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International Journal of Thermal Sciences 50 (2011) 2154e2160

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Double-diffusive natural convective boundary layer flow in a porous mediumsaturated with a nanofluid over a vertical plate: Prescribed surface heat, soluteand nanoparticle fluxes

W.A. Khan a, A. Aziz b,*

aDepartment of Engineering Sciences, National University of Sciences and Technology, Karachi 75350, PakistanbDepartment of Mechanical Engineering, School of Engineering and Applied Science, Gonzaga University, Spokane, WA 99258, USA

a r t i c l e i n f o

Article history:Received 10 March 2011Received in revised form27 May 2011Accepted 30 May 2011Available online 5 July 2011

Keywords:Vertical plateBinary base fluidNanofluidPrescribed surface heat, solute andnanoparticle concentration fluxesBrownian motionThermophoresis

* Corresponding author. Tel.: þ1 509 313 3540; faxE-mail addresses: wkhan_2000@yahoo.com (W.

(A. Aziz).

1290-0729/$ e see front matter � 2011 Elsevier Masdoi:10.1016/j.ijthermalsci.2011.05.022

a b s t r a c t

The Buongiorno model [16] has been used to study the double-diffusive natural convection froma vertical plate to a porous medium saturated with a binary base fluid containing nanoparticles. Themodel identifies the Brownian motion and thermophoresis as the primary mechanisms for enhancedconvection characteristics of the nanofluid. The behavior of the porous medium is described by the Darcymodel. The vertical surface has the heat, mass and nanoparticle fluxes each prescribed as a power lawfunction of the distance along the wall. The transport equations are transformed into four nonlinear,coupled similarity equations containing eight dimensionless parameters. These equations are solvednumerically to obtain the velocity, temperature, solute concentration and nanoparticle concentration inthe respective boundary layers. Results are presented to illustrate the effects of various parametersincluding the exponent of the power law describing the imposed surface fluxes on the heat and masstransfer characteristics of the flow. These results are supplemented with the data for the reduced Nusseltnumber and the two reduced Sherwood numbers, one for the solute and the other for the nanoparticles.

� 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

Convective heat transfer in nanofluids is a topic of majorcontemporary interest in the heat transfer research community.The word “nanofluid” coined by Choi [1] describes a liquidsuspension containing ultra-fine particles (diameter less than50 nm). With the rapid advances in nano manufacturing, manyinexpensive combinations of liquid/particle are now available.These include particles of metals such aluminum, copper, gold, ironand titanium or their oxides. The base fluids used are usually water,ethylene glycol, toluene and oil. Experimental studies e.g. [2e7].show that even with the small volumetric fraction of nanoparticles(usually less than 5%), the thermal conductivity of the base liquidcan be enhanced by 10e50%. The enhanced thermal conductivity ofa nanofluid together with the thermal dispersion of particles andturbulence induced by their motion contributes to a remarkableimprovement in the convective heat transfer coefficient. Thisfeature of nanofluids make them attractive for use in applications

: þ1 509 313 5871.A. Khan), aziz@gonzaga.edu

son SAS. All rights reserved.

such as advanced nuclear systems [8] and cylindrical heat pipes [9].The literature on the thermal conductivity and viscosity of nano-fluids has been reviewed by Trisaksri and Wongwises [10],Wangand Mujumdar [11], Eastman et al. [12], and Kakac and Pra-muanjaroenkij [13], among several others. These reviews discuss indetail the preparation of nanofluids, theoretical and experimentalinvestigations of thermal conductivity and viscosity of nanofluids,and the work done on convective transport in nanofluids. Abenchmark study of thermal conductivity of nanofluids has beenpublished by Buongiorno et al. [14]. This study analyzed theexperimental thermal conductivity data gathered by 30 organiza-tions worldwide and found most of them to be consistent within10%. The study concludes that the thermal conductivity of a nano-fluid increases with the particle concentration and aspect ratio inconformity with the classical Maxwell theory which predicts thatthe effective thermal conductivity ratio k/kf is a function of particlevolume fraction f and the thermal conductivity ratio kp/kf. Thisfunctional dependence is valid for f << 1 and kp/kf < 10.

Several ideas have been proposed to explain the enhanced heattransfer characteristics of nanofluids. For example, Pak and Cho [4]attributed the increased heat transfer coefficients observed innanofluids to the dispersion of suspended particles. Xuan and Li [5]suggested that the heat transfer enhancement was the result of the

Nomenclature

C solutal concentrationDB Brownian diffusion coefficientDT thermophoretic diffusion coefficientDCT Soret diffusivityDTC Dufour diffusivityDSm solutal diffusivity of porous mediumf(h) dimensionless rescaled nanoparticle volume fractiong acceleration due to gravityk effective thermal conductivity of nanofluidkp thermal conductivity of nanoparticleskf thermal conductivity of base fluidK permeability of the porous mediumLd Dufour-solutal Lewis numberLe regular Lewis numberLn nanofluid Lewis numberNb Brownian motion parameterNc regular double-diffusive buoyancy parameterNd modified Dufour parameterNr buoyancy-ratioNt thermophoresis parameterNux local Nusselt numberNur reduced Nusselt numberPr Prandtl numberqm solute wall mass fluxqnp nanoparticle wall mass fluxqw wall heat fluxRax local Rayleigh numbers(h) dimensionless stream functionShx local solutal Sherwood number

Shx,n local nanoparticle Sherwood numberShr reduced solutal Sherwood numberShrn reduced nanoparticle Sherwood numberT local fluid temperatureTN ambient temperatureu,v velocity components along x and y directionsx coordinate along the platey coordinate normal to the plate.

Greek symbolsam thermal diffusivity of porous mediumbT volumetric thermal expansion coefficient of the fluidbC volumetric solutal expansion coefficient of the fluidg dimensionless solutal concentration3 porosity of the mediumf nanoparticle volume fractionfw nanoparticle volume fraction at the wallfN ambient nanoparticle volume fractionh similarity variablem absolute viscosity of the base fluidn kinematic viscosity of the fluidrf fluid densityrp nanoparticle mass density(rc)f heat capacity of the fluid(rc)m effective heat capacity of the porous medium(rc)p effective heat capacity of the nanoparticle materials ratio between the effective heat capacity of the

nanoparticle material and heat capacity of the fluidq dimensionless temperaturej stream function

W.A. Khan, A. Aziz / International Journal of Thermal Sciences 50 (2011) 2154e2160 2155

increase in turbulence induced by the nanoparticle motion. Basedon his experimental data on water and glycerin based nanofluids,Ahuja [15] concluded that the heat transfer enhancement wascaused by the rotation of the nanoparticles. However, after anextensive evaluation of the literature, Buongiorno [16] has shownthat the high heat transfer coefficients in nanofluids cannot beexplained satisfactorily by thermal dispersion [4] or increase inturbulence intensity [5] or nanoparticle rotation [13]. He proposedthat the analytical model for convective transport in nanofluidsmust take into account the Brownian diffusion and thermophoresis,and the increase in the heat transfer coefficient was due to signif-icant decrease in the viscosity of the fluid caused by the largetemperature variations in the boundary layers.

The literature on convective heat transfer in nanofluids hasgrown steadily in the last ten years. However, the number ofanalytical studies on natural convection in nanofluids is relativelysmall compared with those devoted to forced convection. Khanaferet al. [17] analyzed the two-dimensional natural convective flow ofa nanofluid in an enclosure and found that for any given Grashofnumber, the heat transfer rate increased as the volume fraction ofnanoparticles increased. Kim et al. [18] introduced a new frictionfactor to describe the effect of nanoparticles on the convectiveinstability and the heat transfer characteristics of the base fluid.Tzou [19] considered thermal instability of nanofluids in naturalconvection and concluded that the higher turbulence triggered bythe nanoparticles prompted higher heat transfer coefficient thanthe effect of enhanced thermal conductivity. Contrary to theobservations in [17], Putra et al. [20] and Wen and Ding [21] foundthat the heat transfer coefficient decreases, not increases, with theincrease in particle concentration. The difference in conclusions ofthe analytical studies of Khanafer et al. [17] and Kim et al. [18] and

experimental works of Putra et al. [20] and Wen and Ding [21] maybe due to the assumptions made in [17,18] in developing theanalytical models. In a more recent paper, Abu-Nada et al. [22]studied the effect of variable thermal conductivity and variableviscosity on heat transfer in awatereAl2O3 nanofluid confined in anenclosure. They found that at low Rayleigh numbers the Nusseltnumber increased slightly with the increase in the volume fractionof the nanoparticles, but at high Rayleigh numbers the effect wasopposite. This brief survey clearly indicates that a definitiveconclusion regarding the role of nanoparticles in enhancing naturalconvective transport is yet to be reached.

In the past three years, a few papers have considered somefundamental problems in external natural convective heat transferin nanofluids. Kuznetsov and Nield [23] extended the classicalproblem of natural convection of a regular fluid over an isothermalvertical plate to the flow of a nanofluid. They used the modelproposed by Buongiorno [16] which takes into account the Brow-nian diffusion as well as thermophoresis in writing the transportequations. Their similarity analysis identified four parameters gov-erning the transport process, namely a Lewis number Le, a buoy-ancy-ratio number, Nr, a Brownian motion number Nb, anda thermophoresis numberNt. For afixed Prandtl number Pr¼10 andLewis number Le ¼ 10, their numerical results indicated that thereduced Nusselt number decreased with the increase in each of theparameters Nr, Nb, and Nt. The authors later extended the study toa nanofluid saturated porous medium [24], extending the wellknown ChengeMinkowycz problem [25] to a nanofluid. The Buon-giornomodel [16] has also been used by Khan and Pop [26] to studythe boundary layerflowof a nanofluid past a stretching sheet and byKhan and Aziz [27] to investigate the boundary layer flow ofa nanofluid past a vertical surfacewith a constant heat flux. The gap

W.A. Khan, A. Aziz / International Journal of Thermal Sciences 50 (2011) 2154e21602156

between the work of Kuznetsov and Nield [23] and Khan and Aziz[27] has been recently filled by Aziz and Khan [28] who employeda convective boundary condition to study natural convection ina nanofluid past a vertical surface. A recent paper by Kuznetsov andNield [29] has considered double-diffusive (in reality a triple diffu-sive) natural convection process from a vertical surface to a binarybase fluid containing solute (e.g. salt) as well as nanoparticles. Asimilarity solution was derived based on the assumption of surfaceconditions of constant temperature, constant solute concentrationand constant nanoparticle concentration. A companionpaper by thesame authors extended thiswork to a nanofluid in a porousmedium[30]. The present paper extends the work by Kuznetsov and Nield[30] to a situation where the vertical surface has heat, solute andnanoparticles fluxes specified as a power law function of thedistance along the surface. A similarity solution is derived and usedto predict the heat and mass transfer characteristics of the flow.

2. Convective transport model

Consider the two-dimensional (x,y) natural convectionboundary layer flow over a vertical plate as illustrated in Fig. 1.Although only the veocity and thermal boundary layers are shownin Fig. 1 to avoid congestion, there are two additional boundarylayers,namely a solutal concentration boundary layer and a nano-particle concentration boundary layer. At the surface of the plate( y ¼ 0), the heat flux, solutal flux, and the nanoparticle flux eachare prescribed as functions of the distance along the plate (x).Again, for clarity, only the surface heat flux is indicated. Thetemperature, solutal concentration and the nanoparticle concen-tration at large distances from the plate (y/N) are denoted by TN,CN and fN, respectively. The plate is situated in a porous mediumwhich is saturated by a binary fluid with dissolved solute andcontaining nanoparticles in suspension.

The governing equations for mass, momentum, thermal energy,and nanoparticles derived by Kuznetsov and Nield [30] may bewritten as

vuvx

þ vv

vy¼ 0 (1)

vuvy

¼ ð1�fNÞrfNgKm

�bT

vTvy

þ bCvCvy

���rp � rfN

�gK

m

vf

vy(2)

Fig. 1. Vertical surface in a porous medium saturated with a fluid containing soluteand with nanoparticles in suspension.

uvTvx

þ vvTvy

¼ am

v2Tvy2

!þ sDB

"�vf

vy

��vTvy

�þ�DT

TN

��vTvy

�2#

þsDTCv2Cvy2

(3)

uvCvx

þ vvCvy

¼ 3Dsmv2Cvy2

þ 3DCTv2Tvy2

(4)

uvf

vxþ v

vf

vy¼ 3DB

v2f

vy2þ 3

DT

TN

v2Tvy2

(5)

where

am ¼ kmðrcÞf

; s ¼ 3ðrcÞpðrcÞf

; s ¼ ðrcÞmðrcÞf

(6)

where u and v are the velocity components along the x and ydirections, respectively, rf is the density of base fluid, rp is thenanoparticle density, m is the absolute viscosity of the base fluid, amis the effective thermal diffusivity of the porous medium, C is thelocal nanoparticle volume fraction, bT is volumetric thermalexpansion coefficient of the base fluid, bC is the solutal volumetricexpansion coefficient of the fluid, 3 is the porosity of the medium,DB is the Brownian diffusion coefficient, DT is the thermophoreticdiffusion coefficient,DCT is Soret diffusivity,DTC is Dufour diffusivity,g is the acceleration due to gravity, km is the effective thermalconductivity of the porous medium, T is the local temperature, andg is the acceleration due to gravity. The subscript N denotes thevalues at large values of y where the fluid is quiescent.

The boundary conditions for the case of prescribed surface heat,solute, and nanoparticle fluxesmay be written as

y¼ 0 : v¼ 0; �kvTvy

¼ qwðxÞ; �DsmvCvy

¼ qmðxÞ; �DBvf

vy¼ qnpðxÞ

y/N : u¼ 0; v¼ 0; T ¼ TN; C ¼ CN; f¼fN

9=;(7)

where, in view of Darcy’s law, the u component of the velocity aty ¼ 0 can have an arbitrary value i.e. there is a slip at the boundary(u s 0), qw(x) is the heat flux, qm(x) is the solute flux and qnp(x) isthe nanoparticle flux. The case of wall heat flux is more easilyachievable experimentally than the isothermal condition. More-over, a heat flux condition is sometimes approximated in practicalapplications [31]. Similarly, the boundary conditions of prescribedsolute and nanoparticle fluxes are also relevant in themanipulationof nanoparticles through porous membranes [32].

Introducing stream function j(x,y), defined by

u ¼ vj

vyand v ¼ �vj

vx(8)

which satisfies the continuity equation (Eq. (1)) identically leavingthe remaining four equations i.e., Eqs.(2e5) in the following forms:

v2j

vy2¼ ð1�fNÞrfNgK

m

�bT

vTvy

þbCvCvy

���rp�rfN

�gK

m

vf

vy(9)

vj

vyvTvx

� vj

vxvTvy

¼ am

v2Tvy2

!þ sDB

"�vf

vy

��vTvy

�þ�DT

TN

�

��vTvy

�2#þ sDTC

v2Cvy2

(10)

W.A. Khan, A. Aziz / International Journal of Thermal Sciences 50 (2011) 2154e2160 2157

vj

vyvCvx

� vj

vxvCvy

¼ 3Dsmv2Cvy2

þ 3DCTv2Tvy2

(11)

vj

vyvf

vx� vj

vxvf

vy¼ 3DB

v2f

vy2þ 3

DT

TN

v2Tvy2

(12)

3. Similarity analysis

Weconsider the similarity transformations for theprescribedfluxboundary conditions. The following similarity quantities are intro-duced to transform Eqs. (1e5) into ordinary differential equations.

h ¼ yxRa1=3x ; sðhÞ ¼ j

amRa1=3x

;

qðhÞ ¼ T � TNðqwx=kÞRa

1=3x gðhÞ ¼ C � CN

ðqmx=DsmÞRa1=3x

and f ðhÞ ¼ f� fN�qnpx=DB

�Ra1=3x

9>>>>>>>>>>=>>>>>>>>>>;

(13)

with

Rax ¼ xa

"rKgbTmeff

�qwxk

�#1=3(14)

After some algebraic manipulation, the governing Eqs. (8e11)reduce to the following four coupled, nonlinear ordinary differen-tial equations:

s00 � q0 � Ncg0 þ Nr f 0 ¼ 0 (15)

q00 � 2lþ 13

s0qþ lþ 23

sq0 þ Nbf 0q0 þ Ntq02 þ Ndg00 ¼ 0 (16)

g00 � 2lþ 13

Le$s0gþ lþ 23

Le$sg0 þ Ld$q00 ¼ 0 (17)

f 00 � 2lþ 13

Ln$fs0 þ lþ 23

Ln$sf 0 þ NtNb

$q00 ¼ 0 (18)

with boundary conditions

sð0Þ ¼ 0; q0ð0Þ ¼ �1; g0ð0Þ ¼ �1; f 0ð0Þ ¼ �1as h/N; s0 ¼ 0; q ¼ 0; g ¼ 0; f ¼ 0

(19)

where primes denote differentiation with respect to h and theparameters Nr (nanofluid buoyancy-ratio), Nb (Brownian motionparameter), Nt (thermophoresis parameter), Nd (modified Dufourparameter), Nc (regular double-diffusive buoyancy parameter), Le(modified Lewis number), Ln (nanofluid Lewis number), and Ld(Dufour-solutal Lewis number) are defined as follows:

Nb ¼sDB

hxqnpðxÞ=DB

iamRa

1=3x

; Nt ¼ sDT ½xqwðxÞ=k�amTNRa1=3x

Nd ¼ sDTC ½qmðxÞ=Dsm�am½qwðxÞ=k� ; Nc ¼ bC ½qmðxÞ=Dsm�

bT ½qwðxÞ=k�

Nr ¼�rP � rfN

�½qmðxÞ=Dsm�

rNð1�fNÞbT ½qwðxÞ=k�; Ld ¼ DCT ½qwðxÞ=k�

Dsm½qmðxÞ=Dsm�Le ¼ am

3Dsm; Ln ¼ am

3DB

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

(20)

with

qwðxÞ ¼ Awxl qmðxÞ ¼ Amxl qnpðxÞ ¼ Anpxl (21)

In writing Eq. (21), the surface fluxes are assumed to beproportional to xl where l is a real positive exponent.

The quantities of practical interest, in this study, are the localNusselt number Nux, the Sherwood number Shx and the nanofluidSherwood number Shx,n, which are defined as

Nux ¼ xqwðxÞkðTw � TNÞ (22)

Shx ¼ xqmðxÞDsmðgw � gNÞ (23)

Shx;n ¼ xqnpðxÞDBðfw � fNÞ (24)

Following Kuznetsov and Nield [30], the reduced local Nusseltnumber Nur, reduced local Sherwood number Shr and reducednanofluid Sherwood number can be introduced and represented as

Nur ¼ Ra1=3x Nux ¼ 1qð0Þ (25)

Shr ¼ Ra1=3x Shx ¼ 1gð0Þ (26)

Shrn ¼ Ra1=3x Shx;n ¼ 1f ð0Þ (27)

4. Numerical solution

Eqs. (15e18) subject to the boundary conditions, Eq. (19), weresolved numerically using Maple 14.0. This software uses a fourth-fifth order RungeeKuttaeFehlberg method as the default to solvethe boundary value problems numerically. Its accuracy androbustness has been repeatedly confirmed in various heat transferpapers. To facilitate convergence for all values of parameters,Nc, Nr,Nb, Nt, Nd, Le, Ld, and Ln chosen for this study, the unity coefficientof the term f00 was replacedwith (101�100m) and continuation¼mwas used in the dsolve command.Without this modification, Maplereturns the results that do not give the correct asymptotic valuesdictated by Eq. (19) but give results that intersected the h witha steep angle. More information on overcoming the convergenceissues can be found in Maple’s help section under the numericalsolution of difficult ode boundary value problems. The asymptoticboundary conditions given by Eq. (19) were replaced by usinga value of 6 for the similarity variable hmax as follows.

hmax ¼ 6; s0ð6Þ ¼ 0; qð6Þ ¼ 0; f ð6Þ ¼ 0; gð6Þ ¼ 0 (28)

The choice of hmax ¼ 6 ensured that all numerical solutionsapproached the asymptotic values correctly. This is an importantpoint that is often overlooked in publications on boundary layerflows. In a paper devoted exclusively devoted to the appraisal ofpublished results in boundary layer flows, Pantokratoras [33] foundthat many published results on boundary layer flows are erroneousbecause the graphs for the velocity and temperature distributionsin the boundary layers do not approach the correct values asymp-totically. All the errors were the consequence of using smallervalues of h to represent the boundary condition at h ¼ N.

Fig. 3. Effect of surface fluxes exponent l on the boundary layer functions for a mono-diffusive nanofluid (MDNF).

W.A. Khan, A. Aziz / International Journal of Thermal Sciences 50 (2011) 2154e21602158

5. Results and discussion

Because of eight dimensionless parameters (Nc, Nr, Nb, Nt, Nd,Le, Ld, and Ln) and the exponent l, only a selection of the numericalresults will be presented. Fig. 2 shows the similarity solutions forthe dimensionless stream function s(h), dimensionless tempera-ture, q(h), dimensionless solutal concentration g(h) and dimen-sionless nanoparticle concentration fðhÞ for Pr ¼ 10, Le ¼ 1, Ln ¼ 10and Ld ¼ 1 for a mono-diffusive regular fluid (MDRF) i.e.Nc ¼ Nr ¼ Nb ¼ Nt ¼ Nd ¼ 0. Results for l ¼ 0 and 0.5 are showntogether for comparison. The curves for l ¼ 0 correspond to thecase of constant surface fluxes. All four dimensionless quantitiesexperience a decrease when l is increased from 0 to 0.5. Fig. 3shows the similarity solutions for s(h), (h), g(h) and fðhÞ fora mono-diffusive nanofluid (MDNF). The graphs follow the samepattern as in Fig. 2 except that with l¼ 0.5, the temperature and thesolutal concentration are higher compared with the correspondingvalues for l ¼ 0. A comparison between the four boundary layersshows that the thermal and the solutal concentration boundarylayers are of comparable thickness but the nanoparticle concen-tration layer is much thinner.

Sample results for the double-diffusive regular fluid (DDRF) anddouble-diffusive nanofluid (DDNF) are given in Fig. 4 and Fig. 5,respectively. In generating the numerical results for the double-diffusive regular fluid, each of the parameters Nr, Nb and Nt wasassigned a value of 10�5 instead of 0 to avoid singularities in thenumerical computations. Although graphically all boundary layersappear to be comparable in thickness, the numerical data showedthat the nanoparticle concentration layer has the smallest thick-ness. The effect of changing the exponent l from 0 to 0.5 is todecrease the local (at a given value of h) values of the streamfunction, temperature, solutal concentration and the nanoparticleconcentration.

Finally we present the data for the reduced Nusselt number andthe two reduced Sherwood numbers. Fig. 6 shows the reducedNusselt number as a function of l for themono-diffusive (MDRF andMDNF) and double-diffusive (DDRF and DDNF) regular and nano-fluids. Comparing the curves for MDRF and DDRF, it can beenseen that the reduced Nusselt number decreases as each of theparameter Nr, Nb and Ntis increased from 0 for MDRF to 0.2 for

Fig. 2. Effect of surface flux exponent l on boundary layer functions for a mono-diffusive regular fluid (MDRF).

MDNF. The present predictions can be qualitatively compared withthe predictions from the following correlation of Kuznetsov andNield [30] for a vertical surface with constant temperature,constant solutal concentration and constant nanoparticle concen-tration at the surface.

Nurest ¼ 0:465�0:298Nr�0:208Nb�0:351Nt þ 0:069Nc

�0:133Nd (29)

Eq. (29) also shows that the reduced Nusselt number decreasesas Nr, Nb and Nt increase. A similar trend is observed when thecurves for DDRF and DDNF are compared. By comparing the curvesfor MDNF and DDNF, we note that the reduced Nusselt numberdecreases as Nc and Nd are each increased from 0 to 0.3. For each ofthe four cases shown in Fig. 6, the reduced Nusselt number

Fig. 4. Effect of surface fluxes exponent l on the boundary layer functions for a double-diffusive regular fluid (DDRF).

Fig. 5. Effect of surface fluxes exponent l on boundary layer functions for a double-diffusive nanofluid (DDNF).

Fig. 7. Effectof surfacefluxesexponentlon the local Sherwoodnumber fordifferentfluids.

W.A. Khan, A. Aziz / International Journal of Thermal Sciences 50 (2011) 2154e2160 2159

increases as the exponent l increases. This increase in the Nusseltnumber is the direct result of the increase in the imposed heat fluxcaused by the increase in l.

The reduced Sherwood data for the solute transfer is illustratedin Fig. 7. For themono-diffusive cases, the presence of nanoparticlesenhances the Sherwood number because of the contributions of theBrownian motion, thermophoresis and the buoyant motionprompted by the difference in the densities of nanoparticles andthe base fluid. The same enhancement occurs in the double-diffusive cases when a nanofluid is present instead of a regularfluid. For all the four cases exhibited in Fig. 7, the reduced Sherwoodnumber increases as l increases. This increase is caused as a resultof the increase in the surface solutal flux caused by the increase in l.

The nanoparticle Sherwood number data plotted in Fig. 8reveals that the transport of nanoparticles is largely dictated by

Fig. 6. Effect of surface fluxes exponent l on the local Nusselt numbers for differentfluids.

whether the process is mono-diffusive or double-diffusive, thelatter promoting much higher values of the Sherwood numbersthan the former. It is interesting to note that the Brownian motion,thermophoresis and the buoyant motion prompted by the differ-ence in the densities of nanoparticles and the base fluid have hardlyany effect on the nanoparticle Sherwood number when double-diffusion occurs. In the case of the mono diffusion, the nanofluidenhances the nanoparticle Sherwood numbers slightly comparedwith the values obtained with a regular fluid. For all the four casesillustrated in Fig. 8, the nanoparticle Sherwood number increases asthe exponent l increases. This is understandable because theincrease in l increases the imposed nanoparticle flux driving thetransport of the nanoparticles in the convective flow field. Theclassical book by Nield and Bejan [34] on convection in porousmedia is an excellent source for gaining a fundamental

Fig. 8. Effect of surface fluxes exponent l on nanoparticle Sherwood number fordifferent fluids.

W.A. Khan, A. Aziz / International Journal of Thermal Sciences 50 (2011) 2154e21602160

understanding of the basic concepts that form the backbone of thisand numerous other papers on porous media.

6. Conclusions

The main conclusions emerging from this study are as follows:(1) For both mono and double diffusions in regular fluids, the localvalues of stream function, temperature, solute concentration andthe nanoparticle concentration decrease as the exponent l

increases, (2) For mono diffusion in a nanofluid, the local streamfunction and nanoparticle concentration decrease and the localtemperature and solute concentration increase as the exponent lincreases, (3) For double diffusion in nanofluids, the local values ofstream function, temperature, solute concentration and the nano-particle concentration decrease as the exponent l increases, (4) Forboth mono and double diffusions in regular and nanofluids, thereduced Nusselt number and the reduced solute Sherwood numberincrease as the exponent l increases, (5) the highest values ofreduced Nusselt numbers are achieved in mono diffusion ina regular fluid and the lowest values occur with double diffusion innanofluids, (6) the highest values of reduced solute Sherwoodnumbers are achieved with double diffusion in nanofluids and thelowest values occur with mono diffusion in regular fluids, (7) thereduced nanoparticle Sherwood number is significantly higher fordouble diffusion in regular and nanofluids than it is for monodiffusion in regular and nanofluids and in both cases it increases asthe exponent l increases, and (8) the reduced nanoparticle Sher-wood number is largely dependent on the diffusion process (monoor double) and little affected by the fluid (regular or nano).

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