Journal of Thermal Analysis and Calorimetry https://doi.org/10.1007/s10973-018-7326-4 (0123456789().,-volV)(0123456789().,-volV)

Thermal transport of combined electroosmotically and pressure driven nanofluid flow in soft nanochannels Guangpu Zhao1 • Yongjun Jian2 Received: 10 January 2018 / Accepted: 22 April 2018  Akade´miai Kiado´, Budapest, Hungary 2018

Abstract In this paper, the heat transfer characteristics of the nanofluid through a parallel plate soft nanochannel are investigated under the fully developed condition. The flow is actuated by the combined effects of pressure gradient and implied electric field. Based on the ion partitioning effect and the Debye–Hu¨ckel linearization, the analytical solutions for electrokinetic flow in such nanochannel are obtained. Meanwhile, the uniform wall heat flux is utilized in the analysis, and the influences of viscous dissipation and the Joule heating are taken into account. The results for pertinent dimensionless parameters are presented graphically and discussed in brief. The relevant result reveals that the ion partitioning effect can greatly impact the electrostatic potential, velocity and temperature distribution. Furthermore, this ion partitioning effect can exert an influence on heat transfer of the nanofluid. The present study also indicates the possibility of alteration in the nanofluid heat transfer by the use of nanoparticle volume. Keywords Soft nanochannel  Ion partitioning effect  Electrokinetic  Heat transfer  Nusselt number

Introduction Owning to the rapid development of the microelectromechanical systems (MEMS) sensors, micropumps, microvalves and micromedical instruments, the electrokinetic transport in such devices has been received extensive attention [1–3]. Electroosmosis is a basic electrokinetic phenomenon and is widely employed as a pivotal flow actuation technology in diverse applications. This electroosmotic transport lies in the facts that when the polar material surface is contacted with the electrolyte solution, the counter ions of electrolyte solution will be triggered to move toward the charger interface and finally form a layer with a high concentration of counterions, which is usually named as Stern layer. Combining with outer diffuse layer, the electric double layer (EDL) is & Yongjun Jian [email protected] 1

College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, Inner Mongolia, People’s Republic of China

2

School of Mathematical Science, Inner Mongolia University, Hohhot 010021, Inner Mongolia, People’s Republic of China

generated in the vicinity of the charged surface. Once the external electric field is utilized in the EDL, the mobile ions in the diffuse region are actuated to move, and bring about a bulk liquid motion via viscous effect, which is normally referred to as the electroosmotic flow (EOF). An abundance of investigations of EOF about hydrodynamics and thermal transport phenomena have been reported in various geometric domains of micro- and nanochannels [4–7]. In recent years, many scholars pay the attention on a novel nanofluidic channel, known as soft nanochannel, which is made by polyelectrolyte materials on nanochannel walls. Those soft nanochannels are characterized by wallgrafted ion-penetrable charged polyelectrolyte layer (PEL). The charge in PEL is associated with a fixed charge density of ions, which is produced by the ionization of the polyelectrolyte molecules in the PEL. Therefore, the PEL can also be denoted as a fixed charge layer (FCL). The PEL ions exist only within the PEL, while the electrolyte ions can present in the whole soft nanochannel. Consequently, the PEL-electrolyte interface can be believed as a semipenetrable membrane. In particular, the phenomena of the soft nanochannel are analyzed at the nanoscopic level; a much smaller dimension is required. The physical size of the domain places the calculations in flow regime, and

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there is a little scope for continuum or modified continuum modeling in this case. A variety of applications pertaining to the soft nanochannels have been implemented in nanofluidic systems [8–12]. Due to the contribution of the PEL, the Donnan potential within the FCL emerges and alters the EDL electrostatics in the entire system. Also, the FCL imparts an additional drag force on the fluid motion and gives rise to the significant impacts on the EOF. To elucidate these two effects, many relevant researches have been launched. The pioneered work in this context was done by Donath and Voigt [13]. They offered an electrokinetic theory for the soft nanochannels and considered the influences of the induced streaming potential. Ohshima and Kondo [14] executed EOF in a parallel plate channel with ion-permeable surfaces. The similar study was probed by Keh and Ding [15], they analyzed a comprehensive EOF in both capillary tubes and slits, and different polyelectrolytes materials were taken into consideration. Ma and Keh [16] presented the diffusioosmotic flow of an electrolyte solution in the narrow capillary tube and slit. Very recently, Chanda et al. [17] researched the streaming potential and electroviscous effects on soft nanochannels, and a comparison between soft and rigid nanochannels was executed in order to assess the apparent disparities in those two kinds of nanochannels. Beyond Debye–Huckel linearization, Chen and Das [18] probed streaming potential and electroviscous effects in soft nanochannels. Furthermore, in their later research [19], electroosmotic transport in soft nanochannels with PHdependent charge density was discussed and a suppressed EOF velocity with PH-dependent was observed. Patwary et al. [20] carried out the electrochemical energy conversion efficient in soft nanochannels with PH-dependent charge density and demonstrated that the increase in PH and polyelectrolyte layer thickness substantially enhanced the energy conversion efficiency. Their relevant study provides us a wider perspective to explore the soft nanochannels. In view of the boundary slip condition, Matin and Ohshima [21] examined the mixed electroosmotic and pressure-driven flows through a soft charged nanochannel, and a more comprehensive comparison between soft and rigid nanochannel was given briefly. Later, Matin and Ohshima [22] researched the thermal transport features of combined electroosmotic and pressure driven flow under the thermal fully developed condition in such soft nanochannels. Just recently, Li et al. [23, 24] not only examined the alternating current electroosmotic flow in polyelectrolyte-grafted nanochannels, but also probed the investigation of transient alternating current electroosmotic flow of a Jeffrey fluid through a polyelectrolyte-grafted nanochannel. In view of the steric effects, Xing and Jian [25] numerically analyzed the Steric effects on electroosmotic flow in soft nanochannels.

123

Beyond the assumption of the uniform permittivities both inside and outside the PEL, Poddar et al. [26] researched the effects of ion partitioning on the electrokinetics in the polyelectrolyte-grafted nanochannel and compared electrokinetic energy conversion efficiency (EKEC) with the usual case. Since the need of higher heat transfer rates in the industrial application, a new fluid named nanofluid was introduced in 1990s. This functional fluid is consisted with the solid nanoparticles (normally smaller than 100 nm in diameter) and the conventional liquids. Those liquids contain water, oil, ethylene glycol and so on. Many applications involving nanofluids in diverse advanced systems and micro- or nanomechanical devices have been reported in the publications by Hedayati et al. [27], Ahammed et al. [28], Mashaei et al. [29], Sheikholeslami et al. [30], Mahian et al. [31], Hedayati and Domairry [32], Ganguly and Sarkar [33, 34], Malvandi and Ganji [35], Turkilmazoglu [36], Torabi et al. [37], Dickson et al. [38] and so on. From the above review, we can find many researches with regard to soft nanochannels are mainly emphasized on the two aspects: electrokinetic transport and EKEC. The research of nanofluid heat transfer in such nanochannel is rarely reported in the studies. The only study was carried out by Matin and Ohshima [22], and they considered the general fluid as the working fluid. Moreover, it has been verified that the ion partitioning effects can be emerged when the membrane matrix and bulk solvent possess different dielectric permittivity [26], which is an amusing phenomenon. Therefore, we delineate the heat transfer characteristics of the nanofluid in a soft nanochannel in the present paper, considering the ion partitioning effect and constant wall heat flux condition. In order to use the linear Debye–Hu¨ckel approximation, we suppose the wall electric potential is very small. The results for velocity and temperature are presented in closed-form expression. Furthermore, the variations of the Nusselt number are reduced thoroughly. The applications of this nanoscale heat transfer analysis are concerned with the thermal performance of the nanoscale systems, for example, biological cell membranes for medical science [8].

Mathematical formulation Problem definition Consider the situation where flow of an incompressible viscous nanofluid takes place through a long parallel plate soft nanochannel with channel half height of h under the mixed effects of pressure gradient and electrokinetic transport. This nanochannel is consisted of wall-grafted

Thermal transport of combined electroosmotically and pressure driven nanofluid flow in soft…

ion-penetrable charged polyelectrolyte layer (PEL) of thickness d. The bottom and upper walls of this channel bear the same constant charge density. The ion partitioning effect is taken into account (i.e., the permittivities of the electrolyte solution and the PEL are appreciably different). An illustration of the problem is sketched in Fig. 1. The flow is assumed to be steady and hydrodynamically and thermally fully developed. Also it is assumed that the uniform and constant wall heat flux qw is applied along the channel walls, where qw is considered to be negative when directed out the flow. Also, the EDLs formed on the channel walls are assumed not to overlap.

Electrical potential equation and approximate solution The presence of PEL in soft nanochannels gives rise to two mainly changes compared to rigid channels. One is PEL generates a Donnan potential which enormously alters the EDL electrostatics in the entire system, and the other is PEL imparts an additional drag force on the fluid motion, as mentioned in [17]. Furthermore, in order to probe the heat transfer features in such soft nanochannels, we should firstly think over the distribution of electrostatic potential. Similar to the work of [26], according to the principle of the ion partitioning effect, we get the equations dictating the electrostatic potential distributions in dimensionless form as   d2 w1 2zen0 ezw1 ¼ sin h ð1 þ d\y\0Þ ð1Þ dy2 e0 er;c KB Tav     d2 w2 2zen0 ezw2 DWi ¼ sin h exp  dy2 e0 er;PEL KB Tav KB Tav ð2Þ zeN ð1\y\  1 þ dÞ  e0 er;PEL In Eqs. (1) and (2), w1,2 are the electric potential of the electrolyte solution and PEL. e is the proton charge, z and N are the valence and the ionic number concentration of the

qw PEL

Ex

d y x h

PEL

qw Fig. 1 Geometry of the physical problem, coordinate system, polyelectrolyte layer (PEL)

PEL ions, n0 is the ionic number concentration of in the bulk electrolyte, e0 is the permittivity of a vacuum, er,c is the relative permittivity of the electrolyte, and er,PEL is the 2

relative permittivity of the PEL layer. DWi ¼ ðzeÞ 8pri   1 1 ePEL  ec , where Wi is the Born energy and ri is the radius of the ions. To achieve the analytical solution, we use the Debye– Hu¨ckel linearization, which is only valid for jezw=KB Tav j\1 at the average temperature Tav = 300 K and the electric potential being smaller than 25.7 mV. Some dimensionless parameters are given as: y ¼ y=h,  ¼ DW=KB Tav , d ¼ d=h, k ¼ k=h, kFCL ¼ kFCL =h, DW  f ¼ er;PEL er;c , the EDL thickness and the equivalent EDL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  thickness, defined as k ¼ er;c e0 KB Tav 2z2 e2 n0 , kFCL ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  er;PEL e0 KB Tav zNe2 jZ j, wherein f is the ratio of the relative dielectric permittivity of the PEL layer to the dielectric permittivity of bulk electrolyte solution.  d2 w 1  1  y\0Þ ¼ 2w ð3Þ 1 ð1 þ d\ 2  d y k !  d2 w 1 1 1 2  expðDWÞ   w ¼ ð4Þ f k2 2 d y2 k2FCL  ð1\ y\  1 þ dÞ To achieve an analytical solution for those equations, following boundary conditions are needed   dw 1 y¼0 ¼ 0 d y       w   þ ¼ w2 y¼ð1þdÞ 1 y¼ð1þdÞ

ð5aÞ ð5bÞ

    dw 1 dw 1 2 y¼ð1þdÞ   þ ¼ f d d y y y¼ð1þdÞ

ð5cÞ

  dw 2 y¼1 ¼ X dy

ð5dÞ

where X ¼ rzeh=ekB Tav is the dimensionless surface charge density on the walls and r is the surface charge density. Based upon Eq. (5), the dimensionless form of the electrostatic potential distributions can be obtained as    ¼ A1 cos h y w ð1 þ d  y  0Þ ð6Þ 1 k   2 k  ¼ A2 cos hðb w y Þ þ A sin hðb y Þ þ 3 2  ð7Þ wkFCL  ð1  y   1 þ dÞ where

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 w ¼ expðDWÞ rffiffiffiffi 1 w b¼  k f

ð8Þ

  d1  b sin hðbðd  1ÞÞ cos O1 ¼ cos hðbðd  1ÞÞ sin h  k   d1 O2 ¼ sin hðbðd  1ÞÞ sin h  b cos hðbðd  1ÞÞ cos k

  d1  k   d1 h k h

ð9Þ  A2 cos hðbðd  1ÞÞ þ A3 sin hðbðd  1ÞÞ þ k2 ðwk2FCL Þ  A1 ¼  cos hððd  1Þ kÞ X A2 ¼ A3 cot hðbÞ þ b sin hðbÞ   2   sinXb  wkk2 sin h d1 hðbÞ O1 k FCL A3 ¼ cot hðbÞO1 þ O2

Analytical solutions of the velocity field The flow has been considered to be fully developed in a nanochannel with polyelectrolyte walls, subject to the combined effect of applied electric field Ex and axial pressure gradient. Since the flow field depends on the potential distribution inside the channel, there is a distinction between the momentum equation inside and outside the FCL. Combined with the ion partitioning effect, the velocity of the fluid governed by the modified Navier– Stokes equation can be written as 2

d u1 dp  þ qe1 Ex ¼ 0 dx dy2

leff

d2 u2 dp  þ qe2 Ex  lc u2 ¼ 0 dx dy2

ðh þ d  y  0Þ

ð11Þ

ðh  y   h þ dÞ ð12Þ

here u1,2 are the axial velocity outside and inside the FCL, leff is the viscosity of the electrolyte, defined as [39] " #!  2x dp 2=3ðxþ1Þ leff ¼ ð1 þ 2:5/Þ 1 þ 1 / ð13Þ uf D here lf the viscosity of the base fluids, / is the volume fraction of the nanoparticles, dp is the nanoparticle diameter, D is the hydraulic diameter of the microchannel, x ¼ 1=4 and 1 ¼ 280. This viscosity is assumed to be same value both inside and outside the FCL. lc is the drag coefficient of the polyelectrolyte layer. Within the FCL, there will be an additional resistive force quantified as lcu2. This term exhibits that the flow model in such a soft nanochannel is similar to Darcy model in a porous media.

123

h2 dp ue ur ¼ up ¼ lf dx up pffiffiffiffiffiffiffiffiffiffiffi l a ¼ h lc =lf g ¼ f leff

u ¼

u up

 w d2 u1  g  ur g 21 ¼ 0 2 d y k

ue ¼

kB Tav e0 er Ex ez lf

ð1 þ d  y  0Þ

 w d2 u2  g  ur gw 22  la2 u2 ¼ 0 2 d y k

ð14Þ

ð15Þ

 ð1  y   1 þ dÞ ð16Þ

ð10Þ

leff

Substituting dimensionless electrostatic potential (6) and (7), and introducing following dimensionless form, then the momentum equation can be yielded as

where up is a reference pressure-driven flow velocity, ue is the Helmholtz–Smoluchowski electroosmotic velocity, ur is the velocity scale ratio which implies the ratio of the pressure-driven flow velocity to the Helmholtz–Smolupffiffiffiffiffiffiffiffiffiffiffi chowski velocity, a ¼ h lc =lf is termed as drag parameter. Corresponding boundary conditions are as follows d u1  ð17aÞ y¼0 ¼ 0 d y     2 y¼ð1dÞ ð17bÞ u1 y¼ð1dÞ  þ ¼ u d u1  d u2  y¼ð1dÞ   þ ¼ d y d y y¼ð1dÞ  d u2  u2 y¼1 ¼ c y¼1 d y

ð17cÞ ð17dÞ

where c ¼ Ls =h and Ls is the slip length. Solving Eqs. (15) and (16), and employing the condition (17), the exact solutions can be written as   y 1 2 u1 ¼ gy þ ur gA1 cos h  þ B2 ð1 þ d  y  0Þ 2 k ð18Þ yÞ þ D2 sin hða yÞ þ C1 cos hðb yÞ u2 ¼ D1 cos hða  þ C2 sin hðb yÞ þ C3 ð1  y   1 þ dÞ

ð19Þ

where a¼

pffiffiffi ga

pffiffiffi pffiffiffi B2 ¼ D1 cos hð gaðd  1ÞÞ þ D2 sin hð gaðd  1ÞÞ þ E1 ð20Þ C1 ¼

ur gwA2 w  k2 ga2 f

C2 ¼

ur gwA3 w  k2 ga2 f

C3 ¼ 

k2FCL þ ur a2 k2FCL ð21Þ

Thermal transport of combined electroosmotically and pressure driven nanofluid flow in soft…

pffiffiffi pffiffiffi pffiffiffi pffiffiffi E2 ½sin hð gaÞ þ c ga cos hð gaÞ þ E3 cos hð gaðd  1ÞÞ D1 ¼ pffiffiffi  pffiffiffi pffiffiffi  cos hð gadÞ þ c ga sin hð gadÞ pffiffiffi E2  D1 sin hð gaðd  1ÞÞ D2 ¼ pffiffiffi  cos hð gaðd  1ÞÞ

ð22Þ   d1  E1 ¼ C1 cos hðbðd  1ÞÞ  ur gA1 cos h k 1 þ C2 sin hðbðd  1ÞÞ þ C3  gðd  1Þ2    2  1 u gA d1 r 1  gðd  1Þ þ  sin h E2 ¼ b sin hðbð d  1ÞÞ  C 1 a k k E3 ¼ ðbC2 c  C1 Þ cos hðbÞ þ ðC2  bC1 cÞ sin hðbÞ  C3

 thermal conductivity. Prf ¼ cp;f lf kf is the Prandtl number  of the fluid. a ¼ 2Rb kf dp is the nanoparticle Biot number, where Rb is the interfacial resistance. Reb is the Brownian– Reynolds number, which is defined as Reb ¼ ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  18KB Tav pqs dp =tf , with tf as the fluid kinematic viscosity. It is reported that for the suspension of Al2O3 nanoparticles in water, Rb = 0.77 9 10-8 m2 K W-1, m = 2.5 and A = 40,000 [39]. reff is the effective electrical conductivity of the nanofluid, which is defined as reff ¼ rf ð1 þ

ð23Þ

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 soft 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  2  oT1 o T1 o2 T1 ðqCp Þeff u1 ¼ keff þ þ reff Ex2 ox ox2 oy2 ð24Þ  2 du1 þ leff ðh þ d  y  0Þ dy  2  oT2 o T2 o2 T2 ðqCp Þeff u2 ¼ keff þ þ reff Ex2 ox ox2 oy2  2 ð25Þ du2 þ leff þlc u22 dy ðh  y   h þ dÞ where T is the local temperature of the nanofluid, (qcp)eff is the heat capacitance of the nanofluid at the reference pressure, defined as

qCp eff ¼ ðqCp Þf ð1  /Þ þ ðqCp Þs / ð26Þ here (qCp)s is the heat capacitance of nanoparticles, and (qCp)f is the heat capacitance of base fluids. keff is the effective thermal conductivity of nanofluid and is supposed to have the equal value both inside and outside the FCL, defined as [39]: keff ¼

ð28Þ

here rs is the electrical conductivity of nanoparticles and rf is the electrical conductivity of base fluids. Furthermore, coupled with uniform wall heat flux condition, for thermally fully developed problem, we have

Temperature distribution for fully developed flow with uniform wall heat flux



3ðrs =rf  1Þ/ Þ ðrs =rf þ 2Þ  ðrs =rf  1Þ/

 sð1 þ 2aÞ þ 2 þ 2/½sð1  aÞ  1 1=3 kf ð1 þ ARem b Prf /Þ sð1 þ 2aÞ þ 2  /½sð1  aÞ  1

ð27Þ The parameter s ¼ ks =kf is the thermal conductivity ratio of the particle thermal conductivity to the base fluid

oT1 oT2 dTw dTm o2 T1 o2 T2 ¼ ¼ ¼ ¼ const: and ¼ 2 ¼0 ox ox dx dx ox2 ox ð29Þ Under these assumptions, energy Eqs. (23) and (24) yield  2 dTm d2 T1 du1 2 ¼ keff 2 þ reff Ex þ leff ðqCp Þeff u1 dx dy dy ð30Þ ðh þ d  y  0Þ  2 dTm d2 T2 du2 ¼ keff 2 þ reff Ex2 þ leff ðqCp Þeff u2 þlc u22 dx dy dy ðh  y   h þ dÞ ð31Þ Considering an over energy balance for an elemental control volume on a length of duct dx along the centerline of the channel, the axial bulk temperature gradient in the thermally fully developed situation has the form dTm qw þ reff Ex2 h þ Q ¼ ðqCp Þeff hup ðb11 þ b21 Þ dx

ð32Þ

where . Q ¼ leff u2p ðb12 þ b22 Þ h þ leff lc hu2p b23 b11 ¼ b12 ¼ b23 ¼

Z Z

0

u1 ð yÞd y;

1þd  0

1þd Z 1þd 1

d u1 d y

2 d y;

b21 ¼

Z

ð33Þ

1þd

u2 ð yÞd y; 1

b22 ¼

Z

1þd 

1

du2 d y

2 d y;

ð34Þ

u22 ð yÞd y

Introducing the following dimensionless temperature for fully developed flow

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G. Zhao, Y. Jian

hð yÞ ¼

TðyÞ  Tw ; qw h=kf

ð35Þ

here Tw is the wall temperature of channel, qw = h0 (Tw - Tm) is the constant wall heat flux, Tm is the mean temperature of fluid, and h0 is the convective heat transfer coefficient. The dimensionless energy equation and boundary conditions can be described as (  2 ) d2 h1 kf d u1  ¼ G u1  S  Br ð1 þ d  y  0Þ 2 d y keff d y ð36Þ d2 h2 kf ¼ 2 d y keff

(

 2 d u2 G u2  S  Br a2 Br u22 d y

) ð37Þ

ð38aÞ ð38bÞ

dh1  dh2   y¼1þd ¼ d y d y y¼1þd  

dh2  kf or h2 y¼1 ¼ 0 y¼1 ¼ d y keff

ð38cÞ ð38dÞ

where G¼

1 ½ð1 þ S þ Brðb12 þ b22 Þ þ a2 Brb23  ðb11 þ b21 Þ ð39Þ

rf Ex2 h S ¼ S=s qw Br ¼ Br=g

S¼

s ¼ rf =reff

Br ¼

lf u2p qw h

ð40Þ

The parameter S denotes the ratio of Joule heating to heat flux from microchannel wall in physical, usually termed as the dimensionless Joule heat parameter, and the parameter Br is Brinkman number, which indicates the ratio of heat produced by viscous dissipation to heat transported by molecular conduction. Substituting dimensionless velocity field (18) and (19) into non-dimensional energy Eqs. (36) and (37), integrating it twice and applying boundary conditions (38), finally the solution of the temperature field can be obtained as follows h1 ð yÞ ¼

kf yÞ þ K 2 g ff1 ð keff

ð41Þ

yÞ ¼ h2 ð

kf yÞ þ K 4 g ff2 ð keff

ð42Þ

The dimensionless bulk temperature can be defined as

123

hm ¼

1

yþ u2 h2 d

R 1þd 1

yþ u2 d

R 0

1 h1 d y 1þd u

R 0

1 d y 1þd u

¼ kf ðTm  Tw Þ=qw h ð43Þ

The heat transfer rate can be expressed in the form of Nusselt number Nu, which is defined as Nu ¼

2hqw kf ðTw  Tm Þ

ð44Þ

Combining Eqs. (43) and (44), the Nusselt number can be ultimately reduced as Nu ¼ 

 ð1  y   1 þ dÞ dh1  y¼0 ¼ 0 d y   h1 y¼1þd ¼ h2 y¼1þd

R 1þd

kf 2 keff hm

ð45Þ

Due to the expressions of the velocity and the temperature having been known from (18), (19), (41) and (42), respectively, the Nu number can be obtained from (45).

Results and discussion In our present work, the analytical expressions derived in the preceding section are used to describe the electrokinetic transport and heat transfer of the nanofluid (Al2O3–water) in soft nanochannel. The Knudsen number is the vital parameter, which manifests the continuum of the nanofluid, defined as Kn ¼ k=D, where k is the fluid molecule mean path and D is the hydraulic diameter of the nanochannel. It is reported that the mean free path of the water molecules and the Al2O3 nanoparticle is estimated, respectively, 0.1 and 10 nm [39]. In this study, the order of the channel height is 10-6 m. The Knudsen number is evaluated nearly 0.01 and satisfies the continuum assumption. Ex is set to be 10 V m-1. The range of the equivalent EDL thickness kFCL changes from 0.5 to 0.8, and the range of the EDL thickness k is from 0.2 to 0.6. Moreover, based upon the supposition of negative wall heat flux, the value of Brinkman number (Br) is changed from - 0.1 to 0 and the value of Joule heat parameter (S) is deemed to be negative and changes from - 10 to 0. Some typical parameters are taken as follows in Table 1. In Fig. 2, we compare the variation of the dimensionless electric potential and velocity with the earlier research launched by Xing and Jian [25]. In their study, they paid the great attention to how the steric effects can impact the electroosmotic flow in soft nanochannels. From the figures, we can find that a good agreement with the results of article [25] is obtained when steric factor t is zero. That is to say the results in our present study are the special case of the steric effects. Meanwhile, an increased trend of electric potential and velocity with enhanced steric factor v is shown in Fig. 2.

Thermal transport of combined electroosmotically and pressure driven nanofluid flow in soft… Table 1 Thermophysical properties of pure water and Al2O3 nanoparticles Parameters

qf/kg m-3

qs/kg m-3

(Cp)f/J kg-1 K-1

(Cp)s/J kg-1 K-1

lf/kg m-1 s-1

rf/X-1 m-1

rs/X-1 m-1

Values

997.1

3600

4179

765

8.91910-3

0.05

10-12

(a) – 0.33

(b) 0.4

– 0.34

0.35 0.3

– 0.35

0.25

φ

u

– 0.36 – 0.37

0.15 present result, v = 0 Xing and jian [25], v = 0 Xing and jian [25], v = 0.4

– 0.38 – 0.39 – 0.4 –1

0.2

– 0.8

– 0.6

– 0.4

– 0.2

y

present result, v = 0 Xing and jian [25], v = 0 Xing and jian [25], v = 0.4

0.1 0.05 0

0 –1

– 0.8

– 0.6

– 0.4

– 0.2

0

y

Fig. 2 Comparison of numerical solutions in Ref. [25] with present analytical solutions of the dimensionless electric potential and velocity for k = 2, kFCL = 2, d = 0.05, X = - 0.1, f = 1 (a) and k = 1, kFCL = 0.5, d = 0.05, X = 1, f = 1 (b)

Figure 3a depicts the variation of the dimensionless velocity distribution for different values of PEL dimensionless thickness d in the soft nanochannel. We can find that the dimensionless velocity boosts as the increase in the PEL dimensionless thickness. Due to the assumption of constant number ions within the FCL, a bigger FCL dimensionless thickness is easy to give a lower density number, which brings about a lower resistance for the nanofluid in the channels. Figure 3b illustrates the variation of the dimensionless velocity distribution for different values of the equivalent EDL thickness kFCL . From the plots, one can observe that the dimensionless velocity decreases as the enhancement of the equivalent EDL thickness. In physically, the larger kFCL means the less ionic number concentration of the PEL ions, which results in the abatement of electric potential inside of the nanochannels. Thus, the driving force of the fluid appeared in Eq. (7) diminishes accordingly. Figure 3c shows the variation of the dimensionless velocity distribution for different values of volume fraction / in the nanochannel. As it is obvious, we can see the dimensionless velocity of the nanofluid inhabits as the augment of nanoparticles volume fraction. This mainly is attributed to the fact with enhancement of 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. The variation of the dimensionless velocity distribution for different values of the ion partitioning parameter is addressed in Fig. 3d. We can find the velocity will reduce as the ion partitioning parameter increases. The reason is that the smaller values of ion partitioning parameter easily cause enhanced Born energy (denoted by w) at the interface of the FCL and bulk dielectric solution [26]. It augments the ions of the PEL and leads to the bigger electrostatic potential. The variation of the dimensionless temperature distribution for variable values of PEL dimensionless thickness d is presented in Fig. 4a. It should be noted that the wall heat flux is assumed to be negative throughout this study, i.e., qw \0. That implies the temperature of the nanofluid is higher than the temperature of the walls, and the direction of the heat transfer is from the fluid to the walls. Naturally, the vital parameters of the Brinkman number and Joule heating number are negative. It can be found that the dimensionless temperature is encouraged with the escalation of PEL dimensionless thickness. Combining with Eq. (29), the actual temperature of the nanofluid is diminishes. In fact, the motion of the fluid can be affected by the enhanced PEL dimensionless thickness, as

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0.15

0.15

0.1

0.1

u

(b) 0.2

u

(a) 0.2

d = 0.12 d = 0.15 d = 0.20

0.05

0 –1

– 0.8

– 0.6

– 0.4

y

– 0.2

0 –1

0

(c)0.5

– 0.8

– 0.6

y

– 0.4

– 0.2

0

(d) 0.7 φ = 0%

0.4

λ FCL = 0.5 λ FCL = 0.6 λ FCL = 0.7

0.05

0.6

φ = 2% φ = 4%

0.5

0.3

ζ = 0.5 ζ = 0.75 ζ = 1.0

u

u

0.4 0.3

0.2

0.2 0.1 0.1 0 –1

– 0.8

– 0.6

y

– 0.4

– 0.2

0

0 –1

– 0.8

– 0.6

y

– 0.4

– 0.2

0

Fig. 3 Variations of the velocity distribution for different values of PEL dimensionless thickness d (kFCL = 0.5, k = 0.35, / = 2%, t = 1) (a), equivalent EDL thickness kFCL (d = 0.2, k = 0.35, / = 3%, t = 1) (b), volume fraction /(d = 0.2, kFCL = 0.5,

k = 0.35, t = 1) (c) ion partitioning parameter f (d = 0.2, kFCL = 0.5,k = 0.35, / = 2%) (d) at c = 0.002, a = 1, X = 1, w = 0.01, dp = 30 nm and ur = - 1 in the nanochannel

illustrated in Fig. 3. This enhanced convection will contribute to the reduction of the local temperature in nanochannels. Figure 4b provides the variation of the dimensionless temperature distribution for different values of parameter number ur, which indicates the strength of the electroosmotic velocity to comparable pressure-driven velocity. It is clearly seen that the bigger parameter number ur brings about the less dimensionless temperature. As known to all that the reversed pressure-driven velocity (i.e., pressure opposed flow) can attenuate the motion of the nanofluid, consequently it results in the enhanced local temperature of the nanofluid. The variation of the dimensionless temperature distribution for variable values of volume fraction / is plotted in Fig. 4c. We can find the dimensionless temperature

decrease as the improved nanoparticle volume fraction. As expect, the thermal conduction of the fluid can be impacted significantly by adding nanoparticles. For a nanochannel system, this higher effective thermal conductivity of nanoparticles gives rise to the thermal diffusion effect, which accordingly causes an increased performance of the temperature. The variation of the dimensionless temperature distribution for different values of ion partitioning parameter f is presented in Fig. 4d. Just as the explanation of Fig. 2d, the lower ion partitioning parameter strengthens the motion of the nanofluid and boosts the thermal motion from the nanofluid to the walls. Therefore, the dimensionless temperature reduces with decreased ion partitioning parameter. The variation of the Nusselt number with respect to Brinkman number for various values of the PEL

123

Thermal transport of combined electroosmotically and pressure driven nanofluid flow in soft…

(b) 0

(a)0

– 0.2

d = 0.12 d = 0.15 d = 0.20

– 0.2 – 0.4

– 0.4

θ

θ

– 0.6 – 0.8

– 0.6

–1

ur = – 1.0 ur = – 0.8 ur = – 0.5

– 0.8 – 1.2 –1 –1

– 0.8

– 0.6

y

– 0.4

– 0.2

0

(c) 0

– 1.4 –1

– 0.6

– 0.4

y

– 0.2

0

(d) 0

– 0.1

φ = 0%

– 0.1

ζ = 0.5 ζ = 0.75

– 0.2

φ = 2%

– 0.2

– 0.3

φ = 4%

– 0.3

ζ = 1.0

– 0.4

θ

θ

– 0.4 – 0.5

– 0.5

– 0.6

– 0.6

– 0.7

– 0.7

– 0.8 –1

– 0.8

– 0.8

– 0.6

y

– 0.4

0

– 0.2

Fig. 4 Variations of the temperature distribution for different values of PEL dimensionless thickness d (kFCL = 0.5, k = 0.35, / = 2%, t = 1, dp = 20 nm) (a) parameter number ur (d = 0.15, kFCL = 0.5, k = 0.35, / = 2%, t = 1, dp = 15 nm) (b) volume fraction /

– 0.8 –1

– 0.8

– 0.6

y

– 0.4

– 0.2

0

(d = 0.2, kFCL = 0.5, k = 0.35, t = 1, dp = 20 nm) (c) ion partitioning parameter f (d = 0.2, kFCL = 0.5,k = 0.35, / = 2%, dp = 20 nm) (d) at c = 0.002, a = 1, X = 1, w = 0.01 in the nanochannel

3.5

4 3.5

3 S=–5 S=–3 S=–2

2.5

Nu

Nu

3

d = 0.12 d = 0.15 d = 0.20

2

1.5 – 0.1

– 0.08

– 0.06

– 0.04

– 0.02

2.5 2 1.5

0

Br

Fig. 5 Effects of the PEL dimensionless thickness on the nature of variation of Nusselt number with Brinkman number at kFCL = 0.5, c = 0.002, f = 1, a = 1, k = 0.35, X = 1, w = 0.01, / = 3%, S = - 2, dp = 20 nm and ur = - 1 in the nanochannel

1 0.5

0.55

0.6

0.65

λ FCL

0.7

0.75

0.8

Fig. 6 Effects of the Joule heating parameter S on the nature of variation of Nusselt number with the equivalent EDL thickness kFCL at d = 0.2, c = 0.002, f = 1, a = 1, k = 0.35, X = 1, w = 0.01, / = 2%, Br = - 0.01, dp = 20 nm and ur = - 1 in the nanochannel

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G. Zhao, Y. Jian

7 6.5

ζ = 0.5 ζ = 0.75 ζ = 1.0

6

Nu

5.5

4.5 4 3.5 0.35

0.4

0.45

0.5

0.55

0.6

λ Fig. 7 Effects of the ion partitioning parameter f on the nature of variation of Nusselt number with EDL thickness k at d = 0.15, c = 0.002, a = 1, X = 1, w = 0.01, / = 2%, S = - 2, Br = - 0.01, dp = 20 nm and ur = - 1 in the nanochannel

123

4 3.5 3

φ = 0% 2.5

φ = 2% φ = 4%

2 0

0.2

0.4

0.6 dp

0.8

1 ×10–7

Fig. 8 Effects of the volume fraction / on the nature of variation of Nusselt number with nanoparticle diameter at d = 0.2, c = 0.002, f = 1, a = 1, kFCL = 0.5, k = 0.35, X = 1, w = 0.01, S = - 2 and ur = - 1 in the nanochannel

number with increased EDL thickness k can be observed from the figure. Figure 8 evaluates the effects of the volume fraction / on the nature of variation of Nusselt number with nanoparticle diameter (0–100 nm). We can find the heat transfer of the base fluid (i.e., / = 0%) is not dependent on the variation of the nanoparticle diameter. However, with increasing the nanoparticle diameter, the heat transfer of the nanofluid decreases gradually. The reason for this is that the bigger nanoparticle diameter may weaken the motion of the nanofluid and then affects the thermal performance of the nanofluid flow. The similar case can be witnessed as enhanced nanoparticle volume fraction due to the improved viscosity of the nanofluid. Moreover, for / = 4%, Nusselt number is reduced nearly about 44% as compared to that of the base fluid when dp is taken as 100 nm.

Conclusions

5

3 0.3

4.5

Nu

dimensionless thickness is described in Fig. 5. It can be observed that the Nusselt number takes bigger value by increasing of Brinkman number. Increased Brinkman number in negative value manifests enhanced viscous dissipation effect, associated with the negative heat flux of the walls, which tries to increase the wall temperature Tw more than the mean temperature Tm; therefore, the magnitude of (Tw - Tm) increases gradually, and the convective heat transfer coefficient h0 decreases. Meanwhile, a suppressed Nusselt number with increased PEL dimensionless thickness can also be found in the picture. Figure 6 presents influences of the Joule heating parameter S on the nature of variation of Nusselt number with the equivalent EDL thickness kFCL . Since Joule heating parameter has a spatially uniform heating effect, and there is a boost in homogeneous manner with an elevation of Joule heating. Therefore, the temperature gradient in the vicinity of the wall reduces. Ultimately, it causes a decrement of Nusselt number with increasing magnitude of Joule heating parameter. Beside, a subdued tendency of the Nusselt number with the equivalent EDL thickness kFCL can also be witnessed in such case. Figure 7 reflects influences of the ion partitioning parameter on the nature of variation of Nusselt number with the EDL thickness k . By increasing the ion partitioning parameter, we can see the heat transfer of the nanofluid will reduce. The reason of this fact is that the bigger ion partitioning parameter decreases the motion of the nanofluid, the convective heat transfer of the nanofluid decreases accordingly. Moreover, a weaken Nusselt

In the present paper, the flow and thermal behaviors of nanofluid flow (Al2O3-water) in a soft nanofluidic channel by considering the ion partitioning effect and the thermally fully developed condition are investigated theoretically. The flow is actuated by combined effects of pressure gradient and external electric field. Meanwhile, the negative wall heat flux is assumed in our research. The analytical solutions for electrostatic potential, velocity and temperature are derived. Further, the

Thermal transport of combined electroosmotically and pressure driven nanofluid flow in soft…

variations of the Nusselt number are also discussed. The corresponding study reveals that owing to the polyelectrolyte layer (PEL), the velocity distribution and temperature variation of the nanofluid remarkably rely on the values of PEL thickness d and the equivalent EDL thickness kFCL . The enhanced nanoparticle volume fraction / gives rise to the subdued velocity and temperature. It might be interesting that the smaller ion partitioning parameter leads to increased velocity and decreased temperature of the nanofluid. The Nusselt number takes bigger values by growing of the Brinkman number and the Joule heating parameter. Increased ion partitioning parameter reduces the Nusselt number. Nevertheless, increased PEL thickness d and escalated nanoparticle volume fraction / can weaken the heat transfer performance of the nanofluid.

Acknowledgements The work was supported by the National Natural Science Foundation of China (Grant Nos. 11472140, 11772162), the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2016MS0106), the Foundation of Inner Mongolia Autonomous Region University Scientific Research Project (Grant No. NJZY18093), the Foundation of Inner Mongolia University of Technology (Grant No. ZD201714) and the Inner Mongolia Grassland Talent (Grant No. 12000-12102013).

Appendix The coefficients in Eqs. (41) and (42) are presented as follows:  1 f1 ð ð1 þ S þ Brðb12 þ b22 Þ þ a2 Brb23 yÞ ¼ ðb11 þ b21 Þ     1 4 1 y 2 2  g y þ ur gA1 k cos h  þ B2 y 24 2 k "  2 1 2 1 2 4 1 1 y  Br g y þ u  S gA r 1 2 12 2 k    2  k 2 y 1 cos h   y2 2 4 k      y sin h y  2k2 cos h y 2ur g2 A1 k þ K1 y k k

1 f2 ð yÞ ¼ ½ð1 þ S þ Brðb12 þ b22 Þ þ a2 Brb23  ðb11 þ b21 Þ  D1 D2 C1 sin hða yÞ þ 2 cos hðb cos hða yÞ þ yÞ a2 a b  C2 C3 1 2 þ 2 sin hðb yÞ þ y2  S y b 2 2    1 1 1 2 1  y  Br ðD1 aÞ2 cos hð2a y Þ  yÞ þ C1 C2 sin hð2b 2 4a2 2 4   1 1 1 1 þ ðD2 aÞ2 cos hð2a yÞ þ y2 þ D1 D2 sin hð2a yÞ 2 4a2 2 4   1 1 1 þ ðC1 bÞ2 cos hð2b yÞ  y2 2 4b2 2   1 1 1 2 2  y þ ðC1 bÞ cos hð2b y Þ þ 2 4b2 2 ðD1 C1 þ D2 C2 Þab cos hðða þ bÞ yÞ þ ða þ bÞ2 ðD2 C2  D1 C1 Þab þ cos hðða  bÞ yÞ ða  bÞ2 ðD1 C2 þ D2 C1 Þab þ cos hðða þ bÞ yÞ ða þ bÞ2   1 1 1 cos hð2a yÞ  y2 þ D22 2 4a2 2   D1 D2 1 2 1 1 2  C y þ sin hð2a y Þ þ cos hð2b y Þ  2 4a2 2 4b2 2   1 2 1 1 2 C1 C2 þ C2 cos hð2b yÞ  y þ sin hð2b yÞ 4b2 2 4b2 2 1 ðD1 C2  D2 C1 Þab sin hðða  bÞ yÞ þ C32 y2 þ 2 ða þ bÞ2    1 1 1 coshð2a yÞ þ y2  Bra2 D21 2 4a2 2   1 1 1 þ D22 cos hð2a yÞ  y2 2 4a2 2   D1 D2 1 2 1 1 2  C y þ sin hð2a y Þ þ cos hð2b y Þ  4a2 2 2 4b2 2 C1 C2 1 2 2 sin hð2b yÞ þ C3 y þ 4b2  2  1 2 1 1 2  y þ C2 cos hð2b y Þ  2 4b2 2 ðD1 C1 þ D2 C2 Þ cos hðða þ bÞ yÞ þ ða þ bÞ2 ðD1 C1  D2 C2 Þ þ cos hðða  bÞ yÞ ða  bÞ2 ðD1 C2 þ D2 C1 Þ sin hðða þ bÞ yÞ þ ða þ bÞ2 ðD2 C1  D1 C2 Þ sin hðða  bÞ yÞ þ ða þ bÞ2  2C3 D1 2C3 D2 cos hða y Þ þ sin hða y Þ þ K3 y þ a2 a2

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where K1 ¼ 0, K2 ¼ f2 ðd  1Þ  f2 ð1Þ  f1 ðd  1Þ, K3 ¼   df1 ð yÞ  and yÞy¼1þd, K4 ¼ f2 ð1Þ, dy y¼1þd  gð yÞ  K3 . gð yÞ ¼ dfd2 ð y

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