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Available online at www.sciencedirect.com

Journal of Applied Researchand Technology

www.jart.ccadet.unam.mxJournal of Applied Research and Technology 15 (2017) 464–476

Original

Double stratification effects on unsteady electrical MHD mixed convectionflow of nanofluid with viscous dissipation and Joule heating

Yahaya Shagaiya Daniel a,b, Zainal Abdul Aziz a,b,∗, Zuhaila Ismail a,b, Faisal Salah c

a Department of Mathematical Science, Faculty of Sciences, Universiti Teknologi Malaysia, UTM, 81310 Johor Bahru, Johor, Malaysiab UTM Centre for Industrial and Applied Mathematics, Universiti Teknologi Malaysia, UTM, 81310 Johor Bahru, Johor, Malaysia

c Department of Mathematics, Faculty of Science, University of Kordofan, Elobied 51111, Sudan

Received 9 December 2016; accepted 18 May 2017Available online 14 October 2017

bstract

The problem of unsteady mixed convection electrical magnetohydrodynamic (MHD) flow and heat transfer induced due to nanofluid over aermeable stretching sheet using Buongiorno model is investigated. The transverse electric and magnetic fields are considered in the flow field,hile in the heat convection is associated with the thermal radiation, heat generation/absorption, viscous and Ohmic dissipations, and chemical

eaction is incorporated in the mass diffusion. A similarity transformation is used to reduce the boundary layer governing equations which areartial differential equations to nonlinear differential equations and then solved numerically using implicit finite difference scheme. The nanofluidelocity and temperature are sensitive to an increase in the electric field, which resolved the problem of sticky effects due to the magnetic field.estructive chemical reaction increases the level nanoparticles concentration while reversed behave happened in the case of the generative chemical

eaction. Heat source boosts the fluid temperature while as opposite occurred with the heat sink. Thermal and concentration stratifications decreasedhe fluid temperature and the nanoparticles concentration profiles. Buoyancy ratio parameter reduced the Nusselt and Sherwood numbers whereasixed convection parameter increases for higher values. A comparison with the previous study available in literature has been done and found an

xcellent agreement with the published data. 2017 Universidad Nacional Autónoma de México, Centro de Ciencias Aplicadas y Desarrollo Tecnológico. This is an open access article under

he CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

eywords: Magnetic nanofluid; Doubly stratified flow; Mixed convection; Thermal radiation; Electric field; Viscous and Ohmic dissipations

acaiF&n2tcac

. Introduction

Magnetohydrodynamic (MHD) nanofluid is an essentialspect of convective heat transfer and has numerous applica-ions in science and engineering processes (Lin, Zheng, Zhang,

a, & Chen, 2015). Nanofluids is a composition of nanoparti-les suspended in a conventional liquid that has higher thermalonductivity in contrast with the conventional liquid (Choi,995). Such conventional liquid is water, oil and ethylene gly-ol in nature are poor in thermal conductivity, which limitshe heat transfer performance. Due to innovation in technologyia miniaturization of an electronic device requires the further

∗ Corresponding author.E-mail address: zainalaz@utm.my (Z.A. Aziz).Peer Review under the responsibility of Universidad Nacional Autónoma de

éxico.

&bKro

http://dx.doi.org/10.1016/j.jart.2017.05.007665-6423/© 2017 Universidad Nacional Autónoma de México, Centro de CienciasC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

dvance of heat transfer from energy saving. To overcome thishallenges leads to a new class of fluid called nanofluid. Thepplication of magnetic field for electrical conducting nanofluids of recent interest among different researchers (Abolbashari,reidoonimehr, Nazari, & Rashidi, 2014; Daniel, Aziz, Ismail,

Salah, 2017a; Daniel, 2015a, 2015b, 2016, 2017; Hedayat-asab, Abnisa, & Daud, 2017; Malvandi, Hedayati, & Ganji,014; Sandeep & Sulochana, 2015; Waqas et al., 2016). Essen-ial roles in sink-float separation, sealing of materials andonstruction of loudspeakers. The rate of heat transfer withpplication to the nuclear reactions and metallurgical pro-esses is as result of its greater thermal conductivity (Shateyi

Motsa, 2011). Due to the fascinating properties possess

y magnetic nanofluid of liquid and magnetic behavior (Bég,han, Karim, Alam, & Ferdows, 2014). Such materials have

elevance applications via magneto-optical wavelength fibers,ptical switches, optical modulators, nonlinear optical materials,

Aplicadas y Desarrollo Tecnológico. This is an open access article under the

Y.S. Daniel et al. / Journal of Applied Research and Technology 15 (2017) 464–476 465

Nomenclature

a,b constantsB0 magnetic fieldB applied magnetic fieldc stretching sheet constantcf skin friction coefficientDB Brownian diffusion coefficientDT thermophoresis diffusion coefficientE0 electric field factorE1 electric field parameterE applied electric fieldEc Eckert numberf′ dimensionless velocityg magnitude of the gravityGr Grashof numberJ Joule currentk thermal conductivitykf, kp thermal conductivity of the base fluid and

nanoparticleLe Lewis numberM magnetic field parameterNb Brownian motion parameterNr buoyancy ratio parameterNt thermophoresis parameterNu local Nusselt numberPr Prandtl numberqm wall mass fluxqr radiative heat fluxqw wall heat fluxRd radiation parameterRex local Reynolds numbers suction/injectionsc solutal stratificationst thermal stratificationSh local Sherwood numberT temperature of the fluidTW constant temperature at the wallT∞ ambient temperatureT0 Reference temperatureu, v velocity component along x- and y-direction com-

ponentV velocity fluidVW wall mass transfer

Greek symbolsα thermal diffusivityσ* Stefan–Boltzmann constantσ electrical conductivityβ volume expansion coefficientβnf volumetric volume expansion coefficient of the

nanofluid

ε heat generation/absorption parameterη dimensionless similarity variableμ dynamic viscosity of the fluidυ kinematic viscosity of the fluidρ, ρnf densityρp particle density(ρ)f density of the fluid(ρc)f heat capacity of the fluid(ρc)p effective heat capacity of a nanoparticleψ stream functionσ electrical conductivityϕ concentration of the fluidϕW nanoparticle volume fraction at the surfaceϕ∞ ambient nanoparticleϕ0 reference concentrationθ dimensionless temperatureφ dimensionless concentrationτ ratio between the effective heat transfer capacity

of the nanoparticle and the heat capacity of thefluid

τW surface shear stressλ mixed convection parameterγ chemical reaction parameter

Subscripts∞ condition at the free stream

asmh2RQ&G2

oaoPiitTt2rhlpu

W condition at the wall/surface

nd optical gratings. The heat conduction has an unlimitedignificance in numerous industrial heating or cooling equip-ent. Recently, the impact of magnetic field on nanofluids

as also been reported by other researchers (Daniel & Daniel,015; Daniel, Aziz, Ismail, & Salah, 2017b; Freidoonimehr,ashidi, & Mahmud, 2015; Gómez-Pastora et al., 2017; Hayat &asim, 2011; Hayat, Waqas, Khan, & Alsaedi, 2017b; Kumar

Sood, 2017; Mabood, Khan, & Ismail, 2015; Mohammed,omaa, Ragab, & Zhu, 2017; Shagaiy, Aziz, Ismail, & Salah,017).

Stratification is a deposition (formation) of layers whichccur as result of different fluids mixtures, temperature vari-tion, densities difference or concentration differences whichccur in fluids (Rehman, Malik, Salahuddin, & Naseer, 2016).hysically, heat and mass transfer analysis run concurrently, it

s of great interest to explore the mechanism of double strat-fication (thermal and mass stratifications) on the convectiveransport in nanofluid (Hayat, Waqas, Khan, & Alsaedi, 2016d).hese draw the attention of researchers such as (Hayat, Mum-

az, Shafiq, & Alsaedi, 2017a; Mehmood, Hussain, & Sagheer,016; Ramzan, Bilal, & Chung, 2017) due to its frequent occur-ence in various industrial and engineering processes namelyeat rejection into the surrounding (environment) via rivers,

akes and seas, also thermal energy storage systems via solaronds, condenser of power plants, industrial composition, man-facturing processing and heterogeneous composition in food,

4 esearch and Technology 15 (2017) 464–476

aoBsDgaigmbl&mhpsveItIHh2d

ehtosawtetitfi&tppt

2

tsfirnmmy

Nanofluid

y,v

B(x)E(x)B(x)

x,u

Momentum boundary layer

Thermal boundary layer

Concentration boundary layer

Magnetic field Magnetic field

Permeable stretching sheet Force

Electric field

Slit

Fig. 1. Physical configuration of the geometry.

asflJ

iitnmfitscctrvltcg

66 Y.S. Daniel et al. / Journal of Applied R

tmospheric density stratification, etc. The random movementf nanosized particles within the conventional fluids is known asrownian motion, which occurred as result of continuous colli-

ions between the nanosized particles and the fluid molecules.iffusion of particles sized under the influence of temperatureradient is called thermophoresis. Thermophoresis is a remark-ble significance of the Brownian motion of nanosized particlesn liquids with an externally sustained and constant temperatureradient. Buongiorno (2006) developed a two-phase nanofluidodel for investigating thermal energy transport, which has

een successfully used to address various diverse flow prob-ems (Hayat, Muhammad, Shehzad, & Alsaedi, 2016b; Mabood

Khan, 2016) involving nanofluids the aspects of Brownianotion and thermophoretic diffusion. The influence of radiative

eat transfer which is essential in industries at higher tem-erature processes subjected to isothermal and non-isothermalurroundings (Pal & Mandal, 2016). The radiative heat transferia nanofluid play a key role in system devices and renewablenergy sources (Hayat, Qayyum, Waqas, & Alsaedi, 2016c).n view of these technological and engineering widely applica-ions drawn the attentions of some researchers (Bhatti, Zeeshan,jaz, Bég, & Kadir, 2016; Daniel, Aziz, Ismail, & Salah, 2017c;ayat, Gull, Farooq, & Ahmad, 2015; Hayat, Waqas, She-zad, & Alsaedi, 2016e; Hussain, 2017; Jing, Pan, & Wang,017; Khan, Tamoor, Hayat, & Alsaedi, 2017) to explore in thisirection.

The aim of the current study is to investigate the combinedffects of an electric field, magnetic field, thermal radiation,eat generation/absorption, chemical reaction, viscous dissipa-ion, and Joule heating for unsteady mixed convection flowf electrical conducting nanofluid in the presence of thermaltratification and concentration stratifications over a perme-ble stretching sheet. Buongiorno nanofluid model is adoptedhich features the novel aspects of the Brownian motion and

hermophoresis. Additional, water based nanofluid (Maboodt al., 2015) is used. The boundary layer governing equa-ions which are partial differential equations are transformednto a set of nonlinear self-similar ordinary differential equa-ions. Computational analysis is performed through implicitnite difference scheme known as Keller box method (Cebeci

Bradshaw, 2012). The effects of physical sundry parame-ers on the velocity, temperature, nanoparticle concentrationrofiles, skin friction, and Nusselt and Sherwood numbers areresented and discussed extensively with the aids of graphs andables.

. Mathematical formulation

Consider the unsteady mixed convection flow of an elec-rical conducting nanofluid over a permeable stretching sheeturface in the presence of external magnetic and electricelds, thermal radiation, heat generation/absorption, chemicaleaction, viscous dissipation and Joule heating. The coordi-

ate system is selected in such a manner that the x-axis iseasured along the stretching sheet and the y-axis is nor-al to it and the flow is occupied above the surface sheet

> 0 displayed in Figure 1. Two equal and opposite forces

re impulsively applied along the x-axis so that the sheet istretched with the velocity uW(x, t) along the x-axis. As theuid is electrical conducting so the Lorentz force �J × �B where

� = σ( �E + �V × �B) is the Joule current (electrical conduct-ng density), σ is the electrical conductivity and �V = (u, v)s the fluid velocity, �B = ( 0, B, 0 ), �E = ( 0, 0, −E ) ishe transverse magnetic and electric fields. The induced mag-etic field and Hall current effects are ignored under smallagnetic Reynolds number. Both the electric and magneticelds due to viscous dissipation and Joule heating contributed

o the momentum and thermal boundary layer equations. Theheet is main at temperature TW = T0 + A1x(1 − at)−1 and con-entration ϕW = ϕ0 + C1x(1 − at)−1. The temperature and massoncentration of the ambient fluid are assumed to be stratified inhe form T∞ = T0 + A2x(1 − at)−1 and ϕ∞ = ϕ0 + C2x(1 − at)−1

espectively. In addition, the effects of thermal radiation andiscous dissipation are considered in the thermal boundaryayer equation whereas chemical reaction in the concentra-ion boundary layer equation. Using Buongiorno model, theonservative equations in the aforementioned conditions isiven as:

Continuity equation

∂u

∂x+ ∂v

∂y= 0 (1)

x-Momentum equation

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= − 1

ρf

∂P

∂x

+ν(∂2u

∂x2+ ∂2u

∂y2

)+ σ

ρf

(EB − B2u

)+

1

ρ

[(1 − ϕ∞) ρf ∞β (T − T∞) − (

ρp − ρf ∞)β (ϕ − ϕ∞)

]g

(2)

f

esear

a

HscAsv

E

tBsg

tvrrdfcr

ic

(E

q

wts

T

mbS

r

Y.S. Daniel et al. / Journal of Applied R

y-Momentum equation

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y= − 1

ρf

∂P

∂y

+ν(∂2v

∂x2+ ∂2v

∂y2

)+ σ

ρf

(EB − B2v

)+

1

ρf

[(1 − ϕ∞) ρf ∞β (T − T∞) − (

ρp − ρf ∞)β (ϕ − ϕ∞)

]g

(3)

Energy equation

∂T

∂t+ u

∂T

∂x+ v

∂T

∂y= k

(ρc)f

(∂2T

∂x2+ ∂2T

∂y2

)− 1

(ρc)f

(∂qr

∂y

)

+ μ

(ρc)f

(∂u

∂y

)2

+ Q

(ρc)f(T − T∞) +

σ

(ρc)f(uB − E)2

+τ{DB

(∂ϕ

∂x

∂T

∂x+ ∂ϕ

∂y

∂T

∂y

)+ DT

T∞

[(∂T

∂x

)2

+(∂T

∂y

)2]}

(4)

Concentration equation

∂ϕ

∂t+ u

∂ϕ

∂y+ v

∂ϕ

∂y= DB

(∂2ϕ

∂x2 + ∂2ϕ

∂y2

)

+DTT∞

(∂2T

∂x2 + ∂2T

∂y2

)− k1 (ϕ − ϕ∞) (5)

The boundary conditions at the sheet for the physical modelre presented by

y = 0 : u = uw (x, t) , v = vw (x, t) ,

T = TW (x, t) = T0 + A1x(1 − at)−1,

ϕ = ϕW (x, t) = ϕ0 + C1x(1 − at)−1,

y → ∞ : u → 0, T → T∞ = T0 + A2x(1 − at)−1,

ϕ → ϕ∞ = ϕ0 + C2x(1 − at)−1

(6)

ere uW(x, t) = bx/(1 − at) denotes the velocity of the lineartretching sheet, where b is the stretching rate, a is the positiveonstant with the property (at < 1) and dimension of a,b is T−1.1, A2, C1, and C2 are the dimensional constants with dimen-ion L−1 vw = −v0/

√1 − at is the wall mass transfer, when

w < 0 denote the injection while vw > 0 indicates the suction, = E0/

√1 − at is the electric field and E0 is the intensity of

he electric field, and B = B0/√

1 − at is the magnetic field and0 is the intensity of the magnetic field. Where u and v repre-ent the velocity components along the x-and y-axis respectively., P, α = k/(ρc)f , μ, ρ ρf and ρp is the gravitational accelera-ion, the fluid pressure, the thermal diffusivity, the kinematiciscosity, the density, the fluid density and particles densityespectively. We also have DB, DT, τ = (ρc)p/(ρc)f which rep-esents the Brownian diffusion coefficient, the thermophoresis

iffusion coefficient, the ratio between the effective heat trans-er capacity of the ultrafine nanoparticle material and is the heatapacity of the fluid and k1 = k0/(1 − at), Q = Q0/(1 − at) is theate of chemical reaction and heat generation/absorption as (Q0

ch and Technology 15 (2017) 464–476 467

s the heat generation or absorption coefficient and k0 is thehemical reaction coefficient) respectively.

The radiative heat flux qr via Rosseland approximationDaniel & Daniel, 2015; Sparrow & Cess, 1978) is applied toq. (4), such that.

r = −4σ∗

3k∗∂T 4

∂y(7)

here σ* represent the Stefan–Boltzmann constant and k*denotehe mean absorption coefficient. Expanding T4 by using Taylor’series about T∞ and neglecting higher order terms, we have,

4 ∼= 4T 3∞T − 3T 4

∞ (8)

Using Eq. (8) into Eq. (7), we get

∂qr

∂y= −16T 3∞σ∗

3k∗∂2T

∂y2 (9)

Use Eq. (9) in Eq. (4), we obtain

∂T

∂t+ u

∂T

∂x+ v

∂T

∂y= k

(ρc)f

(∂2T

∂x2+ ∂2T

∂y2

)

+ 1

(ρc)f

(16T 3

∞σ∗

3k∗∂2T

∂y2

)+ μ

(ρc)f

(∂u

∂y

)2

+

Q

(ρc)f(T − T∞) + σ

(ρc)f(uB − E)2

+τ{DB

(∂ϕ

∂x

∂T

∂x+ ∂ϕ

∂y

∂T

∂y

)+ DT

T∞

[(∂T

∂x

)2

+(∂T

∂y

)2]}

(10)

Using the order of magnitude analysis for the y-directionomentum equation which is normal to the stretching sheet and

oundary layer approximation (Daniel et al., 2017a; Ibrahim &hankar, 2013), such as

u v

∂u

∂y ∂u

∂x,∂v

∂t,∂v

∂x,∂v

∂y

∂p

∂y= 0

(11)

After the analysis, the boundary layer Eqs. (1)–(5) areeduced to the following as:

∂u

∂x+ ∂v

∂y= 0 (12)

∂u + u∂u + v

∂u = ν

(∂2u

)+ σ (

EB − B2u)+

∂t ∂x ∂y ∂y2 ρf

1

ρf

[(1 − ϕ∞) ρf ∞β (T − T∞) − (

ρp − ρf ∞)β (ϕ − ϕ∞)

]g

(13)

4 esear

i

ψ

u

td

htuv(scGNa(L

ifiE

tctitdf(tη

iiA2e2

l

w

68 Y.S. Daniel et al. / Journal of Applied R

∂T

∂t+ u

∂T

∂x+ v

∂T

∂y= k

(ρc)f

(∂2T

∂y2

)

+ 1

(ρc)f

(16T 3∞σ∗

3k∗∂2T

∂y2

)+ μ

(ρc)f

(∂u

∂y

)2

+ Q

(ρc)f(T − T∞) +

σ

(ρc)f(uB − E)2 + τ

{DB

(∂ϕ

∂y

∂T

∂y

)+ DT

T∞

(∂T

∂y

)2}

(14)

∂ϕ

∂t+ u

∂ϕ

∂y+ v

∂ϕ

∂y= DB

(∂2ϕ

∂y2

)+ DT

T∞

(∂2T∂y2

)−k1 (ϕ − ϕ∞) (15)

The boundary conditions at the sheet are presented by

y = 0 : u = uw (x, t) , v = vw (x, t) ,

T = TW (x, t) = T0 + A1x(1 − at)−1,

ϕ = ϕW (x, t) = ϕ0 + C1x(1 − at)−1,

y → ∞ : u → 0, T → T∞ = T0 + A2x(1 − at)−1,

ϕ → ϕ∞ = ϕ0 + C2x(1 − at)−1

(16)

The equations are reduced into the dimensionless form byntroducing the following dimensionless quantities defined as:

=√

bv

1 − atxf (η) , η = y

√b

v (1 − at),

θ = T − T∞Tw − T0

, φ = ϕ − ϕ∞ϕw − ϕ0

, (17)

The stream function ψ can be given as:

= ∂ψ

∂y, v = −∂ψ

∂x(18)

Using Eqs. (17) and (18) into Eqs. (12)–(16). The equa-ions of momentum, energy and nanoparticle concentration inimensionless form become:

f ′′′ + ff ′′ − f ′2 − δ(f ′ + η

2f ′′

)+ M

(E1 − f ′)

+λ (θ − Nrφ) = 0 (19)

1

Pr

(1 + 4

3Rd

)θ′′ + fθ′ − f ′θ − δ

(st + η

2θ′ + θ

)+Nbφ′θ′ + Ntθ′2 + Ec(f ′′)2

+MEc(f ′ + E1)2 + εθ + stf′ = 0

(20)

φ′′ + Lefφ′ − Lef ′φ − Leδ(sc + η

2φ′ + φ

)+ Nt

Nbθ′′

−Leγφ − Lescf′ = 0 (21)

hsin

ch and Technology 15 (2017) 464–476

The boundary conditions are given by

f = s, f ′ = 1, θ = 1 − st, φ = 1 − sc, at η = 0

f ′ = 0, θ = 0, φ = 0, as η → ∞(22)

ere f′,θ and φ is the dimensionless velocity, tempera-ure, and concentration, respectively, δ = a/b represent thensteadiness parameter, λ = Gr/Re2 is the mixed con-ection parameter (Richardson number) for values ofλ > 0) associate to heated surface and (λ < 0) corre-ponds to cold surface whereas (λ = 0) indicates forcedonvection flow, Gr = gβ(1 − ϕ∞)(TW − T0)ρf∞/ν2ρf is therashof number, Re = uwx/v is the Reynolds number,r = (ρf − ρf∞)(ϕW − ϕ0)/βρf∞(1 − ϕ∞)(TW − T0) is the buoy-ncy ratio parameter, Pr = v/α stand for Prandtl number, Nb =ρc)pDB(ϕw − ϕ∞)/(ρc)f v is the Brownian motion parameter,e = v/DB is the Lewis, Nt = (ρc)pDT (Tw − T∞)/(ρc)f vT∞

s the thermophoresis parameter, M = σB20/bρf is the magnetic

eld parameter, E1 = E0/uWB0 is the electric field parameter,c = u2

W/cp(Tw − T∞) is the Eckert number, st = A2/A1 denotehe thermal stratification parameter, sc = C2/C1 indicate theoncentration stratification parameter, s = v0/

√vb is the suc-

ion (s > 0)/injection (s < 0) parameter and Rd = 4σ∗T 3∞/k∗ks the radiation parameter, ε = Q0/b(ρc)f is the heat genera-ion/absorption as (ε > 0) denotes heat generation and (ε < 0)enotes heat absorption, γ = k0/b is the chemical reaction,or (γ > 0) associates to destructive chemical reaction whileγ < 0) corresponds to generative chemical reaction respec-ively. Where prime represents differentiation with respect to. In our present study, the selection of non-dimensional phys-cal sundry parameters of nanofluids is considered to varyn view of the works of (Abolbashari et al., 2014; Daniel,ziz, Ismail, & Salah, 2017d; Daniel, Aziz, Ismail, & Salah,017e; Hsiao, 2016, 2017; Ibrahim & Shankar, 2013; Maboodt al., 2015; Noghrehabadi, Saffarian, Pourrajab, & Ghalambaz,013).

The skin friction coefficient the local Nusselt number and theocal Sherwood number are

cf = τw

ρu2w (x, t)

, Nu = xqw

k (Tw − T∞),

Sh = xqm

DB (ϕw − ϕ∞), (23)

here

qw=−((

k + 16σ∗T 3∞3k∗

)∂T

∂y

)y=0

, qm=−DB(∂ϕ

∂y

)y=0

,

τw = μf

(∂u

∂y

)y=0

, (24)

ere τw is the shear stress in the stretching surface, qw is theurface heat flux while qm is the surface mass flux, Re = uwx/v

s the Reynolds number and k is the thermal conductivity of theanofluid. For the local skin-friction coefficient, local Nusselt

Y.S. Daniel et al. / Journal of Applied Research and Technology 15 (2017) 464–476 469

Table 1Comparison with previously published works for the values of skin frictioncoefficient −f′′(0) for various values of M, s, and δ when E1 = Nr = λ = 0.

M s δ Khan andAzam (2017)

Ibrahim andShankar (2013)

Hayat et al.(2016a)

Presentresults

0.0 0.0 0.0 – 1.0000 – 1.0000000.5 – 1.2808 1.2808 1.280776

0.5 0.5 – 1.5000 – 1.5000001.0 0.5 – 1.6861 – 1.6861411.5 0.5 – 1.8508 – 1.8507812.5 0.5 – 2.0000 – 2.0000001.0 0.0 – 1.4142 1.4142 1.4142145.0 – – 2.4494 2.4494900.0 0.0 1.0000 – – 1.000000

0.2 1.06801 – – 1.0680120.4 1.13469 – – 1.1346850.6 1.19912 – – 1.1991170.8 1.26104 – – 1.2610401.2 1.37772 – – 1.3777201.4 1.43284 – – 1.4328312.0 1.58737 – – 1.587358

af

3

atdccPnpREnsvces2dnRtid

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ƒ′(η)

E1=0.0

E1= 0.1

E1= 0.2

E1= 0.3

Pr=6.2, Rd=Ec=δ=ε=0.1, Nb=Nt=λ=0.5,Le=5.0,

s=1.5, M=0.1, st=0.1, sc=0.2, Nr=γ=1.0

Fig. 2. Influence of E1 on the velocity profile f′(η).

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

M=0.0

M=0.4

M=0.8

M=1.2

ƒ′(η)

Pr=6.2, Rd=Ec=δ=ε=0.1, Nb=Nt=λ=0.5,Le=5.0,

s=0.5, E1=0.0 1, st=0.1, sc=0.2, Nr=γ=1.0

Fig. 3. Influence of M on the velocity profile f′(η).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ƒ′(η)

δ=0.1

δ=0.3

δ=0.5

δ=0.7

Pr=6.2, M=Nt=st=γ=ε=0.1, E1=0.01,sc=Nb=Rd=Ec =0.2, Le =5.0 , s= λ= Nr=0.5,

T

nd local Sherwood numbers are presented in non-dimensionalorm as

Re1/2cf = f ′′ (0) ,

Nu/Re1/2 = −(

1 + 4

3Rd

) (1

1 − st

)θ′ (0)

Sh/Re1/2 = −(

1

1 − sc

)φ′ (0)

(25)

. Results and discussion

In this paper, the transformed equations momentum, energynd concentration (19)–(21) subjected to the boundary condi-ion (22) are solved numerically by an efficient implicit finiteifference scheme known as Keller box method which is dis-ussed comprehensively in (Cebeci & Bradshaw, 2012). Theomputational analysis is performed with sundry parameters vizrandtl number Pr, magnetic field M, electric field E1, unsteadi-ess parameter δ, mixed convection parameter λ, buoyancy ratioarameter Nr, suction/injection s, thermal radiation parameterd, Brownian motion Nb, thermophoresis Nt, Eckert numberc, heat generation/absorption ε, thermal stratification st, Lewisumber Le, chemical reaction γ , and concentration stratificationc. We have selected a step size of Δη = 0.01 to satisfy the con-ergence condition of 10−5. In Table 1, we see that the presentomputational scheme from the numerical values reveals is in anxcellent agreement with the results reported by previous studiesuch as (Hayat, Imtiaz, & Alsaedi, 2016a; Ibrahim & Shankar,013; Khan & Azam, 2017), under limiting conditions. Table 2isplays the numerical values of local Nusselt and Sherwoodumbers for different values of s, Le, Nt, Nb, st, sc, Ec, E1, M,d, δ, ε, γ , Nr, and λ when Pr = 6.2 are conducted. From this

able, it is examined that the values of local Nusselt number arencreased by increasing the values of s, sc, E1, δ, and λ but theseecrease by increasing Le, Nt, Nb, st, Ec, M, Rd, ε, γ , and Nr.

tir

Fig. 4. Influence of δ on the velocity profile f′(η).

he values of local Sherwood numbers increase by increasing

he values of s, Le, Nt, Nb, Ec, E1, M, Rd, δ, ε, γ , and λ whereasts decreases with increases in the values of st, sc, and Nr. Theesults obtained are shown through Figures 2–20 for the velocity,

470 Y.S. Daniel et al. / Journal of Applied Research and Technology 15 (2017) 464–476

Table 2Numerical results of local Nusselt number Nux/Re1/2 and local Sherwood number Shx/Re1/2 for various values of s, Le, Nt, Nb St, Sc, Ec, E1, M, Rd, δ, λ, Nr, ε and γ

when Pr = 6.2.

s Le Nt Nb St Sc Ec E1 M Rd δ ε γ Nr λ Nu/Re1/2 Sh/Re1/2

0.1 5.0 0.1 0.2 0.1 0.2 0.2 0.1 0.1 0.2 0.1 0.1 1.0 0.1 0.5 1.609856 3.2777810.3 1.914229 3.6847840.5 2.260243 4.1047240.1 2.0 1.836066 1.696388

4.0 1.662801 2.8281667.0 1.533674 4.0573265.0 0.2 1.424676 3.212881

0.3 1.276608 3.2140530.5 1.061147 3.3052510.1 0.1 1.983094 2.809780

0.3 1.308123 3.4127650.5 0.875038 3.4917200.2 0.2 1.554430 3.230745

0.4 1.465419 3.1540990.5 1.449703 3.1383460.1 0.1 1.516611 3.451757

0.4 1.640436 2.8406900.5 1.683675 2.6365160.2 0.1 1.616170 3.231450

0.3 1.498383 3.2648950.5 1.383440 3.2973140.2 0.2 1.567688 3.250943

0.5 1.589580 3.2589481.0 1.595010 3.2737100.1 0.2 1.533827 3.250854

0.5 1.464559 3.2581831.0 1.352284 3.2709990.1 0.3 1.529815 3.250125

0.5 1.475525 3.2558830.7 1.423302 3.2629691.0 0.2 1.400481 3.383713

0.4 1.482374 3.5888700.7 1.586703 3.8658580.1 0.2 1.302533 3.291501

0.3 1.255259 3.3091220.5 1.176202 3.3452120.1 1.5 1.343690 3.524437

2.0 1.336748 3.7611082.5 1.330360 3.9845701.0 0.3 1.334108 3.267329

0.5 1.315966 3.2603831.0 1.263585 3.2424190.1 1.0 1.430948 3.315061

tp

vurinffwei

toilpatatp

emperature, and nanoparticle concentration profiles for sundryarameters respectively.

Figures 2–7 display the velocity profiles f′(η) for thearious embedded parameters viz electric field, magnetic field,nsteadiness parameter, mixed convection parameter, buoyancyatio parameter, and suction/injection parameter. Figure 2ndicates that higher values of electric parameter increase theanofluid velocity. An electric parameter acts as acceleratingorce. The higher electric parameter has the stronger Lorentzorce and the lower electric parameter associated with the

eaker Lorentz force. The stronger Lorentz force creates more

nhancement and resolves sticky effect due to the nanoparticlesn the fluid that signified an increase in the convective heat

lla

1.5 1.483361 3.3568742.0 1.515111 3.400111

ransfer and momentum boundary layer thickness. The effectf magnetic field parameter M on the nanofluid velocity profiles exhibited in Figure 3. The velocity and momentum boundaryayer thickness decreased for higher values of the magneticarameter. Magnetic field results in a resistive type force knowns Lorentz flow where it opposite the flow. This leads to resists inhe fluid flow behavior that amount to a reduction in the velocitynd thinner boundary layer thickness. Figure 4 demonstrateshe effect of unsteadiness parameter δ on the nanofluid velocityrofile. The behavior is due to acceleration case (δ > 0), a

ower rate of fluid flow and thinner momentum boundaryayer thickness. The velocity profile reduces for the highercceleration. In Figure 5, it is noticed that the momentum

Y.S. Daniel et al. / Journal of Applied Research and Technology 15 (2017) 464–476 471

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1s= -0.5

s= -0.2

s= 0.0

s= 0.2

s= 0.5

s<0

s>0

η

ƒ′(η)

Pr=6.2, Rd=Ec=M=st=δ=ε= 0.1, E1=0.01,

sc=0.2, Le= 5.0, Nb =Nt= λ=0.5, Nr=γ=1.0

Fig. 5. Influence of s on the velocity profile f′(η).

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

λ= -1.0

λ= -0.5

λ= 0.0

λ= 0.5

λ= 1.0

λ>0

λ<0

ƒ′(η)

Pr=6.2, Rd=Ec=M=st=δ=ε= 0.1,E1=0.01,

sc=0.2, Le=5.0, Nb=Nt=s=0.5, Nr=γ=1.0

Fig. 6. Influence of λ on the velocity profile f′(η).

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

Nr=0.1

Nr=0.4

Nr=0.7

Nr=1.0

ƒ′(η)

Pr=6.2, Rd=Ec=M=st=δ=ε= 0.1, E1=0.01,

sc=0.2, Le= 5.0, Nb =Nt=s =0.5, λ=γ=1.0

bibn

0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 1. 6 1. 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

Rd=0.0

Rd=0.2

Rd=0.4

Rd=0.6

θ(η)

Pr=6.2, Ec=Nb=sc=0.2,s =λ= Nr=0.5 ,

M=st=Nt=γ=δ=ε=0.1, Le=5.0,E1=1.0,

Fig. 8. Influence of Rd on the temperature profile θ(η).

0 0. 5 1 1. 5 2 2. 5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ(η)

Ec=0.1

Ec=0.3

Ec=0.5

Ec=0.7

η

Pr=6.2, Rd=Nb=sc=0.2,s =λ= Nr=0.5 ,

M=st=Nt=γ=δ=ε=0.1, Le=5.0,E1=1.0,

Fig. 9. Influence of Ec on the temperature profile θ(η).

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

δ=0.1

δ=0.3

δ=0.5

δ=0.7

θ(η)

Pr=6.2, Rd =Nb=sc=0.2,s= λ=Ec= Nr=0.5,

M=st=Nt= γ=ε=0.1, Le= 5.0, E1=1.0,

b

Fig. 7. Influence of Nr on the velocity profile f′(η).

oundary layer thickness for the nanofluid decreases with

ncreasing the values of the suction parameter (s > 0). This cane attributed to a decrease in the fluid velocity due to increasinganofluid suction at the stretching surface. While reverse

oph

Fig. 10. Influence of δ on the temperature profile θ(η).

ehavior occurred for injection parameter (s < 0). The effect

f mixed convection parameter λ on the nanofluid velocity isresented in Figure 6. Corresponds to heated surface (λ > 0),igher values leads to enhancement in the fluid velocity and

472 Y.S. Daniel et al. / Journal of Applied Research and Technology 15 (2017) 464–476

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ(η)

ε= -1.0

ε= -0.5

ε= 0.0

ε= 0.5

ε= 1.0

ε <0

ε >0

Pr=6.2, Rd=Ec=Nb=sc=0.2,s= λ= Nr=0.5,

M=st=Nt= δ=γ=0.1, Le= 5.0, E1=1.0,

Fig. 11. Influence of ε on the temperature profile θ(η).

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ(η)

M=0.0

M=0.1

M=0.2

M=0.3

Pr=6.2, Rd=Ec=s=sc=λ= Nr=0.5, E=1.0,

Nb=0.2,=Nt= st=γ=δ=ε=0.1, Le=5.0,

Fig. 12. Influence of M on the temperature profile θ(η).

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

E1=0.0

E1=0.2

E1=0.4

E1=0.6

θ(η)

Pr=6.2, Rd=Ec=s=sc=λ= Nr=0.5, M=1.0,

Nb=0.2,=Nt= st=γ=δ=ε=0.1, Le=5.0,

ta(d

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ(η)

st=0.0

st=0.1

st=0.2

st=0.3

Pr=6.2, Rd=Ec=s=sc=λ= Nr=0.5, E1=1.0,

Nb=0.2, Nt=M=γ=δ=ε=0.1, Le=5.0,

Fig. 14. Influence of st on the temperature profile θ(η).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ(η)

Nb=0.1

Nb=0.2

Nb=0.3

Nb=0.5

Pr=6.2, δ=ε=Rd=Ec= E1=M=Nt= st=0.1,

Le=1.5,=s=γ=1.0, sc=0.2, λ= Nr=0.5,

Fig. 15. Influence of Nb on the concentration profile φ(η).

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Nt=0.1

Nt=0.2

Nt=0.3

Nt=0.5

φ(η)

Pr=6.2, δ=ε=Rd=Ec=E1=M=st=0.1,

Le=1.5,=s=γ=1.0=sc=Nb=0.2, λ= , Nr=0.5,

tt

Fig. 13. Influence of E1 on the temperature profile θ(η).

hicker momentum boundary layer thickness. This leads to

ssist the nanofluid flow in the upward direction, whereasλ < 0) indicate cold surface opposing the nanofluid flow in theownward direction with thinner momentum boundary layer

Tvv

η

Fig. 16. Influence of Nt on the concentration profile φ(η).

hickness. As (λ = 0) signifies absence of the mixed convection,herefore force convection behavior due to nanofluid velocity.

he effect of buoyancy ratio parameter Nr on the nanofluidelocity is publicized in Figure 7. It is noticed that the nanofluidelocity reduces with a high amount of buoyancy force. The

Y.S. Daniel et al. / Journal of Applied Research and Technology 15 (2017) 464–476 473

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ(η)

Le=1.5

Le=2.0

Le=3.0

Le=5.0

Pr=6.2, δ=ε=Rd=Ec=E1=M=Nt=st=0.1,

s=γ=1.0, sc=Nb=0.2, λ=Nr=0.5,

Fig. 17. Influence of Le on the concentration profile φ(η).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

η

δ=0.1

δ=0.3

δ=0.5

δ=0.7

φ(η)

Pr=6.2, ε=Rd=Ec=E1=M=Nt=st=0.1,

s=γ=1.0, Le=1.5, sc=Nb=0.2, λ=Nr=0.5,

Fig. 18. Influence of δ on the concentration profile φ(η).

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

γ= -0.5

γ= -0.3

γ= 0.0

γ= 0.3

γ= 0.5

γ<0

γ>0

φ(η)

Pr=6.2, δ =ε=Rd=Ec= E1=M=Nt= st=0.1, s=1.0, Le=1.5, Nb=sc=0.2, λ=Nr=0.5,

bttm

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ(η)

sc=0.0

sc=0.1

sc=0.2

sc=0.3

Pr=6.2, δ =ε=Rd=Ec=E1=M=Nt=st=0.1, s=γ=1.0, Le=1.5, Nb=0.2, λ=Nr=0.5,

ReetctwmoHhltttivevohflhflsenhepfrpbt

Fig. 19. Influence of γ on the concentration profile φ(η).

uoyancy force behaves as unfavorable pressure gradient,

herefore, stronger buoyancy force reduced the fluid flow dueo stretching surface in the downwind direction and thinner

omentum boundary layer thickness.

apv

Fig. 20. Influence of Sc on the concentration profile φ(η).

In Figures 8–14 display the effects of thermal radiationd, Eckert number Ec, unsteadiness parameter δ, heat gen-ration/absorption parameter ε, magnetic field parameter M,lectric field parameter E1, and thermal stratification parame-er st on the temperature profiles θ(η). Figure 8 elucidates thehanges that are noticed in nanofluid temperature profiles dueo increase in the values of thermal radiation parameter Rd. It isorth noticing that the nanofluid temperature increases as ther-al radiation increase due to the fact that the conduction impact

f the nanofluid improves in the presence of thermal radiation.ence higher values of radiation parameter mean higher surfaceeat flux and so, enhance the temperature within the boundaryayer region. It is also demonstrated that thermal boundary layerhickness increases with increasing the values of thermal radia-ion. In Figure 9, illuminates that the nanofluid temperature andhermal boundary layer thickness are enhanced due to kineticncrease rises over the energy to enthalpy with an increase in thealues of Eckert number. The influence of unsteadiness param-ter on the temperature profile in Figure 10, is similar to of theelocity in Figure 4. The effect of heat generation/absorptionn the temperature is depicted in Figure 11. It is noted thateat source (ε > 0) gives an increase in the temperature of theuid and the thermal boundary layer thickness. While as theeat sink (ε < 0) provides a decrease in the temperature of theuid and thinner thermal boundary layer thickness. For ε = 0ignify the absence of heat generation/absorption. Figure 12stablishes the impact of magnetic field parameter M on theanofluid temperature profiles. The transverse magnetic fieldas increased the thermal boundary layer thickness. Thus, annhancement in thermal boundary layer thickness and fluid tem-erature. The magnetic field performances as a strong Lorentzorce that boost on the nanofluid temperature in the boundaryegion. The effect of the electric field parameter E1 on the tem-erature profile θ(η) is revealed in Figure 13. The electric fieldehaves as accelerating force that enhances the fluid tempera-ure and thermal boundary layer thickness. A higher value of

n electric field is associated with thicker higher amount tem-erature distribution within the boundary layer region of theicinity of the nanofluid. Figure 14 is a sketch for nanofluid

4 esear

tpvttbdqrt

scpFmirctnflbptomittmeccntLfiptiihWrccct(icscTcc

4

ba(Jneldstd

1

2

3

4

5

6

7

8

C

A

Efisr

R

A

Bég, O. A., Khan, M. S., Karim, I., Alam, M. M., & Ferdows, M. (2014).Explicit numerical study of unsteady hydromagnetic mixed convective

74 Y.S. Daniel et al. / Journal of Applied R

emperature profile with various values of thermal stratificationarameter st. The nanofluid temperature is reduced for higheralues of thermal stratification parameter. It is also noticed thathe case of prescribed surface temperature is achieved whenhermal stratification is absence (st = 0). Physically, the variationetween the surface temperature and the ambient temperature isiminished for a high amount of thermal stratification. Conse-uently, leads lower amount of fluid temperature in the boundaryegion of the nanofluid and thinner thermal boundary layerhickness.

The effects of Brownian motion parameter Nb, thermophore-is parameter Nt, Lewis number Le, unsteadiness parameter δ,hemical reaction parameter γ , and concentration stratificationarameter sc on the concentration profiles φ(η) are shown inigures 15–20. In Figure 15 portrays the influence of Brownianotion parameter on the nanoparticle concentration profiles. It

s noticed that the nanoparticle concentration of the nanofluideduced with increase in Brownian motion of the nanoparti-les. It is worth interesting to note that Brownian motion ofhe nanoparticles at the molecular and nanosized rates is a keyanoscale processes governing their thermal behavior with theuid molecules. Consequently, leads to thinner solutal thermaloundary layer thickness due to Brownian motion. Figure 16roves the effect of thermophoresis parameter on the concentra-ion profiles φ(η). The impact of the thermophoresis parametern the nanoparticles concentration distribution is monotoniceaning the nanoparticles concentration is sensitive to increases

n thermophoresis parameter. The magnitude of the concen-ration gradient at the surface decrease for higher values ofhermophoresis parameter. Hence, due to the fact that as a ther-ophoresis depreciated mass transfer of nanofluids. In Figure 17

xamines the influence of Lewis number on the nanoparticlesoncentration. It is worth noticeable that the nanoparticles con-entration is reduced significantly for higher values of Lewisumber. This reduction in nanoparticles concentration is due tohe change in Brownian motion coefficient. Higher values ofewis number associated to feebler Brownian diffusion coef-cient. Figure 18 demonstrates the behavior of unsteadinessarameter on the concentration profiles, which is similar tohat of Figure 10. In Figure 19 elucidates the effect of chem-cal reaction on the nanoparticles concentration profile. It is ofnterest to note that nanoparticles concentration enhances forigher values of destructive chemical reaction parameter (γ < 0).hereas reverse behavior occurred for generative chemical

eaction parameter (γ > 0). Perhaps, higher values of generativehemical reaction associated with the high rate of generativehemical reaction which generates the fluid species more effi-iently and thus nanoparticles concentration rises. But oppositerend is noticed for destructive chemical reaction parameterγ < 0). For (γ = 0) implies absence of chemical reaction. Thenfluence of concentration stratification on the nanoparticle con-entration is described in Figure 20. An increase in concentrationtratification parameter leads to a decrease in nanoparticles con-entration and solutal concentration boundary layer thickness.his is as result of the fact that fluid near the plate can have a con-

entration lower than the ambient medium. The nanoparticlesoncentration is maximum at (sc = 0).

ch and Technology 15 (2017) 464–476

. Final remarks

In this work, unsteady mixed convection with the com-ined effects of thermal radiation, heat generation/absorptionnd chemical reaction on the electrical magnetohydrodynamicMHD) flow of nanofluid in the presence viscous dissipation andoule heating over a permeable stretching sheet is investigatedumerically. Buongiorno model was used to incorporate theffects of Brownian motion and thermophoresis. The boundaryayer governing equations are converted to nonlinear ordinaryifferential equations using similarity transformations and thenolved numerically with implicit finite difference scheme. Onhe basis of findings with sundry parameters in this work, werew the following main conclusions:

. Velocity and temperature increase with an increase in anelectric parameter.

. Thermal radiation and viscous dissipation increase the tem-perature profiles.

. Heat source increases the temperature while as reverseoccurred with a heat sink.

. Destructive chemical reaction enhances the nanoparticlesconcentration while opposite behave happened in the caseof a generative chemical reaction.

. Reversed behavior occurred with Brownian motion and ther-mophoresis parameter with nanoparticles concentration.

. Thermal stratification reduced the temperature profile andconcentration stratification decreased the nanoparticles con-centration profile.

. Buoyancy ratio parameter decreases both Nusselt and Sher-wood numbers whereas mixed convection parameter increaseboth for higher values.

. The skin friction, local Nusselt and Sherwood numberare sensitive to increase in suction whereas, higher valuesunsteadiness parameter reduces the velocity, temperature,and nanoparticles concentration profile.

onflict of interest

The authors have no conflicts of interest to declare.

cknowledgment

The authors would like to acknowledge Ministry of Higherducation and Research Management Centre, UTM for thenancial support through GUP with vote number 11H90, Flag-hip groups with vote numbers 03G50 and 03G53 for thisesearch.

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