Propulsion and Power Research 2019;8(3):243e252

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Propulsion and Power Research

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ORGINAL ARTICLE

Heat and mass transfer analysis on flow ofWilliamson nanofluid with thermal and velocityslips: Buongiorno model

Yap Bing Khoa, Abid Hussananb,c,*,Muhammad Khairul Anuar Mohameda, Mohd Zuki Salleha

aApplied & Industrial Mathematics Research Group, Faculty of Industrial Sciences & Technology,Universiti Malaysia Pahang, MalaysiabDivision of Computational Mathematics and Engineering, Institute for Computational Science,Ton Duc Thang University, Ho Chi Minh City, VietnamcFaculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Received 11 December 2017; accepted 14 January 2019Available online 23 August 2019

KEYWORDS

Williamson nanofluid;Slip conditions;Heat transfer;Stretching sheet

*Corresponding author.

E-mail addresses: yapbing90@hotmailAnuar Mohamed), zuki@ump.edu.my (Mo

Peer review under responsibility of Be

Production and Hosting by

https://doi.org/10.1016/j.jppr.2019.01.0112212-540X/ª 2019 Beihang University. Prlicense (http://creativecommons.org/license

Abstract The thermal and velocity slips of boundary layer of Williamson nanofluid over astretching sheet are studied numerically. Buongiorno model is used to explore the heat transferphenomena caused by Brownian motion and thermophoresis. Using similarity transformations,the governing equations are reduced to a set of nonlinear ordinary differential equations(ODEs). These equations are solved numerically by using Shooting method. The effects ofnon-Newtonian Williamson parameter, velocity and thermal slip parameters, Prandtl number,Brownian parameter, Schmidt number, Lewis number, Brownian motion parameter, thermo-phoresis parameter on velocity, temperature and concentration fields are shown graphicallyand discussed. The results found that the thickness of boundary layer decreases as the slipand thermal factor parameter increases. Further, present results indicate that the nanofluid

.com (Yap Bing Kho), abidhussanan@tdtu.edu.vn (Abid Hussanan), baa_khy@yahoo.com (Muhammad Khairulhd Zuki Salleh).

ihang University.

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oduction and hosting by Elsevier B.V. on behalf of KeAi. This is an open access article under the CC BY-NC-NDs/by-nc-nd/4.0/).

244 Yap Bing Kho et al.

Nomenclature

a; b positive constant for streB thermal slip parameterC nanoparticle volume fracCp effective heat capacity oCw nanoparticle volume fracCN nanoparticle volume fracDB Brownian diffusion coeffiDT thermophoresis diffusionf dimensionless stream funLe Lewis numberNbt diffusivity ratio parameteNc heat capacities ratio paraPr Prandtl numberRe Reynolds numberSc Schmidt numberSh Shearwood number

temperature and concentration are enhanced with a rise of Williamson parameter. The Nusseltnumber is reduced with an increase of the Lewis and Prandtl numbers.ª 2019 Beihang University. Production and hosting by Elsevier B.V. on behalf of KeAi. This is an open

access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

tching rate

tionf nanoparticletion at the sheettion far from the sheetcientcoefficientction

rmeter

T fluid temperatureTw temperature of fluid near sheetTN ambient temperatureu velocity component along x directionv velocity component along y direction

Greek lettersa nanofluid thermal diffusivityq dimensionless temperaturer density of nanofluidrp nanoparticles densitym dynamic viscosityn kinematic viscosityl non-Newtonian Williamson parameterd velocity slip parameterf dimensionless nanoparticle volume fractionG time constant

1. Introduction

Nanofluid has been discovered and discussed by manyresearchers due to its unique properties and well known asthe suspended colloidal liquid with nano-size metallic ornon-metallic particles. Buongiorno [1] noted that thenanoparticle absolute velocity can be viewed as the sum ofthe base fluid velocity and a relative velocity. He consideredin turn seven slip mechanisms: inertia, Brownian diffusion,thermophoresis, diffusiophoresis, Magnus effect, fluiddrainage, and gravity settling. After examining each of thesein turn, he concluded that in the absence of turbulent effectsit is the Brownian diffusion and the thermophoresis that willbe important. Adding nanoparticles to a base fluid enhancesthe thermal conductivity of the traditional heat transferfluids like water, kerosene mineral oils and ethylene glycol.Thus, nanofluids can potentially be used as heat transferfluids for cooling of electronics, vehicles engine cooling,solar water heating, nuclear reactors, heat exchanger, laserdiodes, oscillating heat pipes. Corcione et al. [2] investi-gated nanofluids natural convection flow inside a differen-tially heated cavity using Buongiorno’s model and foundthat the two phase mixture method is more accurate than thesingle phase model. A numerical study was conducted byGaroosi et al. [3] using Buongiorno’s model. They analyzednatural and mixed convection heat transfer of a nanofluid(Al2O3-water) in a laterally heated square cavity. Eiamsa-ard

et al. [4] study the heat transfer enhancement of TiO2-waternanofluid in a heat exchanger tube equipped with over-lapped dual twisted-tapes. Results of the following papersindicated that the thermophoresis force is important forAl2O3-water and specially TiO2-water nanofluids. In fact fornanoparticles with low thermal conductivity like TiO2, thisforce is very important. Some researchers like Turkyilma-zoglu [5], Qasim et al. [6], Hussanan et al. [7], Swalmehet al. [8] and Afridi et al. [9] who studied the heat transferflow with nanofluids, stated that even in the presence of lowconcentration of nanoparticles, the suspensions can enhancethermal conductivity up to 20%. The enhancement of ther-mal conductivity mainly depends on some factors like thematerial of particles, shape/size of particles, temperature ofthe fluid material, and so on. The purpose of heat transfercoefficient is to determine the factor in forced convectioncooling or heating application, like the heat exchange pro-cess involved in electrical engine systems.

In the work of Krishnamurthy et al. [10], Williamsonnanofluid is categorized as visco-inelastic fluids. Blasius[11] initially studied the problem of velocity boundary layeron a flat plat followed by Sakiadis [12] who investigated thetheoretical aspect of approximate and exact method forperformance of boundary layer flow through a flat surface.Next, Ramesh et al. [13] had studied the convectiveboundary condition on Blasius and Sakiadis flows withWilliamson fluid. Prior to that, Khan and Khan [14] carried

Figure 1 The physical model.

Table 1 Comparison of temperature gradient �q0ð0Þ for Prandtlnumber in viscous case.

Pr Ishak et al. [26]Keller boxmethod

Hayat et al. [27]exact solutions

Present studyshooting method

0.72 0.8086 0.808631 0.8088341 1.0000 1.000000 1.0000083 1.9237 1.923682 1.92367810 3.7207 3.720673 3.720671

Heat and mass transfer analysis on flow of Williamson nanofluid 245

out the investigation on boundary layer flows of Williamsonfluid by using homotopy analysis method (HAM). He foundthat the thickness of boundary layer decreases as Wil-liamson parameter increases. Later, Nadeem and Hussain[15] analyzed the problem on Williamson nanofluid withmagnetohydrodynamic (MHD) flow over a heated surfaceand found out that the thermal conductivity of Williamsonfluid is lower than the MHD Williamson nanofluid.

Thermal energy happens when there is a temperaturedifference between objects. Heat transfers in non-Newtonianfluids have become common and important in some fieldssuch as reactor cooling systems, electronic packing and soon. Kurtcebe and Erim [16] studied the problem of turbinecooling application with heat transfer in non-Newtonianvisco-inelastic fluid and concluded that the upper limit ofvisco-inelastic parameter depends to the Reynolds number.The heat transfer analysis flow in nanofluids over astretching sheet is important and crucial for the optimumquality of the final product especially in polymer extrusionproduction. Ibrahim and Shankar [17] studied the heattransfer and MHD boundary layer flow in nanofluid pastover a stretching sheet. Krishnamurthy et al. [18] investi-gated the effect of chemical reaction on melting heat transferand MHD boundary layer flow of Williamson nanofluid.Meanwhile, the flow features and convective heat transfer ofCu-water nanofluids were investigated by Xuan and Li [19].According to their findings, the heat transfer feature in-creases as the volume fraction of nanoparticles increases.With the same Reynolds number, the suspended nano-particles show higher heat transfer coefficient and remark-ably enhance heat transfer process compared to the basefluid. Heris et al. [20] carried out the experiment onconvective heat transfer laminar flow over oxide nanofluidwith boundary condition of constant wall temperature. Theyconcluded that there are several factors to enhance the heattransfer for nanofluid, as well as increase the chaotic

movements of nanoparticles, thermal conductivity, in-teractions and fluctuations.

Slip condition is considered since the presence of nano-particles causes the interface of slip velocity between fluidand solid boundary. Yang [21] had presented the viscousflow over a solid surface with slip boundary condition. Theinterfacial interaction between fluid and solid is reflected byslip condition contributed by the interaction of intermolec-ular and roughness of wall surface. Noghrehabadi et al. [22]studied the problem of the partial slip boundary conditioneffect on nanofluids with prescribed wall temperature over astretching sheet. The Nusselt and Sherwood number de-creases as the velocity slip parameter increases. Malvandiet al. [23] conducted an examination on the slip effects ofnanofluids in unsteady stagnation point over a stretchingsheet. In their work, an increase in the values of slipparameter caused the values of skin coefficient to drop. Theslip and no slip conditions on the laminar nanofluids byforced convection were numerically studied by Raisi et al.[24]. They found out that only higher Reynolds numberaffects the rate of heat transfer as slip velocity coefficientincreases.

Motivated by above analysis, our aim is to study the slipconditions and heat transfer analysis on Williamson nano-fluid over a stretching sheet is studied.

2. Mathematical formulation

This assumed that the flow of Williamson nanofluid overa stretching sheet on two-dimension is steady state andincompressible. The physical model is shown in Figure 1.The linear velocity on the sheet surface can be defined asUwðxÞZbx while the b and x represent the constant and thecoordinate measured along the stretching sheet respectively.y is the coordinate measured normal to stretching sheet aty Z 0. The wall temperature as Tw Zax2 þ TN and theconstant during the stretching surface is illustrated as thefraction of nanoparticles Cw. The nanoparticle fractionambient values of temperature are denoted as CN when ycontinuous tends to infinity.

The two dimensional boundary layer governing equa-tions of non-Newtonian Williamson nanofluid flow are asfollows:

Figure 2 Velocity profiles for velocity slip parameter, when l Z0:5; BZ 1; NcZ 2:5; NbtZ 2; LeZ 10; ScZ 5 and PrZ 7:

Figure 3 Temperature profiles for velocity slip parameter, whenl Z 0:5; BZ 1; NcZ 2:5; NbtZ 2; LeZ 10; ScZ 5 and PrZ7:

Figure 4 Concentration profiles for velocity slip parameter, whenl Z 0:5; BZ 1; NcZ 2:5; NbtZ 2; LeZ 10; ScZ 5 and PrZ7:

Figure 5 Temperature profiles for thermal slip parameter, when l Z0:5; dZ 0:5; NcZ 2:5; NbtZ 2; LeZ 10; ScZ 5 and PrZ 7:

246 Yap Bing Kho et al.

vu

vxþvv

vyZ0 ð1Þ

uvu

vxþ v

vu

vyZn

v2u

vy2þ

ffiffiffi2

pnG

vu

vy

v2u

vy2ð2Þ

uvT

vxþ v

vT

vyZa

v2T

vy2þ rpCp

rC

"DB

vC

vy

vT

vyþDT

TN

�vT

vy

�2#

ð3Þ

Heat and mass transfer analysis on flow of Williamson nanofluid 247

uvC

vxþ v

vC

vyZDB

v2C

vy2þDT

TN

v2T

vy2ð4Þ

Then subjected to the appropriated boundary conditionsare as below:

uZuwþd�m

�vu

vy

�; vZ0; TZTwþB)vT

vy; CZCw at yZ0

u/0; T/TN; CZCN as y/N

ð5Þ

Following Yasin et al. [25], the similarity transformationsare as below:

uZbxf 0ðhÞ; vZ� ðbnÞ12f ðhÞ; hZ

ffiffiffib

v

ry

qðhÞZ T � TN

Tw � TN

; fðhÞZ C�CN

Cw �CN

ð6Þ

By using similarity transformations, the ordinary differ-ential equations are formed as follow:

f000 ðhÞþlf

000 ðhÞf 00ðhÞ þ f 00ðhÞf ðhÞ � f 02ðhÞZ0 ð7Þq00ðhÞ þPrf ðhÞq0ðhÞ � 2Prf 0ðhÞqðhÞ þNc

Leq0ðhÞf0ðhÞ

þ Nc

ðLeÞðNbtÞq02ðhÞZ0

ð8Þ

f00ðhÞþScf ðhÞf0ðhÞ þ 1

Nbtq00ðhÞZ0 ð9Þ

Where

lZ

ffiffiffiffiffiffiffi2b3

n

sGx; PrZ

n

a; NcZ

rpCp

rCðCw �CNÞ

NbtZTNDBðCw �CNÞDT ðTw � TNÞ ; LeZ

a

DB; ScZ

v

DB

and f , q and f are functions of h and prime denotes de-rivatives with respect to h. The corresponding boundaryconditions take the form:

f ð0ÞZ0; f 0ð0ÞZ1þ df 00ð0Þ; qð0ÞZ1þ bq0ð0Þ;fð0ÞZ1 at yZ0

f 0ðNÞZ0; qðNÞZ0;

fðNÞZ0 as y/0

ð10Þ

The skin friction Cf , local Nusselt number Nu andSherwood number Sh are defined as

Cf Ztw

rU 2w

; NuZxqw

kðTw � TNÞ and ShZxqw

DBðCw �CNÞ ð11Þ

where tw is the skin friction or shear stress at the surface ofthe plate, qw is the heat flux from the surface and qm is themass flux of the nanoparticle volume fraction from the sur-face, and are given by

tw Z m

"vuvyþ Gffiffi

2p

�vuvy

�2#; qw Z � kvTvyyZ0

and qwZ� DBvC

vy yZ0

finally, we obtain

Cf

ffiffiffiffiffiffiRe

pZ f 00ð0Þ þ l

2f 002ð0Þ; Nuffiffiffiffiffiffi

Rep Z� q0ð0Þ

andShffiffiffiffiffiffiRe

p Z�f0ð0Þ;ð12Þ

where ReZbnx

2 is the Reynold’s number.

3. Numerical results

The non-linear partial differential equations are solvedwith similarity transformations and transform to ordinarydifferential equations based on the given boundary condi-tions. After that, the derived equations are solved by usingShooting method. Thus, we may obtain unknown initialconditions hZ0 by using Shooting method and assumed theinitial values for boundary value problem. In this study,there is no consideration of variation in temperature, ve-locity and concentration where we took large infinity con-dition but finite value for h. We chose h � 6 for theconsidered parameters value since it is sufficient to achieveasymptotic boundary conditions for all. Based on the com-parison with previous results of other researchers (in Table1), we concluded that this technique work efficiently andthe results presented here are accurate and reliable.

4. Results and discussion

This section included the graphical results for velocity,temperature and concentration profiles for the involvedvariables such as velocity slip parameter, thermal slipparameter, non-Newtonian, Williamson parameter, Prandtlnumber, heat capacities ratio parameter, diffusivity ratioparameter, Lewis number and Schmidt number. Respec-tively. Figures 2e4 depicted the various values of velocityslip factor parameter for velocity, temperature and concen-tration profile respectively. The graphical results showedthat the value of velocity profiles is reducing as the velocityslip factor parameter increases. On the other hand, the walltemperature also decrease as the velocity slip factorparameter increases. However, the nanoparticles volumefraction increase as the velocity slip factor parameter in-creases. Figures 5 and 6 illustrated the temperature andconcentration profile on thermal slip parameter. It is foundthat the wall temperature and volume fraction of nano-particles decrease due to increase of thermal slip parameter.Figures 7e9 presented the several values of non-NewtonianWilliamson parameter on velocity, temperature and con-centration profile, respectively. The value of velocity pro-files decreases and temperature and concentration profilesincrease at the same time when the Williamson parameterincreases. Also, the skin friction coefficient for Williamsonnanofluid is getting low as the non-Newtonian Williamson

Figure 6 Concentration profiles thermal slip parameter, when l Z0:5; dZ 0:5; NcZ 2:5; NbtZ 2; LeZ 10; ScZ 5 and PrZ 7:

Figure 7 Velocity profiles for Williamson parameter, when B Z0:5; dZ 0:5; NcZ 2:5; NbtZ 2; LeZ 10; ScZ 5 and PrZ7:

Figure 8 Temperature profiles for Williamson parameter, when B Z0:5; dZ 0:5; NcZ 2:5; NbtZ 2; LeZ 10; ScZ 5 and PrZ 7:

Figure 9 Concentration profiles for Williamson parameter, whenB Z 0:5; dZ 0:5; NcZ 2:5; NbtZ 2; LeZ 10; ScZ 5 and PrZ7:

248 Yap Bing Kho et al.

parameter increases. It is good to be used as lubricant incooling system due to suspended nanoparticles could keeplonger in the base fluids and enhance the flow characteristicsof nanofluids.

Figure 10 shown that effect of Prandtl number on tem-perature profile. The increment of Prandtl number cause theslow rate in thermal diffusion, thus the temperature profile is

reducing at all the time. Physically, it meets the logic thatfluids with large Prandtl number have high viscosity andsmall thermal conductivity, which makes the fluid thick, andhence causes a de-crease in the velocity of the fluid.

Figure 10 Temperature profiles for Prandtl number, when l Z 0:5;

B Z 0:5; d Z 0:5; Nc Z 2:5; Nbt Z 2; Le Z 10 and Sc Z 5:

Figure 11 Temperature profiles for heat capacitance ratio parameter,when l Z 0:5; B Z 0:5; d Z 0:5; Nbt Z 2; Le Z 10; Sc Z5 and Pr Z 7:

Figure 12 Temperature profiles for Lewis number, when l Z 0:5;

BZ 0:5; dZ 0:5; NcZ 2:5; NbtZ 2; ScZ 5 and PrZ 7:

Figure 13 Temperature profiles for diffusivity ratio parameter, whenl Z 0:5; B Z 0:5; d Z 0:5; Nc Z 2:5; Le Z 10; ScZ 5 and PrZ7:

Heat and mass transfer analysis on flow of Williamson nanofluid 249

Figure 11 depicted the heat capacities ratio effect on tem-perature profile. It can be seen that the wall temperatureincreases when the heat capacities ratio parameter increases.Also, the boundary layer thickness decreased in this case.Therefore, the greater the value of radiation parameter, the

larger the heat superficial heat flux. Figure 12 illustrated theLewis number effects on temperature profiles. Lewis num-ber also refer as thermal to species diffusivity ratio. Lewisnumber is considered for conditions where the temperature

Figure 14 Concentration profiles for diffusivity ratio parameter,when l Z 0:5; B Z 0:5; d Z 0:5; Nc Z 2:5; Le Z 10; Sc Z5 and Pr Z 7:

Figure 15 Temperature profiles for Schmidt number, when l Z 0:5;

BZ 0:5; dZ 0:5; NcZ 2:5; NbtZ 2; LeZ 10 and PrZ 7:

Figure 16 The effect of thermal slip parameter and Prandtl numberon temperature gradient.

Figure 17 The effect of diffusivity ratio parameter and heatcapacitance ratio parameter on temperature gradient.

250 Yap Bing Kho et al.

and mass fraction are larger by defining temperature anddiffusivities, in which reduced the convective heat transfer.Thus, the wall temperature is decreasing function againstLewis number. Figures 13 and 14 depicted the different

value of diffusivity ratio parameter on temperature andconcentration profiles respectively. Both the profiles drop-ped as the value of diffusivity ratio parameter increases.Figure 15 presented the Schmidt number on concentrationprofiles. The value of the profile getting less and lesser asthe value of Schmidt number increases. This is because dueto the momentum diffusivity effects would increase and at

Figure 18 The effect of Lewis number and Prandtl number ontemperature gradient.

Table 2 The computation results for local skin friction coefficient, loc

d B l Pr Nc Nbt L

0.25 1 0.5 7 2.5 2 10.751.251.750.5 0.2 0.5 7 2.5 2 1

0.40.60.8

1 1 0 5 2.5 2 10.40.81.4

0.5 1 0.5 4 2.5 2 16810

0.5 1 0.5 7 5 2 1101520

0.5 1 0.5 7 2.5 0.3 10.40.60.9

0.5 1 0.5 7 2.5 2 1122

0.5 1 0.5 7 2.5 2 1

Heat and mass transfer analysis on flow of Williamson nanofluid 251

the same time, it will slow down the effects of mass transferrate leading to lower concentration profile.

Figure 16 presented the effects of thermal slip parameterand Prandtl number on temperature gradient. When thevalue of thermal slip parameter is small, the temperaturegradient is high. However, with increasing the Prandtlnumber will lead the heat transfer rate decreases sharply.The effect of diffusivity ratio parameter and heat capaci-tance ratio parameter on temperature gradient was shown inFigure 17. Obviously the temperature gradient rises as thevalue of heat capacities ratio parameter and diffusivity ratioparameter increases. Figure 18 depicted the effect of Lewisnumber and Prandtl number on temperature gradient. Basedon the diagram, the gradient of wall temperature increaseswhen the Prandtl number and Lewis number increases. Theskin friction coefficient, gradient temperature and concen-tration with all the parameters have been shown in Table 2.

5. Conclusions

The problem of slip conditions on heat transfer analysisand flow of Williamson nanofluid over a stretching sheet isstudied numerically. The important findings are summarizedas below:

al Nusselt number and local Sherwood number respectively.

e Sc f 00ð0Þ � q0ð0Þ � f0ð0Þ0 5 �0.824627 0.560312 1.214397

�0.521313 0.634114 0.995300�0.391317 0.653846 0.879335�0.316124 0.659675 0.801841

0 5 1.531496 0.7312681.111056 0.8904110.871426 0.9811790.716725 1.039801

0 5 �0.430631 0.653863 0.952791�0.442676 0.648285 0.935371�0.457476 0.641978 0.914200�0.489301 0.631466 0.869951

0 5 0.5937710.6326110.6567980.673705

0 5 0.6346170.6112960.5873040.563051

0 5 0.479166 0.2516990.494843 0.4469980.512197 0.6586690.524636 0.811730

5 0.538971.5 0.579227

0.599359.5 0.6112960 2 0.409463

4 0.7800506 1.0629688 1.299821

252 Yap Bing Kho et al.

� The results indicate that the nanofluid temperature andconcentration are enhanced with a rise of Williamsonparameter.

� An increase in the velocity slip parameter leads to in-crease in the nanoparticles volume fraction

� The higher the non-Newtonian Williamson parameter,the greater the thickness of boundary layer.

� The temperature profile was reducing at all the time andcaused by increment of Prandtl number.

� An increase of velocity slip parameter would lead to anincrease in the heat transfer rate.

� The wall temperature would increase when the heat ca-pacities ratio parameter increased while temperaturegradient decreases if increased in diffusivity ratioparameter.

� Temperature gradient is reduced with an increase of theLewis and Prandtl numbers.

� An increase in thermal slip parameter would cause a dropin the temperature gradient.

� Williamson nanofluid has lower skin friction coefficientwhen the non-Newtonian Williamson parameterincreased.

Acknowledgements

The authors gratefully acknowledge the financial sup-ports received from Ministry of Higher Education, Univer-sity Malaysia Pahang (UMP), Malaysia, through votenumber RDU 150101 and RDU 170358. The second authorwould like to acknowledge Ton Duc Thang University, HoChi Minh City, Vietnam for the financial support.

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