Streaming potential and heat transfer of nanofluids in microchannels in the presence of magnetic field

Author’s Accepted Manuscript Streaming potential and heat transfer of nanofluids in microchannels in the presence of magnetic field Guangpu Zhao, Yongjun Jian, Fengqin Li

www.elsevier.com/locate/jmmm

PII: DOI: Reference:

S0304-8853(16)30025-7 http://dx.doi.org/10.1016/j.jmmm.2016.01.025 MAGMA61049

To appear in: Journal of Magnetism and Magnetic Materials Received date: 14 October 2015 Revised date: 5 January 2016 Accepted date: 9 January 2016 Cite this article as: Guangpu Zhao, Yongjun Jian and Fengqin Li, Streaming potential and heat transfer of nanofluids in microchannels in the presence of magnetic field, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.01.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

Streaming potential and heat transfer of nanofluids in microchannels in the presence of magnetic field Guangpu Zhao, Yongjun Jian, Fengqin Li School of Mathematical Science, Inner Mongolia University, Hohhot, Inner Mongolia 010021, China

ABSTRACT:

In this work, we investigate the heat transfer characteristics of thermally developed

nanofluid flow through a parallel plate microchannel under the combined influences of externally applied axial pressure gradient and transverse magnetic fields. The analytical solutions for electromagnetohydrodynamic

(EMHD) flow

in

microchannels

are obtained

under the

Debye–Hückel linearization. The classical boundary condition of uniform wall heat flux is considered in the analysis, and the effect of viscous dissipation as well as Joule heating is also taken into account. In addition, in virtue of the velocity field and temperature field, the Nusselt number variations are induced. The results for pertinent dimensionless parameters are presented graphically and discussed in briefly. Keywords: Streaming potential; Magnetic field; Nanofluid; Heat transfer; Nusselt number.

1. Introduction Over the last decade, transport processes pertinent to microfluidics and nanofluidics based lab-on-a-chip have received serious attention due to their wide range of applications, such as in the areas of microelectronics and MEMS, inkjet printing, biomedical separation and diagnostic techniques, microactuators, microsensors and heat exchangers [1-3]. Traditionally, flow manipulation in many fluidic devices is achieved by the application of the pressure gradient. * Corresponding author. Tel. : +86 471 4991251 8313

E-mail address: [email protected] (Y.J. Jian).

2

However, in the microfluidic and nanofluidic regime, flow can also be actuated by shear-driven, surface tension-driven, electrokinetics and EMHD effects [4-9]. In this respect, electroosmosis has been extensively utilized as a driving force to manipulate liquid flows and to transport. This phenomenon can be referred to the electric double layer (EDL), which forms due to the interaction of ionized solution with static charge on dielectric surfaces. On application of the axial electric field on the EDL, the mobile ions in the EDL tend to migrate, which gives rise to a fluid body force, resulting in a bulk liquid motion via viscous effect. This is usually referred to as EOF. Fundamental investigations of EOF on hydrodynamics and thermal transport phenomena have been conducted by many researches [10-17]. The presence of the EDL causes another electrokinetic flow phenomenon, and this is usually referred as the streaming potential effect. Although in the absence of the applied electrical field, a reverse electrokinetic transport may be induced on account of the streaming potential field. By virtue of the pressure-driven or EMHD force [18], the mobile ions in the EDL can be triggered to migrate, and accumulate at one end of the channel relative to the other. In such case, a reverse electrical potential can be induced, known as streaming potential. The interaction between the streaming potential and the net mobile charge inside the nanochannel gives rise to another EOF, the direction of which is usually opposite to that of the imposed pressure-driven flow. As a consequence, the overall net flow rate reduces, and this reduction can be characterized in term of an enhanced viscosity, with the phenomenon being commonly referred to as the electroviscous effect [19-26]. Many relative studies have demonstrated that the streaming current can be employed to provide a simple and effective means of transferring mechanical energy of the pressure-driven transport and chemical energy of the EDL to electrical power of the streaming current. Corresponding researches can be found in the papers [27-32].

3

Due to the need of higher heat transfer rates in currently industrial applications, a new class of enhanced heat transfer fluids is immensely favored. Commonly known as nanofluids, these kinds of suspensions are produced by dispersing nanometer-scale particles into base liquids such as water, oil and ethylene glycol. Many applications involving nanofluids in advanced cooling systems, micro/nanochanical devices have been given in the publications by Das et al. [33], Wang [34], Wong and Leon [35], Saidur et al. [36], Shahbeddin et al. [37], Sheikholeslami et al. [38], Mahian et al. [39], Sarkar et al. [40], Ganguly and Sarkar [41], Sarkar and Ganguly [42], Malvandi and Ganji [43], and Turkilmazoglu [44] and so on. Most of the studies in the literature related to steaming potential mainly focused on the effects of EDL on liquid flow, and the influences of the streaming potential on heat transfer subjected to the pressure and magnetic fields can barely be emphsized. However, it is imperative to study the fluid flow and convective heat transfer characteristics in microchannels because of their significant engineering applications. Tan and Liu [45] numerically assessed the combined effects of wall slip and streaming potential on liquid flow and heat transfer in parallel plate microchannels with the high zeta potential assumption, and the Newtonian fluid was utilized in their study as the working fluid. In the present paper, for the first time, the impacts of the nanofluid on thermally developed flows through the microchannel, by considering the combined effects of applied pressure gradient and EMHD effects are delineated, and we expect to provide a valuable guideline towards thermodynamic idealization of the microfluidic system.

2. Mathematical formulation 2.1. Problem definition We resort to an incompressible viscous nanofluid flow through a long parallel plate

4

microchannel with channel half height of H under the combined effects of applied magnetic field and pressure-gradient. An illustration of the problem is shown in Fig. 1. The flow is assumed to be steady, hydrodynamically and thermally fully developed. The bottom and upper wall of the channel are both made of hydrophobic materials and are subjected to the same constant zeta potential. At the same time, the uniform and constant wall heat flux qw is employed along the channel walls, where qw is considered to be positive when directed into the flow. Further, the EDLs formed on the channel walls are assumed not to overlap in the present study.

Fig. 1. Geometry of the physical problem.

2.2. Electrical potential equation and approximate solution For a symmetric (z : z) electrolyte solution, the local volumetric net charge density ρe(Y) is given as

e (Y )  ze(n  n )

(1)

A solution for the electrostatic potential distribution ψ(Y) can be described by the following Poisson-Boltzmann (PB) equation

5

 2 (Y ) / Y 2   e (Y ) / 

(2)

where  is the fluid permittivity, z is the valence number of ions, e is the elementary electric charge,

n and n are the number densities of the electrolyte cations and anions, respectively, and are given by the Boltzmann distribution, i.e.

n  n0 exp[ 

ez (Y ) ] k BTav

(3)

here n0 is the bulk volume concentration of positive or negative ions which is independent of the surface electrochemistry, kB is the Boltzmann constant, and Tav is the absolute temperature over the entire channel. Supposing the electrical potential is small enough, so that the Debye–Hückel linearization can be utilized for Eq. (3)

exp[ 

ez (Y ) ez (Y ) ] 1 k BTav k BTav

(4)

by inserting Eq. (1), (3) and (4) into Eq. (2), we get the equation dictating the electrostatic potential distribution as

d 2 (Y ) / dY 2  k 2 (Y )

with k  (2n0 z 2e2 / kBTav )1 / 2

(5)

The corresponding boundary conditions are

 (Y ) Y  H   0 d (Y ) / dY Y  0  0

(6)

where ψ0 is the wall zeta potential, k is the Debye-Hückel parameter and 1/k denotes the thickness of the EDL. By solving Eq. (5) and (6), then using Eq. (2), the net charge density can be written as

e (Y )  k 2 0 cosh(kY ) / cosh(kh)

(7)

2.3. Analytical solutions of the velocity field The government equations for nanofluids flow can be expressed as

  U  0

(8)

6













nf [U t  (U  )U ]  P  eff 2U  F

(9)

 where U  (U ,V ,W ) is the flow velocity vector, t is the time, ρnf is the effective density of the nanofluid, which is given by

nf   s  (1   )  f

(10)

where  is the volume fraction of the nanoparticles , ρs is the density of a solid and ρf is the density of a fluid. µeff is the effective viscosity of the nanofluid, given by Brinkman [46]

eff 

f (1   ) 2.5

(11)

where µf is the viscosity of the base fluid . 

P is the pressure and F is the net bodyforce acting on the fluid, which is essentially contributed by

the interactions between the induced electrical field and the applied magnetic field. It is defined as

    F  e E  J  B

(12)

 In general, the electric current density J is given by ohm’s law, that is

    J   ( E  U  B)

(13)

 where E  ( ES ,0,0) , ES is the intensity of induced electric field (streaming potential) along X  direction, B is the magnetic field along Y direction and its strength is B0,  is the electrical conductivity of the nanofluid. In this paper, since the flow has been considered to be fully developed, there is no velocity gradient along the axial X direction. Further, considering the condition of 2H << W, hence velocity gradient along the Z direction can also be neglected, i.e. V = W = 0, and U = U(Y). The nanoparticles are assumed to be of uniform in shape and size. In addition, we have supposed that the magnetic Reynolds number is so small that the induced magnetic field

7

can be negligible relative to the imposed one. In this case, the magnetic field is independent of the flow velocities. Therefore, the velocity of the nanofluid governed by Navier-Stokes equation can be simplified as



dP d 2U  eff  e ES  B02U  0 dX dY 2

(14)

No-slip boundary condition on the wall of microchannel and symmetrical condition about Y axis are

U (Y ) Y   H  0 dU (Y ) dY

Y 0

0

(15a, b)

In the above equations, the following dimensionless parameters are introduced

E Y U  E Px H 2 UP  u U e   0 0 ES  s y Up f f H E0 Where

Px=-dp/dx,

Up

is

a

reference

K  kH Ha  B0 H

pressure-driven

flow

 f

velocity,

(16) Ue

is

the

Helmholtz-Smoluchowski electroosmotic velocity, E0 is the characteristic electric field, and K is called electrokinetic width of the EDL denoting the ratio of half-height H of microchannel to Debye length 1/k. Ha is the Hartmann number, which gives an estimate of the magnetic forces compared to the viscous forces. Inserting dimensionless parameters (16) into Eq. (14), the dimensionless momentum equation and relevant boundary conditions have the following form 2 d 2u cosh( Ky )  Ha u    ur K 2 ES 0 2 dy cosh( K )

u ( y) y  1  0

du dy

y 0

0

(17)

(18)

where    f eff ur  U e U p Ha  Ha  Applying the boundary condition (15), the velocity of the nanofluid can be yielded

u( y)  C cosh( Hay)  B cosh( Ky)  A where

(19)

8

A

B



(20a)

Ha2

ur K 2 ES

(20b)

2

( Ha  K 2 ) cosh( K )

C

A  B cosh( K ) cosh( Ha)

(20c)

2.4. Calculation of the streaming potential To obtain the streaming potential ES, the net ionic current through the channel section (per unit width) equaling to zero should be utilized, i.e. H

i  2ez  (nU   nU  )dY  I S  I C  0 0

(21)

where I S is streaming current, I C is conduction current, and U  are the ionic velocities expressed as

U  U 

ezES f

(22)

with f  being the ionic friction coefficient for cations and anions ( f   f   f  6rion , where rion is the ionic radius, with some degree of approximation, we assume that the cationic and anionic radii are approximately equal to around 1.5Å). Using Eq. (3), (19) and (22) into Eq. (21), the dimensionless streaming potential can be finally get as

ES 

M R  ur N

(23)

where

M

1 K Ha

N

2

tanh(K ) 

C1 2

Ha cosh( Ha)

[cosh( Ha) tanh(K )  C3 sinh( Ha)]

 C1K sin(2 K ) C12 K [ 1  ]  [cosh( Ha) tanh(K )  C3 sinh( Ha)] 2 cosh 2 ( K ) 2K cosh( Ha)

(24a)

(24b)

9

R

K BTav E 0 f 0U p

C1 

(24c)

K K  Ha 2

2

C3 

Ha K

(24d)

2.5. Temperature distribution for fully developed flow with uniform wall heat flux Based upon the above obtained velocity field, the energy equation with volumetric joule heating, and energy dissipation can be used to give the temperature distribution in the microchannel. Analytical solution of energy equation is executed with the situation of constant wall heat flux. The governing equation for thermal energy transport by considering axial conduction can be presented as

T  2T  2T dU 2 ( C p ) nf U  keff ( 2  2 )   ( ES2  B02U 2 )  eff ( ) X X Y dY

(25)

where (ρcp)nf is the effective heat capacitance of the nanofluids at the reference pressure, which is calculated by the equation given by Xuan and Li [47] as

( C p )nf   ( C p ) s  (1   )( C p ) f

(26)

where (CP)s and (CP)f are the heat capacities per unit volume of the solid nanoparticles and the base fluid respectively. keff is the effective thermal conductivity of the nanofluid, which is presented by Maxwell   k  2k f  2 (k f  k p )   keff  k f  p    k p  2k f   ( k f  k p )  

(27)

where kp is the thermal conductivity of the nanopaticle, kf is the thermal conductivity of base fluid. T is the local temperature of the nanofluid, the second term at the right hand side of (25) denotes the

10

volumetric heat generation due to Joule heating effect and the third term is the viscous dissipation. Corresponding boundary conditions in X direction can be expressed as dT dY

Y 0

 0 keff

dT dY

Y H

 qw (0r T

yH

 Tw )

(28a, b)

where qw =h (Tw-Tm) is the constant wall heat flux, h is the convective heat transfer coefficient.

Furthermore, coupled with uniform wall heat flux condition, for thermally fully developed problem, we have  2T T dTw dTm 0 =const. and   X 2 X dX dX

(29)

Under these assumptions, energy equation (25) yields

dTm  2T dU 2 ( C p ) nf U ( )  keff   ( ES2  B02U 2 )  eff ( ) 2 dX Y dY

(30)

From an overall energy balance for an elemental control volume on a length of duct dX along the centerline of the channel, the Eq. (30) can be written as



H

0

H

( C p ) nf UdTmdY   keff 0

H H  2T dU 2 2 2 d X d Y   ( E  B U ) d X d Y  eff ( )2 dXdY S 0 2   0 0 Y dY

(31)

Rearranging (31), the axial bulk temperature gradient in the thermally fully developed situation has the form dTm qw  ES2 H  Q  dX ( C p ) nf Hum

(32)

where um is axial mean velocity, which can be given by integration of U(Y) across the section of the microchannel

um 

1 H



H

0

1

U (Y )dY U P  u ( y)dy  U P 1 0

(33a)

11

and

Q  HB02U P2 2  eff U P2 3 H 1

1   u ( y)dy  0

(33b)

C B sinh( Ha)  sinh( K )  A K Ha

(33c)

1 2

1 1 1 sinh(2 Ha） ]  B 2 [1  sinh(2 K） ]  2 BCC1[cosh( Ha) sinh( K ) 2 2K 2 Ha (33d) 2 AC 2 AB 2  C3 cosh( K ) sinh( Ha)]  A  sinh( Ha)  sinh( K ) K Ha 1

 2   u 2 ( y )dy  C 2 [1  0

1

3   ( 0

du 2 1 1 1 1 ) dy  (C Ha) 2 [ sinh(2 Ha）- ]  ( BK ) 2 [ sinh(2 K）- ]  2 BCC1 HaK dy 2 4K 2 4 Ha

(33e)

[cosh( K ) sinh( Ha)  C3 cosh( Ha) sinh( K )]

Introducing the following dimensionless temperature for fully developed flow

 ( y) 

T (Y )  Tw qw H / k f

(34)

The dimensionless energy equation and boundary conditions can be described as 2 d 2 k f du  [(1  S   )u / 1  ( S  Ha Bru 2 )  Br ( ) 2 ] 2 dy keff dy

d dy

y 0

0

where S 

d dy

ES2 H qw

y 1



kf keff

Br 

(or 

 f U P2 qw H

y 1

 0)

(35)

(36a, b)

2

Br  Br    Ha Br 2  Br3

(37)

The parameter S denotes the ratio of Joule heating to heat flux from microchannel wall in physical, usually is termed as the dimensionless Joule heat parameter, and the parameter Br is Brinkman number, which expresses the ratio of heat produced by viscous dissipation to heat transported by molecular conduction. Substituting dimensionless velocity field (19) into dimensionless energy equation (35), integrating it twice and applying the boundary condition (36), finally, the solution of

12

the temperature field can be written as bellow

 ( y) 

kf

{

keff 1

(1  S   )

1

[

C Ha

2

cosh( Ha y ) 

2 B 1 1 1 1 cosh( Ky )  Ay 2 ]  Sy 2  Ha Br[ C 2 ( y 2  2 K 2 2 2 2

1 2 1 2 1 B ( y  cosh(2 Ky ))  2 BCC1 (C1 (cosh( Ky ) cosh( Ha y )  2 2 4K 2 4 Ha 1 1 C3 sinh( Hay ) sinh( Ky ))  C2C3 (cosh( Ky ) cosh( Hay )  sinh( Hay ) sinh( Ky )))  A2 y 2  C3 2 2

2 AC

cosh(2 Hay )) 

2 AB 1 1 cosh( Ky )]  Br[(C Ha) 2 ( cosh(2 Ha y )  y 2 )  ( BK ) 2 2 2 K 4 Ha 8 Ha 1 1 1 ( 2 cosh(2 Ky )  y 2 )  2 BCC1 HaK (C2 (cosh( Ky ) cosh( Ha y )  sinh( Ky ) sinh( Hay ))  8K 4 C3 2

cosh( Ha y ) 

C3 (C1 (cosh( Ky ) cosh( Ha y )  C3 sinh( Hay ) sinh( Ky )))]  C4 } (38) where

C2 

Ha

(39a)

2

Ha  K 2

C4  {

(1  S   )

1

[

C Ha

2

cosh( Ha) 

2 B 1 1 1 1 1 1 cosh( K )  A]  S  Ha Br[ C 2 (  cosh(2 Ha))  B 2 2 K 2 2 2 2 4 Ha2 2

1 1 (  cosh(2 K ))  2 BCC1 (C1 (cosh( K ) cosh( Ha)  C3 sinh( Ha) sinh( K ))  C2C3 (cosh( K ) cosh( Ha)  2 4K 2 1 1 2 AC 2 AB 1 1 sinh( Ha) sinh( K )))  A2  cosh( Ha)  2 cosh( K )]  Br[(C Ha) 2 ( cosh(2 Ha)  )  2 2 C3 2 K 4 Ha 8Ha ( BK ) 2 (

1 1 1 cosh(2 K )  )  2 BCC1 HaK (C2 (cosh( K ) cosh( Ha)  sinh( K ) sinh( Ha))  C3 (C1 (cosh( K ) 2 8K 4 C3

cosh( Ha)  C3 sinh( Ha) sinh( K )))]}

(39b) The dimensionless bulk temperature can be defined as 1

1

0

0

 m   u( y) ( y)dy /  u( y)dy  k f (Tm  Tw ) / qw H The heat transfer rate can be expressed in the form of Nusselt number Nu, which is defined as

(40)

13

Nu 

2 Hqw keff (Tw  Tm )

(41)

Combing Eq. (39) and Eq. (40), the Nusselt number can be derived as

Nu  

kf 2 keff  m

(42)

With the aid of the velocity and temperature distributions having been known from Eq. (19) and Eq. (38) respectively, the Nu number can be calculated from Eq. (42) eventually.

3. Results and discussion The analytical expressions for the dimensionless velocity, the dimensionless temperature and the Nusselt number are utilized to simulate the heat transfer characteristics of nanofluid (Al2O3-water) in microchannel in the presence of magnetic field. We consider the thermophysical properties of nanofluids at reference of temperature (25 oC), and typical parameters can be taken as follows: Tav is 298 K, kB is 1.38110-23J K-1, ψ0 is -0.025V, H =50µm, ρs =3600 kg m-3, ρf =997.1 kg m-3, µf =8.9110-3 kg m-1 s-1. E0 is set to be 1000V m-1, Ue is assumed to be 10-4m s-1, and the value of K is set as 30. The order of electrical conductivity of fluids σ approximately changes from 2.210-4–106 S m-1 [48]. By calculating from Eq. (16), the range of Hartmann number (Ha) is from 3.610-6 to 3, corresponding imposed magnetic field B0 is varied from 40mT–0.44T [49-50]. Moreover, the order of Brinkman number (Br) is set to be 0~0.1 and the order of Joule heat parameter (S) is 0~10, which are shown to be in accordance with the physically acceptable values established in the studies [4, 5, 41]. In addition, the parameter ur is taken as 1.0, indicating the strength of comparable pressure-driven velocity and electroosmotic velocity. Fig. 2 depicts the variations of the dimensionless streaming potential field as the function of the

14

Hartmann number (Ha) for different volume fraction of the nanoparticles. We can see that dimensionless streaming potential exhibits distinct enhancement with the Ha. The reason is that the larger Hartmann number easily tends to trigger the smaller dimensionless velocity (as evident from Fig. 3(a)), which leads to a consequent reduction in the strength of advective transport of ionic charges, and hence gives rise to the decrement in the magnitude of the streaming potential. Meanwhile, effects of volume fraction of the nanoparticles on streaming potential are witnessed, and the profiles of streaming potential increase with the increase in the nanoparticle volume fraction in Fig. 2.

Fig. 2. Variations of the dimensionless streaming potential with the Hartmann Number for different values of volume fraction when ur = 1.0 and K=30 in the microchannel.

Fig. 3 shows the variations of velocity distribution with Hartmann number and nanoparticle volume fraction. We can observe that the velocity distribution of the nanofluid decreases as Ha increases. In fact, the overall velocity profile is a superposition of the pressure driven flow and the counteracting EMHD flow. The reason of the aforementioned is that EMHD force acts as a retarding force in axial direction transport compared with the effective driving pressure gradient, as

15

illustrated in Fig. 3(a). From Fig. 3(b), we can see the velocity of the nanofluid deceases with the increment of the nanoparticle volume fraction. This mainly is attributed to the fact that with the enhancement of the nanoparticle volume fraction  , the effective viscosity of the nanofluid increases in response to the shear rate, which leads to a greater dispersion in the velocity profile.

Fig. 3. Effects of the Hartmann Number when ur = 1.0, K=30, and  =3% (a) and volume fraction when ur = 1.0, Ha=2 and K=30 (b) on dimensionless velocity distribution.

Fig. 4 depicts the variations of dimensionless temperature with Hartmann number and nanoparticle volume fraction. The results of Fig.4 (a) show that, the increase in Ha causes an increase in the dimensionless temperature of the nanofluid inside the microchannel. Owning to the increasing magnitude of Ha, the significantly reduced flow velocity near the channel center greatly impedes the advective heat transfer. Hence, the local temperature of the nanoliquid rises, due to enhanced transverse diffusive heat transfer from the channel walls. The final outcome is the wall temperature decreases and the dimensionless temperature increases. The influence of nanoparticle volume fraction on the dimensionless temperature is demonstrated in Fig. 4(b). It is found that dimensionless temperature increases with increase in nanoparticle concentration. Due to the higher effective thermal conductivity of nanoparticles, the thermal diffusion effects in the nanofluid

16

evidently enhance compared with the base fluid (  =0). Consequently, the larger nanoparticle volume fraction results in the larger local temperature, and the dimensionless temperature increases.

Fig. 4. Effects of the Hartmann Number when ur = 1.0, K=30,  =3% and Br=0.01 (a) and volume fraction when ur = 1.0, Ha=2, K=30 and Br=0.01 (b) on dimensionless temperature profiles.

Fig. 5 depicts the Nusselt number variations with the Hartmann number for different values of nanoparticle volume fraction. In such situation, the Nusselt number is found to increase with Ha. This behavior can be explained by the Fig. 4(a). The larger Ha brings in the larger dimensionless temperature of the fluid, and causes an increase in the bulk mean temperature relative to the microchannel wall temperature. This gives rise to the decrement of the quantity (Tw-Tm). Considering the condition of constant wall heat flux, given by qw=h (Tw-Tm), the convective heat transfer coefficient h increases, and Nusselt number increases coherently with Ha. In addition, the thermal conductivity of the nanofluid is higher than that of base fluid (  =0) due to the presence of the nanoparticles, as a consequence, the heat transfer performance is correspondingly improved, as illustrated in Fig. 5.

17

Fig. 5. Effects of the volume fraction on the nature of variation of Nusselt number with the Hartmann number when ur = 1.0, K=30 and Br=0.01.

Fig. 6(a) delineates the effects of Joule heating on the Nusselt number with the Hartmann number. Regarding the spatially uniform heating effect of Joule heat, the increase in Joule heating implies the increase in the liquid temperature in a homogeneous manner. Thereby, the temperature gradient in the vicinity of the wall reduces. This reduction ultimately causes a decrement of Nusselt number with increasing of Joule heating irrespective of value of Ha. Fig. 6(b) evaluates the effects of Joule heating on the Nusselt number as a function of the nanoparticle volume fraction. It is worthy of being mentioned that the Nusselt number decreases with the enhancement of the Joule heating parameter, and the heat transfer feature can be reduced by increasing nanoparticle volume fraction.

18

Fig. 6. Effects of the Hartmann number when ur = 1.0, K=30 and  =3% (a) and volume fraction when ur = 1.0, K=30 and Ha=2 (b) on the nature of variation of Nusselt number with dimensionless Joule heat parameter.

Fig. 7(a) shows the effects of Brinkman number on the Nusselt number with the Hartmann number. Since the bigger Brinkman number enhances the viscous dissipation effect, which tends to increase the wall temperature Tw much more than the mean temperature Tm. The result of this will increase the magnitude of the quantity (Tw-Tm). As explained in the preceding section, the convective heat transfer coefficient h decreases. Accordingly, Nusselt number decreases with Br coherently no matter of value of Ha. In addition, Fig. 7(b) also depicts a decrease tendency for Nusselt number and an improvement of the heat transfer performance by increasing nanoparticle volume fraction.

Fig. 7. Effects the Hartmann number when ur = 1.0, K=30 and  =3% (a) and volume fraction when ur = 1.0, K=30

19

and Ha=2 (b) on the nature of variation of Nusselt number with Brinkman number.

4. Conclusions In this study, the flow and heat transfer characteristics under the assumption of thermally fully developed nanofluid flow (Al2O3-water) in a microchannel have been critically analyzed. The flow is actuated by combined effects of external pressure gradient and EMHD force. The closed form expressions for velocity and temperature are derived. Furthermore, the Nusselt number variations are also discussed. The influences of pertinent flow parameters such as the dimensionless electrokinetic parameter (K), Hartmann number (Ha), nanoparticle volume fraction (  ), Brinkman number (Br) and Joule heating parameter (S) are investigated qualitatively. The results show that the streaming potential is found to increase with Ha. The amplitude of velocity decreases with an increase in the particle concentration. The dimensionless temperature of the nanofluid increases with Ha and  . The Nusselt number is found to increase with Ha. Furthermore, the Nussult number decreases with an increase in S and Br for different Ha. More interestingly, the distinct heat transfer performances with nanoparticle volume fraction can be witnessed.

Acknowledgments The work was supported by the National Natural Science Foundation of China (Nos.11472140, 11562014 and 11362012), the Program for Young Talents of Science and Technology in Universities

of

Inner

Mongolia

Autonomous

Region

(No.NJYT-13-A02),

the

Inner

Mongolia Grassland Talent (No. 12000-12102013), Opening fund of State Key Laboratory of Nonlinear Mechanics.

References

20

1.

K. Ohno, K. Tachikawa, A. Manz, Microfluidics: applications for analytical purposes in chemistry and biochemistry, Electrophoresis 29 (2008) 4443–4453.

2.

H. Becker, C. Gartner, Polymer microfabrication methods for microfluidic analytical applications, Electrophoresis 21 (2000) 12–26.

3.

K. Nandy, S. Chaudhuri, R. Ganguly, I.K. Puri, Analytical model for the magnetophoretic capture of magnetic spheres in microfluidic devices, J. Magn. Magn. Mater. 320 (2008) 1398–1405.

4.

S. Chakraborty, D. Paul, Microchannel flow control through a combined electromagnetohydrodynamic transport, J. Phys. D: Appl. Phys. 39 (2006) 5364.

5.

R. Chakraborty, R. Dey, S. Chakraborty, Thremal characteristics of electromagnetohydrodynamic flows in narrow channels with viscous dissipation and Joule heating under constant wall heat flux, Int. J. Heat Mass Transfer 67 (2013) 1151–1162.

6.

M. Buren, Y.J. Jian, L. Chang Electromagnetohydrodynamic flow through a microparallel channel with corrugated walls, J. Phys. D: Appl. Phys. 47 (2014) 425501.

7.

M. Buren, Y.J. Jian, Electromagnetohydrodynamic (EMHD) flow between two transversely wavy microparallel plates, Electrophoresis 36 (2015) 1539–1548.

8.

D.Q. Si, Y.J. Jian, Electromagnetohydrodynamic (EMHD) micropump of Jeffrey fluids through two parallel microchannels walls with corrugated, J. Phys. D: Appl. Phys. 48 (2015) 085501.

9.

G.P. Zhao, Y.J. Jian, L. Chang, M. Buren, Magnetohydrodynamic flow of generalized Maxwell fluids in a rectangular micropump underan AC electric field, J. Magn. Magn. Mater. 387 (2015) 111–117.

10. P. Dutta, T.C. Warburton, A. Beskok, Numerical simulation of mixed electroosmotic/ pressure driven flows in complex micro-geometries, J. Numer. Heat Transf. 41(2002) 131–148. 11. D. Maynes, B.W. Webb, Fully developed electro-osmotic heat transfer in microchannels, Int. J. Heat Mass

21

Transfer 46 (2003) 1359–1369. 12. K. Horiuchi, P. Dutta, Joule heating effects in electroomotically driven microchannels, Int. J. Heat Mass Transfer 47 (2004) 3085–3095. 13. R. Dey, D. Chakraborty, S. Chakraborty, Extended Graetz problem for combined electroosmotic and pressure-driven flows in narrow confinements with thick electrical double layers, ASME. J. Heat Transfer133 (2011) 4724–4733. 14. Y.J. Jian, Q.S. Liu, L.G. Yang, AC electroosmotic flow of generalized Maxwell fluids in a rectangular microchannel, J. Non-Newtonian Fluid Mech. 166 (2011) 1304–1314. 15. Q.S. Liu, Y.J. Jian, L.G. Yang, Time periodic electroosmotic flow of the generalized Maxwell fluids between two micro-parallel plates, J. Non-Newtonian Fluid Mech. 166 (2011) 478–486. 16. J. Su, Y.J. Jian, L. Chang, Thermally fully developed electroosmotic flow through a rectangular microchannel, Int. J. Heat Mass Transfer 55 (2012) 62856290. 17. S.X. Li, Y.J. Jian, Z.Y. Xie, Q.S. Liu, F.Q. Li, Rotating electro-osmotic flow of third grade fluids between two microparallel plates, Colloid Surf. A: Physicochem. Eng. Aspects 470 (2015) 240247. 18. M. Farooque, S. Chakraborty, Hydroelectrical energy conversion in narrow confinements in the presence of transverse magnetic fields with electrokinetic effects, Phys. Fluids 21 (2009) 122003. 19. S. Chakraborty, S. Das, Streaming-field-induced convective transport and its influence on the electroviscous effects in narrow fluidic confinement beyond the Debye-Hückel limit, Phys. Rev. E 77 (2008) 037303. 20. S. Das, S. Chakraborty, Influence of streaming potential on the transport and separation of charged spherical solutes in nanochannels subjected to particle–wall interactions, Langmuir 25 (2009) 9863–9872. 21. S. Das, S. Chakraborty, Effect of conductivity variations within the electric double layer on the streaming potential estimation in narrow in narrow fluidic confinements, Langmuir 26 (2010) 11589–11596.

22

22. J. Chakraborty, S. Ray, S. Chakraborty, Role of streaming potential on pulsating mass flow rate control in combined electroosmotic and pressure-driven microfluidic devices, Electrophoresis 33 (2012) 419–425. 23. A. Bandopadhyay, S. Chakraborty, Combined effects of interfacial permittivity variations and finite ionic sizes on streaming potentials in nanochannels, Langmuir 28 (2012) 17552–17563. 24. J. Dhar, U. Ghosh, S. Chakraborty, Alterations in streaming potential in presence of time periodic pressure-driven flow of a power law fluid in narrow confinements with nonelectrostatic ion–ion interactions, Electrophoresis 35 ( 2014) 662–669. 25. S. Chanda, S. Sinha, S. Das, Streaming potential and electroviscous effects in soft nanochannels: towards designing more efficient nanofluidic electrochemomechanical energy converters, Soft Matter 10 (2014) 7558–7568. 26. G. Chen, S. Das, Streaming potential and electroviscous effects in soft nanochannels beyond Debye–Hückel linearization, J. Colloid Interface Sci. 445 (2015) 357–363. 27. T. Nguyen, Y. Xie, L.J. de Vreede, A. van den Berg, J.C.T. Eijkel, Highly enhanced energy conversion from the streaming current by polymer addition, Lab Chip 13 ( 2013) 3210. 28. B.S. Kilsgaard, S. Haldrup, J. Catalano, A. Bentien, High figure of merit for electrokinetic energy conversion in Nafion membranes, J. Power Sources 247 (2014) 235–242. 29. A. Siria, P. Poncharal, A.L. Biance, R. Fulcrand, X. Blase, S.T. Purcell, L. Bocquet, Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube, Nature 494 ( 2013) 455–458. 30. Y.B. Xie, X.W. Wang, J.M. Xue, K. Jin, L. Chen, Y.G Wang, Electric energy generation in single track-etched nanopores, Appl. Phys. Lett. 93 (2008) 163116. 31. A.M. Duffin, R.J. Saykally, Electrokinetic power generation from liquid water microjets. J. Phys. Chem. C 112 (2008) 17018–17022.

23

32. Y.B. Xie, D. Bos, L.J. de Vreede, H.L. de Boer, M.J. van der Meulen, M. Versluis, A.J. Sprenkels, A. van den Berg, J.C.T. Eijkel, High-efficiency ballistic electrostatic generator using microdroplets, Nat. Commun. 5 (2014) 3575. 33. S.K. Das, N. Putra, W. Roetzel, Temperature dependence of thermal conductivity enhancement for Nanofluids, ASME J. Heat Transfer 125 (2003) 567–574. 34. X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J. Therm. Sci. 46 (2007) 1–19. 35. K.F.V. Wong, O.D. Leon, Applications of nanofluid: current and future. Adv. Mech. Eng. (2010) Article ID: 519659. 36. R. Saidur, K.Y. Leong, H.A. Mohammad, A review on applications and challenges of nanofluids, Renew. Sustain. Energy Rev. 15 (2011) 1646–1668. 37. K. Shahbeddin, Mohammadian, Y.W. zhang, Anlysia of nanofluid effects on thermoelectric cooling by micro-pin-fin heat exchangers, J. Appl. Therm. Eng.70 (2014) 282–290. 38. M. Sheikholeslami, M. Hatami, D.D. Ganji, Nanofluid flow and heat transfer in a rotating system in the presence of a magnetic field, J. Mol. Liq. 190 (2014) 112–120. 39. O. Mahian, I. Pop, A.Z. Sahin, H.F. Oztop, S. Wongwises, Irreversibility analysis of a vertical annulus using TiO2/water nanofluid with MHD flow effects, Int. J. Heat Mass Transfer 64 (2013) 671–679. 40. S. Sarkar, S. Ganguly, G. Biswas, Buoyancy driven convection of nanofluids in an infinitely long channel under the effect of a magnetic field, Int. J. Heat Mass Transfer 71 (2014) 328–340. 41. S. Ganguly, S. Sarkar, Thermally developing combined electroosmotic and pressure-driven flow of nanofluid in a microchannel under the effect of magnetic field, Chem. Eng. Sci. 126 (2014) 10–21. 42. S. Sarkar, S. Ganguly, Fully developed thermal transport in combined pressure and electroosmotically driven

24

flow of nanofluid in microchannel under the effect of a magnetic field, Microfluid Nanofluid 18 (2015) 623–636. 43. A. Malvandi, D.D. Ganji, Magnetic field effect on nanoparticles migration and heat transfer of water/alumina nanofluid in channel, J. Magn. Magn. Mater. 362 (2014) 172–179. 44. M. Turkilmazoglu, Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids, Chem. Eng. Sci. 84 (2012) 182–187. 45. D.K. Tan, Y. Liu, Combine effects of streaming potential and wall slip on flow and heat transfer in microchannels, Int. J. Heat Mass Transfer 53 (2014) 39–42. 46. H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. 47. Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, ASME J. Heat Transfer 125 (2003) 151–155. 48. H. Kabbani, A. Wang, X. Luo, S. Qian, Modeling Redox-based magnetohydrodynamics in three-dimensional microfluidic channels, Phys. Fluids 19 (2007) 083604. 49. J. Jang, S.S. Lee, Theoretical and experimental study of MHD micropump, Sens. Actuators A Phys. 80 (2000) 84–89. 50. M. Rivero, S. Cuevas, Analysis of the slip condition in magnetohydrodynamic (MHD) micropumps, Sens. Actuators B Chem. 166 (2012) 884–892.

Figure Captions Fig. 1. Geometry of the physical problem.

Fig. 2. Variations of the dimensionless streaming potential with the Hartmann Number for different values of volume fraction when ur = 1.0 and K=30 in the microchannel.

25 Fig. 3. Effects of the Hartmann Number when ur = 1.0, K=30 and  =3% (a) and volume fraction when ur = 1.0, Ha=2 and K=30 (b) on dimensionless velocity distribution. Fig. 4. Effects of the Hartmann Number when ur = 1.0, K=30,  =3% and Br=0.01 (a) and volume fraction when ur = 1.0, Ha=2, K=30 and Br=0.01 (b) on dimensionless temperature profiles.

Fig. 5. Effects of the volume fraction on the nature of variation of Nusselt number with the Hartmann number when ur = 1.0, K=30 and Br=0.01. Fig. 6. Effects of the Hartmann number when ur = 1.0, K=30 and  =3% (a) and volume fraction when ur = 1.0, K=30 and Ha=2 (b) on the nature of variation of Nusselt number with dimensionless Joule heat parameter. Fig. 7. Effects the Hartmann number when ur = 1.0, K=30 and  =3% (a) and volume fraction when ur = 1.0, K=30 and Ha=2 (b) on the nature of variation of Nusselt number with Brinkman number.

Research Highlights:



Flow and heat transport characteristics of nanofluids are analytically explored.



The nanofluid is under the combined effects of the streaming potential and the magnetic field.



The flow is confined to a microchannel under the Debye-Hückel approximation.



The impacts of pertinent parameters on the velocity, temperature and Nusselt number are

discussed.