Scholarly article on topic 'Double diffusive magnetohydrodynamic heat and mass transfer of nanofluids over a nonlinear stretching/shrinking sheet with viscous-Ohmic dissipation and thermal radiation'

Double diffusive magnetohydrodynamic heat and mass transfer of nanofluids over a nonlinear stretching/shrinking sheet with viscous-Ohmic dissipation and thermal radiation Academic research paper on "Nano-technology"

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Abstract of research paper on Nano-technology, author of scientific article — Dulal Pal, Gopinath Mandal

Abstract The study of magnetohydrodynamic (MHD) convective heat and mass transfer near a stagnation-point flow over stretching/shrinking sheet of nanofluids is presented in this paper by considering thermal radiation, Ohmic heating, viscous dissipation and heat source/sink parameter effects. Non-similarity method is adopted for the governing basic equations before they are solved numerically using Runge-Kutta-Fehlberg method using shooting technique. The numerical results are validated by comparing the present results with previously published results. The focus of this paper is to study the effects of some selected governing parameters such as Richardson number, radiation parameter, Schimdt number, Eckert number and magnetic parameter on velocity, temperature and concentration profiles as well as on skin-friction coefficient, local Nusselt number and Sherwood number.

Academic research paper on topic "Double diffusive magnetohydrodynamic heat and mass transfer of nanofluids over a nonlinear stretching/shrinking sheet with viscous-Ohmic dissipation and thermal radiation"

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ORIGINAL ARTICLE

Double diffusive magnetohydrodynamic heat Q2 and mass transfer of nanofluids over a nonlinear stretching/shrinking sheet with viscous-Ohmic dissipation and thermal radiation

Q1 Dulal Pala,n, Gopinath Mandalb

^Department of Mathematics, Institute of Science, Visva-Bharati University, Santiniketan, West Bengal 731235, India Siksha Satra, Visva-Bharati, Sriniketan, West Bengal 731236, India

Received 18 May 2015; accepted 30 September 2015

KEYWORDS

Nanofluids;

Magnetohydrodynam-

Heat and mass transfer; Thermal radiation; Convection; Ohmic dissipation

Q3 Abstract The study of magnetohydrodynamic (MHD) convective heat and mass transfer near a stagnation-point flow over stretching/shrinking sheet of nanofluids is presented in this paper by considering thermal radiation, Ohmic heating, viscous dissipation and sink parameter/sink effects. Non-similarity method is adopted for the governing basic equation before they are solved numerically using Runge-Kutta-Fehlberg method using shooting technique. The numerical results are validated by comparing the present results with previously published results. The focus of this paper is to study the effects of some selected governing parameters such as Richardson number, radiation parameter, Schimdt number, Eckert number and magnetic parameter on velocity, temperature and concentration profiles as well as on skin-friction coefficient, local Nusselt number and Sherwood number.

© 2017 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V.

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

1. Introduction

Mixed convective heat and mass transfer phenomena arise in industrial and technological applications in the presence of magnetic field. Thus the study of mixed convection boundary layer flow of an electrically conducting nanofluid has been

"Corresponding author. Tel/fax: +91 3463 261029. E-mail addresses: dulalp123@rediffmail.com (Dulal Pal), gopi_math1985@rediffmail.com (Gopinath Mandal).

Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.

http://dx.doi.org/10.1016/jjppr.2017.01.003

2212-540X © 2017 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

60 61 62

Nomenclature Greek letters

C concentration of the fluid (unit: kg/m3) Z buoyancy ratio

C* dimensionless concentration of the fluid À heat generation/absorption parameter

Cf skin friction coefficient Unf effective dynamic viscosity of the nanofluid

CP specific heat at constant pressure (unit: J/(kg • K)) Vf dynamic viscosity of the fluid (unit: N • s/m2)

C free stream concentration (unit: kg/m3) vf kinematic viscosity of the fluid (unit: Pa • s)

C concentration at the wall (unit: kg/m3) Pnf effective density of the nanofluid (unit: kg/m3)

Bo strength of magnetic field (unit: T) G electrical conductivity of the fluid (unit: S/m)

Dm specific diffusivity (unit: J/(kg • K)) * G Stefan-Boltzmann constant (unit: W • m~ 2 • K_4)

Ec Eckert number Kf thermal conductivity of the fluid (unit: W/(m • K))

Gr local Grashof number e dimensionless temperature of the fluid

K* Rosseland mean spectral absorption coefficient (unit: V stream function

m-1) Knf effective thermal conductivity of the nanofluid

M power-law stretching/shrinking parameter (unit: J/ anf effective thermal diffusivity of the nanofluid

(mol • K)) af fluid thermal diffusivity

mw wall mass flux (unit: kg • s_ 1 • m_ 2) Pr coefficient of thermal expansion

Nr thermal radiation parameter (unit: W/m2) Pc coefficient of thermal expansion of concentration

Nux local Nusselt number Prnf thermal expansion of nanofluid

Pr Prandtl number BCnf concentration expansion of nanofluid

qr thermal radiative heat flux (unit: J/m3) Pf thermal expansion coefficient of the fluid

qw wall heat flux (unit: W/m2) Ps thermal expansion coefficient of the nanoparticle

Qo dimensional heat generation/absorption coefficient (1 ( 2 solid volume fraction of the nanoparticles

(unit: W/(m2 • K)) n similarity variable

Rex local Reynolds number s magnetic parameter

Ri Richardson number G electrical conductivity of fluid

S suction/injection parameter Tw wall skin friction

Sc Schimdt number

T temperature of the fluid (unit: K) Subscripts

T ± 1 free stream temperature (unit: K)

T w temperature at the wall (unit: K) nf nanofluid

u velocity component in x-direction (unit: m/s) J f liquid

uw stretching/shrinking sheet velocity (unit: m/s) j s solid

U free stream velocity of the nanofluid (unit: m/s)

v velocity component in y-direction (unit: m/s)

x y direction along and perpendicular to the plate, respec-

tively (unit: m)

considered in this paper. Nanofluid is a suspension of solid nanoparticles or fibers of diameter 1-100 nm in basic fluids such as water, engine oil, ethylene glycol etc. Nanoparticles which are present in base fluids made from various materials (Choi et al. [1]). Recent research on nanofluid has revealed that nanoparticles (diameter less than 50 nm) may change characteristics of the fluid since thermal conductivity of nanoparticles particles was higher than convectional fluids such as water, ethylene glycol, and engine oil which are widely used as heat transfer fluids in thermal system. Nanofluids contains solid nanoparticles dispersion in a base fluid (such as water, oil, and ethylene glycol). The common nanoparticles those are in use are aluminum, copper, iron and titanium or their oxides. Experimental studies have shown that the thermal conductivity of the base liquid can be enhanced by 5%-15% with the small volumetric fraction of nanoparticles less than 5%. The enhanced thermal conductivity of nanofluid contributes to a remarkable improvement in the convective heat transfer coefficient. This feature of nanofluids has attracted researchers to use it in application such as advanced

nuclear system since convective heat transfer mechanisms is a kind of heat exchanger.

Chio et al. [2] found that these nanofluids have better conductivity and convective heat transfer coefficient compared with the base fluid. Due to better performance of heat exchange, great potential and features, nanofluids can be used in several industrial applications such as in chemical production, transportation, car cooling systems, cooling of heat sinks, cooling of electronic chips, power generation in power plant and in nuclear system to obtain high rates of heat extraction from reactors. Many researchers, Das et al. [3], and Kakac and Pramuanjaroenkij [4] have made a comprehensive literature review in their books and review papers by discussing the heat transfer characteristics in nanofluids besides identifying future research in convective heat transfer of nanofluids. Bahiraei and Hangi [5] presented a review of flow and heat transfer characteristics of magnetic nanofluids.

The study of magnetohydrodynamics (MHD) boundary layer flow of a nanofluid over a stretching surface has become the basis of several industrial, scientific and engineering applications.

Uddin [6] developed a model for bio-nano-materials processing for the hydromagnetic transport phenomena from a stretching or shrinking nonlinear nanomaterial sheet with Navier slip and convective heating. Malvandi and Ganji [7] studied magnetohy-drodynamic mixed convective flow of Al2O3 water nanofluid inside a vertical micro-tube. Hamad et al. [8] studied the magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate. They found that for a fixed value of nanoparticle volume fraction, increasing in magnetic parameter leads to decrease in the velocity profile. Hamad et al. [9] also studied the effect of magnetic field on free convection flow of a nanofluid past a vertical semi-infinite flat plate. Heidary [10] numerical studied the effects of magnetic field on nano-fluid forced convection in a channel. Sheikholeslami et al. [11] examined the hydromagnetic CuO-water nanofluid flow and convective heat transfer considering Lorentz forces effects. The effect of thermal radiation on magnetic convection on boundary layer flow on nanofluid was studied by Mat et al. [12]. The convective heat transfer in a nanofluid past a vertical plate has been studied by Kuznestov and Nield [13]. Mahdy [14] analyzed unsteady mixed convection boundary layer flow and heat transfer of nanofluids due to stretching sheet.

In addition combined magnetic field and viscous dissipation on boundary layer flow on temperature distribution are investigated by several researchers. The effect of viscous and Joules dissipation on MHD flow, heat and mass transfer past a stretching surface embedded in porous medium was studied by Anjali Devi and Ganga [15]. Ibrahim and Shankar [16] analyzed the effects magnetic field, slip boundary condition and thermal radiation on boundary layer flow and heat transfer of a nanofluid over a permeable stretching sheet.

In recent times, heat and mass transfer problem with the effects of chemical reaction have attracted the attention of many researchers. Abdul-Kahar et al. [17] examined numerically the steady boundary layer flow of a nanofluid past a porous vertical stretching surface in the presence of chemical reaction and heat radiation using scaling group transformation. Kameswaran et al. [18] focused on heat and mass transfer in nanofluid flow over stretching and shrinking sheets with chemical reaction in the presence of magnetic field. An analysis has been carried out to study the steady two-dimensional stagnation-point flow of a nanofluid over a stretching/shrinking sheet by Bachok et al. [19]. Zaimi et al. [20] investigated the steady two-dimensional flow and heat transfer over a stretching/ shrinking sheet in a nanofluid using two-phase Buongior-no'snanofluid model. Recently, Mansur et al. [21] studied magnetohydrodynamic stagnation point flow of a nanofluid over a permeable stretching/shrinking sheet with suction. They showed that the solutions for a shrinking sheet are non-unique those differ from the results for stretching sheet. Rashad et al. [22] studied the free convective boundary layer flow of a non-Newtonian fluid over a permeable vertical cone embedded in a porous medium saturated with a nanofluid. Sheikholeslami et al. [23] investigated the effects of magnetohydrodynamic on Cu-water nanofluid flow and heat transfer by means of CVFEM.

Thus the aim of the present study is to analyze mixed convection heat and mass transfer at a stagnation point flow over stretching and shrinking sheets of different nanofluids (i.e., copper (Cu), alumina (Al2O3), titanium dioxide (TiO2)) in the presence of thermal radiation, Ohmic heating, viscous dissipation, sink parameter/sink and transverse magnetic field with base fluid water (Pr—6.8), using non-similarity method and Runge-Kutta-Fehlberg method using shooting technique.

2. Formulation of the problem

We consider two-dimensional mixed convective flow of viscous incompressible electrically conducting nanofluids in the vicinity of a stagnation-point flow over a stretching/ shrinking sheet in the presence of transverse magnetic field, thermal radiation, sink parameter/sink and chemical reaction. The sheet has constant linear velocity uw(x) — cxm (for stretching sheet) and u^(x) — -cxT (for shrinking sheet) and velocity of the free stream flow is U(x) — axm, where a, c and m are constants, x is the coordinate measured along the stretching/shrinking surface. The flow takes place at y > 0, where y is the coordinate measured normal to the stretching/ shrinking surface. It is assumed that the temperature and concentration at the stretching/shrinking surface takes the constant values Tw and Cw, while those of the ambient nanofluid takes the constants value Tx and Cx, respectively (Figure 1).

It is also assumed that the electric field owing to polarization of charges and Hall effects are neglected. The base fluid and the nanoparticles are expected to be in thermal equilibrium and no slip occurs between them. Under these assumptions, the boundary layer equations of motion, energy and mass-diffusion under the influence of uniform transverse magnetic field and Ohmic dissipation the presence of sink parameter or sink, viscous dissipation and thermal radiation are as follows:

Y + dx — 0, (1)

du du u— + v— = U (x) ox dy

dU (x) Vnfd-

+ Pnfdx2

(u - U (x))-

Pnf Pnf

+ (C- cm)>

g (T - T1)

Table 1 Thermo physical properties of fluid and nanoparticles (Oztop and Abu-nada [27]).

Physical properties Fluid phase (water) Cu Al2O3 TiO2

Cp /(J/(kg ■ K)) 4179 385 765 686.2

p/(kg/m3) 997.1 8933 3970 4250

K/(W/(m ■ K)) 0.613 400 40 8.9538

^<10-5/K-1 21 1.67 0.85 0.9

Table 2 Comparison of results for F''(0,0) when m = 1 and p=Ri = £ = Z=Nr=X=Ec=Sc = 0 for stretching sheet when Pr=

m Cortell [28] Hamad and Ferdows [29] Present results

0.2 0.7668 0.7659 0.7668

0.5 0.8895 0.8897 0.8895

1.0 1.0000 1.0043 1.0000

3.0 1.1486 1.1481 1.1486

10.0 1.2349 1.2342 1.2348

dT dT d2T Qo frr 1 dqr

dx dy n dy (PCp)nf (PCp)nf dy

du\ 2 gB02 ,

+ (u - U(x))2 dy) Pnf

dC dC d2 C

u T"+ v X" = Dm "ry (4)

dx dy dy2

subject to the boundary conditions for stretching/shrinking sheets:

u — uw (x) — 7 cxm, v — vw, T — Tw — T1 + bx2m, C — Cw — Ci+ dx2m, at y — 0; (5)

u—U(x) — axn, v — 0, T — T1, C—Cœ, as y —1.

Here u and v are the velocity components along the x- and y-directions, respectively. U(x) stands for the stagnation-point free stream velocity. T is the temperature and C is the concentration of the nanofluid, g is the electrical conductivity of fluid, B0 is the strength of the magnetic field, g is the acceleration due to gravity, Q0 is the heat generation or absorption coefficient, Dm is the species diffusivity, a, b, c, d are positive constants and vw is the wall mass flux with vw < 0 for suctions and vw>0 for injection, respectively. Further, p is the fluid density, pnf is the coefficient of viscosity of the nanofluid, pc is the concentration expansion coefficient of nanofluid, pT is the thermal expansion coefficient of nanofluid Knf is the thermal conductivity of the nanofluid.

Also, anf the thermal diffusivity of the nanofluid, pnf is the effective density of the nanofluid, pTnf is the thermal expansion of nanofluid, ( pc)nf is the concentration expansion of nanofluid, (pCp)nf is the heat capacitance of the nanofluid, which are defined as follows:

anf —

(pCp)nf'

(1 - P)25!

pnf — (1 - P) pf + pps

{pP^nf — (1 - p) (pP^f + p(ppT

■ P (pPC

(pPc)nf — (1 -P) {pPc)f

(pCp)nf — (1 - P)(pCP)f + P(pCP)s

(ks + 2Kf) - 2p (Kf - Ks)

(.Ks + 2Kf) + P (Kf - Ks)

where p is the solid volume fraction of the nanofluid, p is the reference density of the fluid fraction, ps is the reference density of the solid fraction, is the viscosity of the fluid fraction, Kf is the thermal conductivity of the fluid, and Ks is the thermal conductivity of the solid fraction.

Applying Rosseland approximation (Magyari and Pan-tokratoras [24]) to optically thick media and the net radiation heat flux qr [W • m-2] by the expression

qr=~ 3K* grad eh ^

where K* [m-1] is the Rosseland mean spectral absorption

coefficient and eh [W • m_ 2] is the blackbody emission power

which is given in terms of the absolute temperature T by the

Stefan-Boltzmann radiation law eh = a*T4 with the Stefan-

Boltzmann constant a* = 5.6697 x 10"8 W • m"2 • K"4 It is

further assumed that the term T"4 due to radiation within the

flow can be expressed as a linear function of temperature itself.

Hence T4 can be expanded as Taylor series about and can be approximated after neglecting the higher order terms as,

;4T 1 T - 3T

Using Eqs. (8) and (9), we get qr 16g*TI, d2T

3K* dy2

We now look for a similarity solution of the Eqs. (1)-(4) with boundary conditions (5) and (6) in the following form:

gB02X puw (x)'

2vfxuw (x)

e (S; n) — ,

T —T

(m + 1)

C*(S; n) —

f (S; n); n —

(m + 1) uw (x)

C —C

where Vf is the kinematic viscosity of the fluid and the stream function ^ is defined in the usual way as

,— ÈV

dy, v = " dx, which identically satisfies the Eq. (1). Substituting Eqs. (7)-(11) into Eqs. (2)-(4), we get following nonlinear ordinary differential equations:

m + 11 ,,, m+ 1 // / 2 p3

--F H--FF " mF +— Rid

2 P1P2 2 P2

" - £ (F" -) + 13 Ri (0 + ZC*)

P2 c P2

, , ,dF ,,dF

— (1 - m) ^ F--F —

V dS dS

Table 3 Comparison of results for -0''(0,0) when — ■ —Ri — S—Z—Nr—X —Sc — 0, Pr— : 5 for stretching sheet.

Ec — 0 Ec—0.1

m Cortell [28] Hamad and Ferdows [29] Present results Cortell [28] Hamad and Ferdows [29] Present results

0.75 3.1250 3.1250 3.1253 3.0170 3.0156 3.0156

1.50 3.5677 3.5672 3.5679 3.4557 3.4566 3.4566

7.00 4.1854 4.1848 4.1854 4.0657 4.0659 4.0659

10.0 4.2560 4.2560 4.2559 4.1353 4.1354 4.1354

m + 1 ("£ +Nr) // m + 1 , , XS

——--Kf-0 + —— F0 - 2mF 0 + — 0

2— Pr 2 —4

m + 1 Ec //2 Ec ,

+--F + —F -

2 —j — 4 — 4 V c

N / /d0 /dF'

m+11 c,/0+ m±lFC* -2mF C* 2 Sc 2

= (1 - m) d F—- C*0 '-F

■ \ d S dS

with the corresponding boundary condition as obtained from Eqs. (5) and (6) in the form:

F — S, F — +1, 0 — 1; C* — 1 at n — O,

F = -,

0-0, C*-1 as n-1,

The non-dimensional constants appearing in Eqs. (12)-(16) are the thermal Grashof number Gr, buoyancy ratio Z, Richardson number Ri, Prandtl number Pr, radiation parameter Nr, sink parameter/sink parameter X (X > 0 for source and X<0 for sink), Eckert number Ec, Schmidt number Sc, mass flux parameter S (S>0 corresponds to the suction and S<0 corresponds to injection). They are respectively defined as

gßT (Tw - T 1) X3

Pr — VI, af

Nr= -■

—5ßC (Cw - C1) —2ßT (Tw - T 1) '

16a*T 3

b(Cp f

Sc — f, D

fK (pcp)fbo2'

2uw (x)^x-(m -1)=2

(m + X)sfcvj

—5 — 1 - — + —

(pßC )s (pßC )f :

Here prime denotes the differentiation with respect to r). The important physical quantities in this study are skin-friction or shear stress coefficient Cf, local Nusselt Number Nux, and Sherwood number Sh„ which are defined by

_ diA _ Tw

Tw — Mnf{dï)y — o, Cf — PfUWw,

qw = Knf

dyj y — o 3K*

y — 0

Kf (Tw - Ti)

mw — PnfDm\ -d-

dyJy =

Shx — -

y — 0

í-x-l

Using Eq. (11) in Eqs. (19)-(21) skin-friction coefficient, local Nusselt number and Sherwood number can be expressed as (Aurangzaib et al. [25]) Q4

1 fm+ 1\2

Rex2 Cf —

(1 - — r\ 2

Knf fm + 1

Rex 2 Nux —---

F (S, 0), (1 + Nr) 0 (S, 0),

Rex -1=2 Shx —-( m±l\1C*'{S, 0)

where Rex — xuw(x)/vf is the local Reynolds number based on the stretching/shrinking velocity uw(x).

(17) 3. Local non-similarity method

—1 —(1 - —)2'5,

—3 — 1 - — + —

(pßT f

—4 — 1 - — + —

1 i r s

—2 — 1 - — + —-,

(PßT )s

(PCp )s (PCp)f '

We now discuss the local non-similarity method to solve Eqs. (12)—(14). Truncation of this equation up to the second level gives almost accurate results comparable to the solutions by the other methods. To do this, we introduce the following new functions (Pal and Mondal [26]):

Table 4 Values of F"(£,0), 0(£,0), C*'(£,0) when Z=a/c =Ri=Nr=X=Ec=Sc = 0 and Pr=6.8 for stretching sheet.

f Sc m — F"(f,0) 0(f,0) C*'(f,0)

1 11 0.0 -1.4242 - 3.8026 - 1.2159

2 -1.7320 - 3.7179 - 1.1275

2 -1.7320 - 3.7179 -1.7956

2 -1.8333 -3.1186 - 1.4518

0.1 - 1.7404 - 2.4548 - 1.4781

Region of non-zero velocity

Figure 1 Flow configuration.

dF d0 ÔC* G = % ' R = % ' H = "df

Introducing the above functions (23) into Eqs. (12)—(14), we get

m + 11 m m + 1 « /2 m3

--F H--FF - mF + — R¿0

2 (P2 2 P2

- - Ç (F - a) + ^ Ri (0 + fC*)

P2 V ^ P2

= (1 - m) f (f'G - FG),

m + 1 [17 + » m + 1 ' ' M

-0 <+ —— F0 -2mF 0 + — 0

2— Pr 2 —4

m+ 1 Ec ''2 Ec ( ' a\2

+ --F + -dF--)

2 —1 —4 —4 V c/

= (1 - m) f (f'r - 0 G),

m±l1 c*'' + mH1 FC*' -2mF'C* = (1 -m)^F H- C*'G, 2 Sc 2 \ J

Differenting the Eqs. (24)-(26) with respect to £ and neglecting the terms involving the derivative functions G and R with respect to £, we obtain:

m + 11 m m + 1 / » »s , / p*

—--G (FG + GF ) - 2mF'G + — RiR

2 P1 P2 2 P2

Figure 2 Variation of velocity profile with different value of f for Cu-water.

- -1 fa + (f - aJ} + ^ {Ri (R + ZH) } -2 L V cJ i -2

= (1 -m){ (f'G' -F''G) + f(<G2-G'G) }, (27)

m + 1 fe + Nr) '' m + i

-—R (FR + G0 )

2—4 Pr 2

- 2m (F'R + G 0) + - (fR + 0) + m±lJ^ 2F G'

-4 2 -1 -4

Ec f / ' a\2 / ' '

+ A (F-d + 2f(F - a)G

= (1 -m) {f (GR-r'g) + (F'R -0'G },

m + 1 1 » m + 1 ( ' .a , '

———H (FH + GC*)-2m(F H + G C*)

2 Sc 2 V /

= (1 -m) {£ (G'H-H'G) +(F'H- C*'G )}, (29)

Differentiating the boundary conditions (15) and (16) with respect to £ and using Eq. (18), we get

G (£, n) = 0; G' (£, n) = 0; R (£, n) = 0; H (£, n) = 0, at n = 0

G' (£; n)-0; R(£; n)-0, H(£, n)-0, as n-i (31)

4. Results and discussions

In this paper, we have analyzed the effects of Ohmic heating, thermal radiation and heat source/sink on mixed convective MHD boundary layer flow near a stagnation-point over an infinite vertical stretching/shrinking plate with suction/ injection effects in nanofluids. The governing partial differential equations were transformed into a set of nonlinear ordinary differential equation which are then solved numerically by Runge-Kutta-Fehlberg method with shooting techni-

Figure 3 Variation of temperature profile with different value of f for Cu-water.

Figure 5 Variation of velocity profile with different value of f for Cu-water.

ques. Here, we assigned physically realistic numerical values to the governed parameters in order to gain an insight into the velocity, temperature and concentration profiles behavior. The thermo physical properties of fluid and nanoparticles are presented in Table 1 (Oztop and Abu-nada [27]). The present results for F (f,0) and 0 (f,0) are compared with those of Cortell [28], Hamad and Ferdows [29] for different values of m and Ec which are presented in Tables 2 and 3. Comparison of the present results with the previously published works gives an excellent agreement which testify the validity of our computed results. The Computed values of F''(f,0), 0'(f,0) and C '(f,0), are presented in Table 4 for different values of f, Sc, m and y for stretching sheet. It is seen that for adding Cu nanoparticle in the base fluid (water) F''(f,0) and 0'(f,0) Q5 increase but C '(f,0) decreases (Figure 1).

Figures 2-4 depict the effects of power-law parameter m on velocity F'(f,0), temperature 0(f,0) and concentration C (f,0) profiles, respectively for Cu-water nanofluid for stretching and shrinking sheets. Figure 2 shows that velocity profile decreases for stretching sheet as well as

shrinking as value of m increases and match the boundary condition F'(f,0) — a/c (— 0.01) as q — <x>.

Figures 3 and 4 show that the temperature and concentration profiles of Cu-water decrease for stretching sheet as the value of m increases whereas reverse trend is observed for shrinking sheet for both profiles. Figures 5-7 illustrate the effects of magnetic field parameter f on velocity, temperature and concentration profiles respectively for stretching/shrinking sheet for Cu-water nanofluid. Figure 5 shows that by increasing the magnetic parameter f, the velocity profile decreases for stretching sheet whereas for shrinking it increases in 0<n< 1.8 but reverse effect is seen when n> 1.8. Figure 6 shows that by increasing the magnetic field parameter, temperature profile increases for both stretching and shrinking sheets. Figure 7 shows that by increasing the magnetic field parameter the concentration profile also increases for stretching sheet but for shrinking sheet it decreases. It is interesting to see that concentration profile for stretching sheet drastically increases as value of f exceeds 1.0. The reason behind these phenomenon of the

Figure 7 Variation of concentration profile with different values of f for Cu-water.

Figure 10 Variation of concentration profile with different values of Ri for Cu-water.

Figure 8 Variation of velocity profile with different value of Ri for Cu-water.

Figure 11 Variation of velocity profile with different value of Nr for Cu-water.

Figure 9 Variation of temperature profile with different value of Ri for Cu-water.

profiles is that the effects of transverse magnetic field on an electrically conducting fluid gives rise to a resisting force called Lorentz force which has the tendency to slow down the motion of the fluid and increases its temperature and concentration boundary layer.

The effects of buoyancy parameter Rion velocity, temperature and concentration profile of Cu-water nanofluid (having 10% Cu-nanoparticle) for stretching and shrinking sheets are shown in Figures 8-10. From Figure 8, it is observed that an increase in Ri, results in the increase in velocity profile for stretching sheet. For shrinking sheet an interesting phenomenon is observed that the velocity profile increases near the boundary wall is increased whereas it decreases near the free stream when the value of Richardson number Ri is increased. Figures 9 and 10 depict that temperature and concentration profiles both decrease with increasing the value of Richardson number Ri for stretching/shrinking sheet. It is interesting to see from Figures 9 and 10 that no overshoot is formed for shrinking sheet.

Figures 11-13 illustrate the effects of thermal radiation parameter Nr on velocity, temperature and concentration profiles for Cu-water nanofluid for stretching and shrinking sheets. From Figure 11, it is seen that by increasing the thermal radiation parameter Nr, the velocity profile of stretching sheet increases whereas the velocity profile of shrinking sheet decreases in the region 0<n<2.25 and increase when n>2.25. It is also seen that the increase in the velocity for stretching sheet is negligible

Figure 12 Variation of temperature profile with different value of Nr Figure 15 Variation of temperature profile with different value of Ec for Cu-water. for Cu-water.

Figure 13 Variation of concentration profile with different values of Figure 16 Variation of concentration profile with different values of Nr for Cu-water. Ec for Cu-water.

Figure 14 Variation of velocity profile with different value of Ec for Cu-water.

comparing to shrinking sheet whereas a significant effect is observed in temperature profiles by the changing in the values of thermal radiation parameter Nr, as seen in Figure 12. Further, as the values of the thermal radiation Nr increases, though the temperature profile for stretching sheet increases for all values of

and n but for shrinking sheet it decreases for a certain value of n, then it increases. Figure 13 shows that as Nr increases the concentration profiles nearly coincide for stretching sheet whereas there is increase in the concentration profiles for shrinking sheet without forming any overshoot.

Figures 14-16 display the behavior of velocity, temperature and concentration profiles of Cu-water for stretching as well as shrinking sheet for different values of Eckert number Ec. It is interesting to note that the profiles of velocity, temperature and concentration as shown in Figures 14-16 and similar pattern of the profiles as shown in Figures 11-13, except that the effects of Eckert number Ec is just reverse of the effects of thermal radiation Nr for shrinking sheet, i.e. there is increase in the profiles (velocity, temperature, and concentration) by increasing the Eckert number Ec as observed in the Figures 14-16 for shrinking sheet whereas there is not much significant effects of Eckert number noticed for stretching sheet as seen from these figures. Figure 17 shows the effect of Schmidt number Sc on concentration profile of Cu-water for stretching and shrinking sheets. It is observed from this figure that the concentration profile

Figure 17 Variation of concentration profile with different values of Sc for Cu-water.

Figure 20 Variation of the skin-friction coefficient with f for different types of nanofluids.

Figure 18 Variation of temperature profile with different values of X for Cu-water.

Figure 21 Variation of the local Nusselt number with f for different types of nanofluids.

Figure 19 Variation of the local Nusselt number with Nr for different f for Cu-water nanofluid.

Figure 22 Variation of the Sherwood number with f for different types of nanofluids.

decreases with the increase in the value of Sc for both types of sheets. It is also interesting to see that the concentration thickness is higher for shrinking sheet than that of stretching sheet.

Figure 18 depicts the effects of the heat source/sink parameter X on temperature profile of Cu-water nanofluid. Temperature profile increases by increasing X for both stretching/shrinking sheets. Figure 19 depict the effects of the thermal

radiation parameter Nr on the local Nusselt number of Cu-water nanofluid for different values of magnetic field parameter f. From this figure it is found that the local Nusselt number increases along with Nr for stretching sheet but slightly decreases for shrinking sheet. It is also seen from this figure that as f increases the local Nusselt number decreases for stretching as well as for shrinking sheet.

Figures 20-22 show the effects of magnetic parameter f on skin-friction coefficient, local Nusselt number and Sherwood number for stretching and shrinking sheets for three types of nanofluids, namely Cu-water, Al2O3-water and TiO2-water nanofluids. It is found that as the magnetic field f increases, the skin-friction coefficient whereas local Nusselt number and Sherwood number all decreases for the case of stretching sheet whereas reverse trend is observed for shrinking sheet. It is also observed that all the profiles for three different nanofluids nearly coincide for shrinking sheet.

5. Conclusions

In this paper we have investigated the mixed convective MHD heat and mass transfer effects of stagnation point flow of nanofluid (copper Cu, alumina Al2O3, titanium dioxide TiO2) over stretching/shrinking sheets in the presence of thermal radiation, viscous dissipation and heat source/sink with base fluid water (Pr= 6.8). From the present study, the following conclusion is drawn:

(i) Velocity, temperature and concentration profiles of Cu-water decrease for stretching sheet but reverse reverse effect is seen in temperature and concentration profiles for shrinking sheet with increasing power-law parameter m.

(ii) With the increase in magnetic field parameter £, nano-fluid velocity profile decreases and there is increase in the concentration profiles also for stretching sheet whereas reverse trend is found for shrinking sheet for Cu-water nanofluid.

(iii) Increase in buoyancy parameter Ri results in increase in the nanofluid velocity profile and decrease in the temperature and concentration profiles for stretching and shrinking sheets for Cu-water nanofluid.

(iv) The effect of increasing the thermal radiation parameter Nr there is increase in the nanofluid velocity, temperature and concentration profiles for stretching sheet and same trend is seen for shrinking sheet for value of n (i.e. closer to the sheet) for Cu-water nanofluid.

(v) With the increase in Eckert number Ec, nanofluid velocity and temperature profiles increase for stretching and shrinking sheets whereas reverse trends are observed on concentration profiles for both stretching and shrinking sheets for Cu-water nanofluid.

(vi) Concentration profile for Cu-water closer to sheet decreaseswith increasing Schmidt number Sc for both types of stretching/shrinking sheet.

(vii) Temperature profile of Cu-water increases by increasing X for both stretching and shrinking sheets.

(viii) As magnetic parameter £ increases, local Nusselt number decreases for any value of Nr for stretching and shrinking sheets.

(ix) As £ increases skin-friction coefficient, local Nusselt number and Sherwood number decrease for the case of stretching sheet for all three types of nanofluids whereas reverse trend is observed in skin-friction coefficient and local Sherwood number for shrinking sheet.

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