenomenon was observed in the absenceof Joule heating, but from Fig. 7 it is clear that the magnitude ofJoule heating does not affect the overall behavior of Nusselt num-ber. So it can be expressed that neglecting viscous dissipation mayaffect accurate flow measurement in flows having great Brinkmannumber.

Fig. 10 illustrates Nusselt number values versus 1/K at differentvalues of Br in the absence of Joule heating. Generally speaking, toincrease Brinkman number is to decrease Nusselt number. The ef-fect of Brinkman number on Nusselt number is more significant athigher values of dimensionless Debye–Huckel parameter. At smallvalues of K which correspond to great values of 1/K, the effect ofdimensionless Debye–Huckel parameter on Nusselt number be-comes insignificant. This is due to the fact that at small values ofK the velocity profile is nearly similar to that for Poiseuille flowand variation of dimensionless Debye–Huckel parameter does

Fig. 8. Nusselt number vs. G1 at different values of S in the absence of viscousheating.

Fig. 10. Nusselt number vs. 1/K at different values of Br in the absence of Jouleheating: (a) pressure assisted flow and (b) pressure opposed flow.

not notably affect profile shape and consequently the energy gen-erated by viscous heating or transferred by the flow. Contrary tosmall values of K, at great values of K the magnitude of dimension-less Debye–Huckel parameter drastically affects Nusselt number.

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At great values of K, the effect of increasing values of K is to in-crease Nusselt number for wall heating case and also for the casewithout viscous dissipation, while the opposite is true for wallcooling case. The effect of K on Nusselt number is more pro-nounced in the presence of viscous heating and also for pressureopposed flow which takes greater velocity gradients. The reasonis that although for great values of K the velocity distribution isnearly uniform over much of the duct cross section, there is a hugevelocity gradient at the solid surface. So, although the effect of vis-cous heating on the bulk temperature is not significant, the walltemperature severely increases and consequently the Nusseltnumber will be much different. It should be noted that for allnon zero values of Brinkman number as K goes to infinity the Nus-selt number value approaches zero, regardless of the magnitude ofJoule heating. The reason is that as a result of the enormous valueof viscous heating, the wall temperature is much greater than thebulk temperature. This behavior of Nusselt number is accompaniedby occurrence of a singularity in Nusselt number values of the wallheating cases as a result of changing the sign of dimensionless bulktemperature from negative for small values of K to positive forgreat values of dimensionless Debye–Huckel parameter.

Here, it will be useful to perform a typical dimensional analysisand study the effects of channel size on viscous heating effects. It isassumed that a pressure gradient of 1000 Pa m�1 at favorable andopposed directions exists and the mean velocity is fixed atU = 1 cm s�1. The relative permittivity of medium, viscosity andthe Debye length are assumed to be 80, 10�3 kg m�1 s�1 and1 nm, respectively. The liquid electrical resistivity is set to104 Xm and the wall heat flux is specified to be 1 kW m�2. The zetapotential is considered 25 mV and other parameters have the samevalues which were considered in Section 2.1. As the channel heightis increased the applied electric field is changed accordingly in or-der to keep the mean velocity constant. The dependence of Nusseltnumber on the half channel height is shown in Fig. 11, where thecases when viscous heating is neglected are shown by symbols.At smaller values of channel height, the flow is actually driven byelectrokinetic effects. Therefore, the Nusselt number for both pres-sure assisted and pressure opposed flows are the same. As thechannel height increases the electrokinetic effects become moreand more limited to the region near the wall and the pressureinfluences become more important. For pressure opposed flow,as channel height increases, due to increasing pressure effects,the velocity decreases in the core region, while it increases near

Fig. 11. Nusselt number vs. half channel height where symbols show the caseswhen viscous heating is neglected.

the wall. As a result, the energy convected by the flow decreasesin the core of channel, while it decreases near the wall, leadingto a smaller difference between the temperatures of the wall andthe bulk flow and ultimately a higher Nusselt number. As observed,the effect of increasing H is to decrease Nusselt number for pres-sure assisted flow. This is because the influence of pressure onvelocity profile is the opposite of that for pressure opposed flow.For H = 1 lm, neglecting viscous heating leads to about 5% overes-timating the Nusselt number for both cases. The influence of vis-cous heating is decreased with increasing H for pressure assistedflow and neglecting viscous heating for H = 100 lm only resultsin an overestimating of about 2%. This is due to the fact that aschannel height increases, as a result of increasing pressure effects,the velocity gradient at the wall becomes smaller. For pressure op-posed flow, the effect of viscous dissipation on Nusselt number in-creases with increasing channel height and reaches about 15% atH = 100 lm. As mentioned previously, the temperature differencebetween the wall and the bulk flow decreases with channel sizefor pressure opposed flow and may reach zero for larger valuesof H, resulting in a singularity in Nusselt number values. Nearthe singularity point, the value of Nusselt number becomes se-verely sensitive to temperature variations and even a little temper-ature variation caused by viscous heating may notably affectNusselt number.

4. Conclusions

In the present study, the fully developed combined pressureand electroosmotically driven flow in parallel plate microchannelshas been studied. The classical boundary condition of uniform wallheat flux was considered in the analysis and the effects of viscousheating as well as Joule heating were taken into account. Closedform expressions were obtained for the transverse distributionsof electrical potential, velocity and temperature and also for Nus-selt number. The problem was found to be governed by fourparameters: dimensionless pressure gradient, dimensionless De-bye–Huckel parameter, dimensionless Joule heating term andBrinkman number. The main results of this study can be summa-rized as follows:

� The Brinkman number has a significant effect on Nusselt num-ber. Generally speaking, to increase Brinkman number is todecrease Nusselt number. Although the magnitude of Jouleheating can affect Brinkman number dependency of Nusseltnumber, but the general trend remains unchanged.� Depending on the value of flow parameters, a singularity may

occur in Nusselt number values even in the absence of viscousheating, especially at great values of dimensionless Joule heat-ing term.� For a given value of Brinkman number, as dimensionless

Debye–Huckel parameter increases, the effect of viscous heat-ing increases. The reason is that although viscous heating effecton the bulk temperature is not significant, the wall temperaturedrastically increases as a result of the huge velocity gradientattainment at the solid surface and consequently the Nusseltnumber will be much different.� For a given value of Brinkman number, as dimensionless

Debye–Huckel parameter goes to infinity, the Nusselt numberapproaches zero, regardless of the magnitude of Joule heating.� The effect of Brinkman number on Nusselt number for pressure

opposed flow is more notable than purely electroosmotic flow,as a result of existing greater velocity gradients especially atthe solid surface, while the opposite is true for pressure assistedflow.

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