Available online at www.sciencedirect.com

ScienceDirect

Journal of the Nigerian Mathematical Society 35 (2016) 128–141www.elsevier.com/locate/jnnms

Similarity solution of 3D Casson nanofluid flow over a stretchingsheet with convective boundary conditions

C. Sulochana∗, G.P. Ashwinkumar, N. Sandeep

Department of Mathematics, Gulbarga University, Gulbarga-585106, India

Received 27 August 2015; received in revised form 30 December 2015; accepted 7 January 2016Available online 23 February 2016

Abstract

In this study, we analyzed the three-dimensional magnetohydrodynamic Newtonian and non-Newtonian fluid flow. Heat andmass transfer over a stretching surface in the presence of thermophoresis and Brownian motion is investigated. The transformedgoverning equations are solved numerically via Runge–Kutta based shooting technique. We obtained good accuracy of thepresent results by comparing with the exited literature. The influence of dimensionless parameters on velocity, temperature andconcentration profiles along with the friction factor, local Nusselt and Sherwood numbers are discussed with the help of graphsand tables. It is found that an increase in the stretching ratio parameter enhances the heat and mass transfer rate. The heat and masstransfer rate in non-Newtonian fluid is comparatively high while compared with the heat and mass transfer rate in Newtonian fluid.c⃝ 2016 The Authors. Production and Hosting by Elsevier B.V. on behalf of Nigerian Mathematical Society. This is an open access

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

Keywords: Newtonian and Non-Newtonian fluids; Thermophoresis; Brownian motion; Stretching sheet

1. Introduction

The study of boundary layer flow and heat transfer over a stretching surface has attracted the attention of manyresearchers due to its huge industrial and engineering applications. In the field of industry, metallurgical processes suchas drawing of continuous filaments through quiescent fluids, annealing and tinning of copper wires, manufacturingof plastic and rubber sheets, crystal growing, and continuous cooling and fiber spinning, in addition to wide-rangingapplications in many engineering processes, such as extrusion of polymer, wire drawing, manufacturing foods andpaper, in textile and glass fiber production etc. During the manufacturing of these sheets, the melt issues from a slitand it is stretched to achieve the desired thickness. The final product depends on two characteristics first is the rateof cooling in the process and the other is stretching rate. In view of these applications, Sakiadis [1] initiated thestudy of boundary layer flow past a stretching surface. Later, Crane [2] extended the idea for the two-dimensional

Peer review under responsibility of Nigerian Mathematical Society.∗ Corresponding author.

E-mail address: math.sulochana@gmail.com (C. Sulochana).

http://dx.doi.org/10.1016/j.jnnms.2016.01.0010189-8965/ c⃝ 2016 The Authors. Production and Hosting by Elsevier B.V. on behalf of Nigerian Mathematical Society. This is an open accessarticle under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141 129

boundary layer flow of an incompressible viscous fluid flow past a linearly stretching surface. This problem was laterextended by Wang [3] and discussed for three-dimensional case. Afterwards, many investigations were made to gaininsight information regarding the flow over a stretching/shrinking sheet in various situations. Such situations includethe considerations of porous surfaces, MHD, suction/injection, thermal radiation, slip effects, convective boundarycondition, etc. was studied by Samir et al. [4], Vajravelu et al. [5], Mohan Krishna et al. [6], Mabood et al. [7], Rajuet al. [8], Mahapatra et al. [9], Sandeep and Sulochana [10]. Magyari and Keller [11] studied the boundary layerflow due to a non-linearly stretching surface. Ishak et al. [12] presented the concept of unsteady two dimensionalmixed convection boundary layer flow and heat transfer through vertical stretching surface. Wubshet and Bandari [13]analyzed the boundary layer flow and heat transfer on a penetrable stretching surface due to a nanofluid with theinfluence of magnetic field and slip boundary conditions.

The study of non-Newtonian fluids has keen interest among the researchers because of its variety of applications inengineering, chemical and petroleum industries. In the class of non-Newtonian fluids, Casson fluid has unique charac-teristics, which have wide application in food processing, in metallurgy, drilling operation and bio-engineering opera-tions, etc. Hayat et al. [14] elaborated the concept of mixed convection stagnation point flow of Casson fluid under theeffect of convective boundary conditions. Flow through a porous linearly stretching surface of MHD three-dimensionalCasson fluid was analyzed by Nadeem et al. [15]. Hayat et al. [16] studied the variable thermal radiation effect over athree-dimensional stretched flow of Jeffery fluid. Nadeem et al. [17] examined the magneto-hydrodynamic boundarylayer flow of a Casson fluid past an exponentially shrinking surface. Haq et al. [18] investigated the effect of MHDand convective heat transfer on Casson nanofluid flow past a linearly shrinking surface.

Jayachandra Babu et al. [19] discussed the effect of radiation on stagnation-point flow of a micropolar fluid over anonlinearly stretching surface with suction/injection effects. Pramanik [20] presented the concept of Casson fluid flowand heat transfer over a porous exponentially stretching sheet in the presence of thermal radiation. Bhattacharyya [21]studied the Boundary layer stagnation point flow of a Casson fluid and heat transfer over a stretching/shrinking sur-face. The effect of a convective boundary condition on the two-dimensional boundary layer flow of a nanofluid overa linear stretching surface is numerically presented by Makinde and Aziz [22]. Zaimi et al. [23] illustrated the two-dimensional boundary layer flow and heat transfer of a nanofluid over a nonlinearly permeable stretching/shrinkingsurface with Brownian motion and thermophoresis effects. Nadeem and Lee [24] discussed the effects of Brownianmotion parameter and thermophoresis parameter on the steady boundary layer flow of a nanofluid past an exponentialstretching surface. Hayat et al. [25] discussed the effects of convective boundary conditions on MHD flow of nanoflu-ids in porous medium past an exponentially stretching surface. Ferdows et al. [26] examined the effect of Brownianmotion parameter and thermophoresis parameter on MHD mixed convective boundary layer flow of a nanofluid overa porous exponentially stretching sheet. The researchers [27,28] studied the heat and mass transfer characteristics ofnon-Newtonian fluid through stretching sheet. Makinde et al. [29] elaborated the influence of thermophoresis and ra-diation on heat transfer of MHD flow with varying viscosity past a heated plate immersed in a porous medium. Khanet al. [30] and Khan et al. [31] discussed the mixed convective heat and mass transfer of power law nanofluid and thirdgrade nanofluid past on a heated vertical surface and heated stretching surface.

In this study, we analyzed the three-dimensional magnetohydrodynamic Newtonian and non-Newtonian fluidflow, heat and mass transfer over a stretching surface in the presence of thermophoresis and Brownian motion isinvestigated. The transformed governing equations are solved numerically via Runge–Kutta based shooting technique.The influence of dimensionless parameters on velocity, temperature and concentration profiles along with the frictionfactor, local Nusselt and Sherwood numbers are discussed with the help of graphs and tables.

2. Mathematical formulation

Consider a three-dimensional, steady, incompressible flow of a Casson fluid over a stretching sheet. It is consideredthat the sheet is stretched along xy-plane while fluid is placed along the z-axis. It is assumed that induced magneticfield is negligible. Thermophoresis and Brownian motion effects are taken into account. Here, we assumed that thesheet has stretched with the linear velocities u = ax and v = by along the xy-plane, respectively, with constants a andb (see Fig. 1). Moreover, it is considered that a constant magnetic field is applying normal to the fluid flow. The heatand mass transfer process is taken in to account. The rheological equation of state for an isotropic flow of a Casson

130 C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141

Fig. 1. Physical model and coordinate system.

fluid can be expressed as follows:

τi j =

2

µB +

pz√

2π

ei j , π > πc

2

µB +pz

√2π

ei j , π < πc.

(1)

In the above equation π = ei j ei j and ei j denotes (i, j)th component of the deformation rate, π be the product of thecomponent of deformation rate itself, πc be critical value of this product based on the non-Newtonian model, µB bethe plastic dynamic viscosity of the Casson fluid and pz be the yield stress of the fluid.

The equation of continuity, momentum, energy and mass transfer are as follows:

∂u

∂x+

∂v

∂y+

∂w

∂z= 0, (2)

u∂u

∂x+ v

∂u

∂y+ w

∂u

∂z= ν

1 +

1β

∂2u

∂z2 −σ B2

0

ρu −

1 +

1β

ν

k1u, (3)

u∂v

∂x+ v

∂v

∂y+ w

∂v

∂z= ν

1 +

1β

∂2v

∂z2 −σ B2

0

ρv −

1 +

1β

ν

k1v, (4)

u∂T

∂x+ v

∂T

∂y+ w

∂T

∂z=

k

ρcp

∂2T

∂z2 + τ

DB

∂C

∂z

∂T

∂z+

DT

T∞

∂T

∂z

2

, (5)

u∂C

∂x+ v

∂C

∂y+ w

∂C

∂z= DB

∂2C

∂z2 +DT

T∞

∂2T

∂z2

, (6)

where u, v and w are the velocity components in the x, y and z-direction, respectively, β = µB√

2πc/pz is the Cassonfluid parameter, B0 is the magnetic induction, T is the temperature, ν be the viscosity, k1 is the permeability, ρ is thedensity of the fluid, k is the thermal conductivity of the fluid, τ is the ratio of the heat capacitances, DB and DT arethe Brownian motion and thermophoretic diffusion coefficients and cp is specific heat capacitance.

The associated boundary conditions of Eqs. (2)–(6) are as follows:

u = Uw(x) = ax, v = Vw(x) = by, w = 0,

−k∂T

∂z= h f

T f − T

, −DB

∂C

∂z= hS (CS − C) at z = 0

(7)

u → 0, v → 0, w → 0, T → T∞, C → C∞ as z → ∞} . (8)

C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141 131

In the above expression, Uw and Vw are stretching velocities in x and y directions, respectively. h f is the convectiveheat transfer coefficient, hs is the convective mass transfer coefficient and T f , C f are the convective fluid temperatureand concentration below the moving sheet.

Introducing the following similarity transformations

u = ax f ′(η), v = byg′(η), w = −√

aν( f (η) + cg(η))

θ(η) =T − T∞

T f − T∞

, φ (η) =C − C∞

C f − C∞

, η = z

a/ν

(9)

Eq. (2) is automatically satisfied, and Eqs. (3)–(6) becomes1 +

1β

f ′′′

− ( f ′)2+ ( f + cg) f ′′

−

M + K

1 +

1β

f ′

= 0, (10)1 +

1β

g′′′

− (g′)2+ ( f + cg)g′′

−

M + K

1 +

1β

g′

= 0, (11)

1Pr

θ ′′+ ( f + cg)θ ′

+ Nbθ ′φ′+ Nt

θ ′

2= 0, (12)

φ′′+ Le ( f + cg) φ′

+Nt

Nbθ ′′

= 0. (13)

The transformed boundary conditions are

f = 0, g = 0, f ′= 1, g′

= c,θ ′

= −Bi1(1 − θ(0)), φ′= −Bi2 (1 − φ (0)) at η = 0

(14)

f ′= 0, g′

→ 0, θ → 0, φ → 0 as η → ∞. (15)

In the above expression M =σ B2

0ρa is a magnetic parameter, K =

νak1

is the porosity parameter, Pr =ρνcp

k

is Prandtl number, R =16σ T 3

∞

3kk∗ is the radiation parameter, Nt =τ DT (T f −T∞)

νT∞is the Thermophoresis Parameter,

Nb =τ DB(C f −C∞)

νis the Brownian motion parameter. Bi1 =

h fk

√ν/a

, Bi2 =

hsDB

√ν/a

are the Biot numbers

and c = b/a is the stretching parameter.Expression for skin friction coefficient C f on the surface along the x and y-directions, which are denoted by C f x

and C f y , respectively are as follows:

C f x =τwx

ρu2w

, C f y =τwy

ρu2w

(16)

where τwx , and τwy are the wall shear stress along x and y-directions, respectively.

Re1/2x C f x =

1 +

1β

f ′′(0), Re1/2

x C f y =

1 +

1β

g′′(0) (17)

where Rex = ux (x)x/ν is the local Reynolds number which depends on the stretching velocity uw(x).The local Nusselt and Sherwood numbers in non-dimensional form are given by

Re−1/2x Nu = −θ ′(0), Re−1/2

x Sh = −φ′(0). (18)

3. Method of solution

The coupled nonlinear ordinary differential equation (10)–(13) with respect to the boundary conditions (14) and(15) are solved numerically using Runge–Kutta based shooting technique (Sandeep and Sulochana [10]). The systemof nonlinear ordinary differential equation (10)–(13) with the boundary conditions (14) and (15) forms highly

132 C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141

Fig. 2. Velocity field for different values of magnetic field parameter.

non-linear coupled differential equations. In order to solve these coupled non-linear differential equations we adoptedbvp4c technique in Matlab package. We consider f = f1, f ′

= f2, f ′′= f3, g = f4, g′

= f5, g′′= f6,

θ = f7, θ′= f8, φ = f9, φ

′= f10. Eqs. (10)–(13) are transformed into systems of first order differential equations

as follows:

f ′= f2

f ′

2 = f3

f ′

3 =β

1 + β

( f2)

2− ( f1 + c f4) f ′′

+

M + K

1 +

1β

f2

,

f ′

4 = f5f ′

5 = f6

f ′

6 =β

1 + β

( f5)

2− ( f1 + c f4) f6 +

M + K

1 +

1β

f5

,

f ′

7 = f8

f ′

8 = − Pr( f1 + c f4) f8 + Nbf8 f10 + Nt ( f8)

2

,

f ′

9 = f10

f ′

10 = −Le ( f1 + c f4) f10 +Nt

NbPr

( f1 + c f4) f8 + Nbf8 f10 + Nt ( f8)

2

.

(19)

Subject to the following initial conditions

f1(0) = f4(0) = 0, f2(0) = 1, f5(0) = λ, . . . etc.. (20)

Here, we assumed some unspecified initial conditions in Eq. (20), Eq. (19) is then integrated numerically as an initialvalued problem to a given terminal point. All these simplifications are done by using Matlab package.

4. Results and discussion

In order to get a physical insight into the problem, a parametric study is conducted on Newtonian and Casson fluidsto illustrate the effects of different governing parameters viz., the Magnetic field parameter M , Brownian motionparameter Nb, Thermophoresis parameter Nt, Biot numbers Bi1, Bi2, stretching ratio parameter c, on the nature of theflow, heat and mass transfer.

The non-dimensional velocities f ′ (η) and g′ (η), temperatureθ (η), and concentration ϕ (η) for different valuesof magneticfield parameter M for both Newtonian and Casson fluids are shown in Figs. 2–5. It is evident that the

C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141 133

Fig. 3. Velocity field for different values of magnetic field parameter.

Fig. 4. Temperature field for different values of magnetic field parameter.

velocities f ′ (η) and g′ (η) declines with rise in magnetic parameter M . But reverse results has been observed with anincrease in the magneticfield parameter. The magnetic field opposes the rate of transportation. Generally, an increase inthe magneticfield results a strong reduction in the dimensionless velocity. This is due to the fact that the magneticfieldintroduces a retarding body force known as Lorentz force. This happens because of the interaction of the magneticfieldand electric fields for the motion of an electrically conducting fluid, and the stronger Lorentz force produces muchmore resistance to the transport phenomena. As the Lorentz force is a resistive force which opposes the fluid motion,so heat is produced and as a result, the thermal boundary layer thickness and concentration (volume fraction) boundarylayer thickness become thicker for stronger magnetic field.

Figs. 6 and 7 have been plotted to demonstrate the effects of thermophoresis parameter Nt on the dimensionlesstemperature and concentration profiles. It is observed that as the thermophoresis parameter Nt increases, the temper-

134 C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141

Fig. 5. Concentration field for different values of magnetic field parameter.

Fig. 6. Temperature field for different values of thermophoresis parameter.

ature of the fluid increases. We noticed the similar type of results in concentration profiles with an increase in thevalue of Nt. It is interesting to mention here that the influence of thermophoresis is significantly high in Newtonianfluid while compared with the non-Newtonian fluid. Figs. 8 and 9 depict to analyze the temperature and concentrationprofiles for various values of Brownian motion parameter Nb. It is noticed that the dimensionless temperature θ (η)

increases with an increase in the Brownian motion parameter. But we noticed an opposite results in the concentrationprofiles with an increase in the Brownian motion parameter.

Figs. 10–12 displayed to show the influence of Biot numbers Bi1, Bi2 on the temperature and concentration profiles.It is evident that the temperature field increases rapidly near the boundary with an increase in the Biot number Bi1.It is also observed that the convective heating of the sheet is enhanced as Biot number Bi1 increases. Because theconcentration distribution is driven by the temperature field, one anticipates that a higher Biot number would promote

C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141 135

Fig. 7. Concentration field for different values of thermophoresis parameter.

Fig. 8. Temperature field for different values of Brownian motion parameter.

a deeper penetration of the concentration. This anticipation indeed realized in Figs. 11 and 12 which predicts higherconcentration at higher values of the Biot numbers.

Figs. 13 and 14 depicted to examine the effects of stretching ratio parameter c on the velocity profiles of bothNewtonian and Casson fluids. It is seen from the figures that the usual decay occurs to the horizontal velocity profilesf ′ (η) for various values of stretching parameter c, while vertical velocity is enhanced with the hike in the stretchingparameter c. The stretching parameter c is the ratio of horizontal stretching to the vertical stretching. Figs. 15 and16 reveal the influences of stretching ratio parameter c on the temperature and concentration profiles. It is observedthat the stretching ratio parameter has quite opposite effect on the temperature and concentration profiles, a rise instretching ratio parameter c, decays the temperature and concentration profiles.

136 C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141

Fig. 9. Concentration field for different values of Brownian motion parameter.

Fig. 10. Temperature field for different values of Biot number.

Table 1 shows the validation of the present results with the published results. We have observed an excellentagreement of the present results with the published results. This proves the validity of the present results along with theaccuracy of the numerical technique we used in this study. Tables 2 and 3 shows the influence of the non-dimensionalgoverning parameters on friction factors along with the local Nusselt and Sherwood numbers for Newtonian andnon-Newtonian fluids respectively. It is evident that with an increase in the magnetic field parameter we have noticeddepreciation in the friction factor along with the heat and mass transfer rate. We noticed the similar type of result infriction factor case with an increase in the stretching ratio parameter. But an increase in the stretching ratio parameterhelps to enhance the heat and mass transfer rate. A raise in the values of Biot numbers, thermophoresis and Brownianmotion parameters does not shown significant influence on friction factor. But we noticed mixed response in heat and

C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141 137

Fig. 11. Concentration field for different values of Biot number.

Fig. 12. Concentration field for different values of Biot number.

mass transfer rate. Casson parameter have tendency to reduce the heat transfer rate and enhance the friction factor andmass transfer rate.

5. Conclusions

A three-dimensional magnetohydrodynamic Newtonian and non-Newtonian fluid flow, heat and mass transferover a stretching surface in the presence of thermophoresis and Brownian motion is investigated. The transformedgoverning equations are solved numerically via Runge–Kutta and Newton’s method. The influence of dimensionlessparameters on velocity, temperature and concentration profiles along with the friction factor, local Nusselt andSherwood numbers are discussed with the help of graphs and tables. The conclusions are as follows:

138 C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141

Fig. 13. Velocity field for different values of stretching parameter.

Fig. 14. Velocity field for different values of stretching parameter.

Table 1

Comparison of the values of

1 +1β

f ′′(0) for different values of M, K , β when c = 0.5.

M K β Nadeem et al. [27] Mahanta and Shaw [28] Present results

0 0 ∞ 1.0932 1.093257 1.0932520 0 5 1.1974 1.197426 1.1974250 0.5 1 1.8361 1.836083 1.836082

10 0 ∞ 3.3420 3.342023 3.34202010 0 5 3.6610 3.660730 3.66073010 0.5 1 4.8310 4.830598 4.830596

C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141 139

Fig. 15. Temperature field for different values of stretching parameter.

Table 2Variation in physical quantities for Newtonian fluid.

M Nt Nb Bi1 Bi2 c β C f x C f y −θ ′(0) −φ′(0)

1.0 −1.829575 −1.829575 0.301751 0.2446502.0 −2.083805 −2.083805 0.288260 0.2363673.0 −2.310564 −2.310564 0.277004 0.229586

0.5 −1.829575 −1.829575 0.293770 0.2132081.0 −1.829575 −1.829575 0.274299 0.1445151.5 −1.829575 −1.829575 0.255629 0.089302

1.0 −1.829575 −1.829575 0.285647 0.2722752.0 −1.829575 −1.829575 0.253739 0.2860513.0 −1.829575 −1.829575 0.222739 0.290593

0.4 −1.829575 −1.829575 0.229439 0.2690310.6 −1.829575 −1.829575 0.282101 0.2609990.8 −1.829575 −1.829575 0.318272 0.255507

0.4 −1.829575 −1.829575 0.320008 0.2214060.6 −1.829575 −1.829575 0.316782 0.2847450.8 −1.829575 −1.829575 0.314360 0.332272

0.5 −1.489532 −0.695215 0.261896 0.2239241.0 −1.564022 −1.564022 0.318272 0.2555071.5 −1.682498 −2.647314 0.379145 0.289266

1 −1.731420 −1.892121 0.320008 0.2214062 −1.544321 −1.744322 0.271241 0.2948733 −1.324151 −1.212141 0.240514 0.299200

• Magneticfield parameter have tendency to control the flow and reduce the friction factor.• The heat and mass transfer rate in non-Newtonian nanofluids is comparatively high while compared with the

Newtonian fluid.• Brownian motion parameter have tendency to enhance the mass transfer rate.• Rise in the stretching ratio parameter increase the local Nusselt and Sherwood numbers.• Biot numbers have tendency to control the heat and mass transfer rate.• The enhancement in the thermal boundary layer thickness of Newtonian fluid is significant while compared with

the non-Newtonian fluid.

140 C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141

Fig. 16. Concentration field for different values of stretching parameter.

Table 3Variation in physical quantities for Casson fluid.

M Nt Nb Bi1 Bi2 c C f x C f y −θ ′(0) −φ′(0)

1.0 −1.056839 −1.056839 0.351302 0.2763922.0 −1.203687 −1.203687 0.342317 0.2705753.0 −1.334672 −1.334672 0.334282 0.265384

0.5 −1.056839 −1.056839 0.343652 0.2495661.0 −1.056839 −1.056839 0.324731 0.1908951.5 −1.056839 −1.056839 0.306227 0.143374

1.0 −1.056839 −1.056839 0.336545 0.2999852.0 −1.056839 −1.056839 0.306797 0.3117653.0 −1.056839 −1.056839 0.277110 0.315664

0.4 −1.056839 −1.056839 0.248208 0.2925150.6 −1.056839 −1.056839 0.311373 0.2849950.8 −1.056839 −1.056839 0.356310 0.279683

0.4 −1.056839 −1.056839 0.358055 0.2405070.6 −1.056839 −1.056839 0.354790 0.3137540.8 −1.056839 −1.056839 0.352273 0.370115

0.5 −0.928995 −0.433610 0.298763 0.2468401.0 −0.975414 −0.975414 0.356310 0.2796831.5 −1.049260 −1.650940 0.414916 0.311562

Acknowledgments

The authors wish to express their thanks to the very competent anonymous referees for their valuable commentsand suggestions. Third author acknowledge the UGC for financial support under the UGC Dr. D.S. Kothari FellowshipScheme.

References

[1] Sakiadis BC. Boundary layer behavior on continuous solid surfaces: 1. Boundary layer equations for two-dimensional and axisymmetric flow.AIChE J 1961;7:26–8.

[2] Crane LJ. Flow past a stretching plate. Z Angew Math Phys 1970;21:645–7.

C. Sulochana et al. / Journal of the Nigerian Mathematical Society 35 (2016) 128–141 141

[3] Wang CY. The three-dimensional flow due to a stretching flat surface. Phys Fluids 1984;27:1915–7.[4] Samir Kumar N, Sumanta S, Tapas Ray M. Unsteady MHD boundary-layer flow and heat transfer of nanofluid over a permeable shrinking

sheet in the presence of thermal radiation. Alexandria Eng J 2014;53:929–37.[5] Vajravelu K, Prasad KV, Ng Chiu-On. Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid

properties. Nonlinear Anal RWA 2013;14:455–64.[6] Mohan Krishna P, Sandeep N, Sugunamma V. Effects of radiation and chemical reaction of MHD convective flow over a permeable stretching

surface with suction and heat generation. Walaliak J Sci Technol 2015;12:831–47.[7] Mabood F, Khan WA, Ismail AIM. MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: A numerical

study. J Magn Magn Mater 2015;374:569–76. http://dx.doi.org/10.1016/j.jmmm.2014.09.013.[8] Raju CSK, Sandeep N, Sulochana C, Sugunamma V, Jayachandra Babu M. Radiation, Inclined Magnetic field and cross-diffusion effects on

flow over a stretching over a stretching surface. J Nigerian Math Soc 2015;34:169–80.[9] Mahapatra TR, Nandy SK, Gupta AS. Magnetohydrodynamic stagnation point flow over a stretching surface. Internat J Non-Linear Mech

2009;44:124–9.[10] Sandeep N, Sulochana C. Dual solution for unsteady mixed convection flow of MHD Micropalar fluid over a stretching/shrinking sheet with

non-uniform heat source/sink. Int J Eng Sci Technol 2015;18:1–8.[11] Magyari E, Keller B. Heat and mass transfer in the boundary layer on an exponentially stretching continuous surface. J Phys 1999;32:577–85.[12] Ishak A, Nazar R, Pop I. Boundary layer flow and heat transfer over an unsteady stretching vertical surface. Meccanica 2009;44:369–75.[13] Ibrahim W, Shankar B. MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal

and solutal slip boundary conditions. Comput & Fluids 2013;75:1–10.[14] Hayat T, Shehzad SA, Alsaedi A, Alhothuli MS. Mixed convective stagnation point flow of Casson fluid with convective boundary condition.

Chin Phys Lett 2012;29:256–307.[15] Nadeem S, Rizwan Ul Haq, Akbar NoreenSher, Khan ZH. MHD three dimensional Casson fluid flow past a porous linearly stretching sheet.

Alexandria Eng J 2013;52: http://dx.doi.org/10.1016/j.aej.2013.08.005.[16] Hayat T, Shehzad SA, Alsaedi A. Three-dimensional stretched flow of Jeffery fluid with variable thermal conductivity and thermal radiation.

Appl Math Mech 2013;34:823–33.[17] Nadeem S, Haq RizwanUl, Lee C. MHD flow of a Casson fluid over an exponentially shrinking sheet. Sci Iran B 2012;19:1550–3.[18] Haq RizwanUl, Nadeem S, Khan ZH, Okedayo ToyinGideon. Convective heat transfer and MHD effects on Casson nanofluid flow over a

shrinking sheet. Cent Eur J Phys 2014;12:862–71. http://dx.doi.org/10.2478/s11534-014-0522-3.

[19] Jayachandra BabuM, Radha Gupta, Sandeep N. Effect of radiation and viscous dissipation on stagnation-point flow of a micropolar fluid overa nonlinearly stretching surface with suction/injection. J Basic Appl Res Int 2015;7:73–82.

[20] Pramanik S. Casson fluid flow and heat transfer past an exponentially porous stretching surface in presence of thermal radiation. Ain ShamsEng J 2014;5:205–12.

[21] Bhattacharya K. Boundary layer stagnation point flow of Casson fluid and heat transfer towards a shrinking/stretching sheet. Front Heat MassTransfer 2013;4:1–9. http://dx.doi.org/10.5098/hmt.v4.2.3003.

[22] Makinde OD, Aziz A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary conditions. Int J Therm Sci 2011;50:1326–32.

[23] Zaimi K, Ishak A, Pop I. Boundary layer flow and heat transfer over a nonlinearly permeable stretching/shrinking sheet in a nanofluid. SciRep 2014;4:4404. http://dx.doi.org/10.1038/srep04404.

[24] Nadeem S, Lee C. Boundary layer flow of nanofluid over an exponentially stretching surface. Nanoscale Res Lett 2012;7:1–6.http://dx.doi.org/10.1186/1556-276x-7-94.

[25] Hayat T, Imtiaz M, Alsaedi A, Mansoor R. MHD flow of nanofluids over an exponentially stretching sheet in porous medium with convectiveboundary conditions. Chin Phys B 2014;23:1–9.

[26] Ferdows M, Khan MS, Alam MM, Sun S. MHD mixed convective boundary layer flow of a nanofluid through a porous medium due to anexponentially stretching sheet. Math Probl Eng 2012;2012:1–21. http://dx.doi.org/10.1155/2012/408528.

[27] Nadeem S, Haq RizwanUl, Akbar NS, Khan ZH. MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet. AlexandriaEng J 2013;52:577–682.

[28] Mahanta G, Shaw S. Three-dimensional Casson fluid flow past a porous linearly stretching sheet with convective boundary conditions.Alexandria Eng J 2015;54:653–9.

[29] Makinde OD, Khan WA, Culham JR. MHD variable viscosity reacting flow over a convectively heated plate in a porous medium withthermophoresis and radiative heat transfer. Int J Heat Mass Transfer 2016;93:595–604.

[30] Khan WA, Culham JR, Makinde OD. Combined heat and mass transfer of third-grade nanofluids over a convectively-heated streatchingpermeable surface. Can J Chem Eng 2015;93:1880–8.

[31] Khan WA, Culham R, Makinde OD. Hydromagnetic blasius flow of power-low nanofluids over a convectively heated vertical plate. Can JChem Eng 2015;93:1830–7.