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Journal of the Nigerian Mathematical Society ■ (IUI) HI-III

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Similarity solution of 3D Casson nanofluid flow over a stretching sheet 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 2016

Abstract

In this study, we analyzed the three-dimensional magnetohydrodynamic Newtonian and non-Newtonian fluid flow. Heat and mass transfer over a stretching surface in the presence of thermophoresis and Brownian motion is investigated. The transformed governing equations are solved numerically via Runge-Kutta based shooting technique. We obtained good accuracy of the present results by comparing with the exited literature. The influence of dimensionless parameters on velocity, temperature and concentration profiles along with the friction factor, local Nusselt and Sherwood numbers are discussed with the help of graphs and tables. It is found that an increase in the stretching ratio parameter enhances the heat and mass transfer rate. The heat and mass transfer rate in non-Newtonian fluid is comparatively high while compared with the heat and mass transfer rate in Newtonian fluid. © 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 many researchers due to its huge industrial and engineering applications. In the field of industry, metallurgical processes such as drawing of continuous filaments through quiescent fluids, annealing and tinning of copper wires, manufacturing of plastic and rubber sheets, crystal growing, and continuous cooling and fiber spinning, in addition to wide-ranging applications in many engineering processes, such as extrusion of polymer, wire drawing, manufacturing foods and paper, in textile and glass fiber production etc. During the manufacturing of these sheets, the melt issues from a slit and it is stretched to achieve the desired thickness. The final product depends on two characteristics first is the rate of cooling in the process and the other is stretching rate. In view of these applications, Sakiadis [1] initiated the study 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.2016.01.001

0189-8965/© 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/)._

Please cite this article in press as: Sulochana C, et al. Similarity solution of 3D Casson nanofluid flow over a stretching sheet with convective boundary conditions. Journal of the Nigerian Mathematical Society (2016), http://dx.doi.Org/10.1016/j.jnnms.2016.01.001

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boundary layer flow of an incompressible viscous fluid flow past a linearly stretching surface. This problem was later extended by Wang [3] and discussed for three-dimensional case. Afterwards, many investigations were made to gain insight information regarding the flow over a stretching/shrinking sheet in various situations. Such situations include the considerations of porous surfaces, MHD, suction/injection, thermal radiation, slip effects, convective boundary condition, etc. was studied by Samir et al. [4], Vajravelu et al. [5], Mohan Krishna et al. [6], Mabood et al. [7], Raju et al. [8], Mahapatra et al. [9], Sandeep and Sulochana [10]. Magyari and Keller [11] studied the boundary layer flow due to a non-linearly stretching surface. Ishak et al. [12] presented the concept of unsteady two dimensional mixed 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 the influence 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 in engineering, chemical and petroleum industries. In the class of non-Newtonian fluids, Casson fluid has unique characteristics, which have wide application in food processing, in metallurgy, drilling operation and bio-engineering operations, etc. Hayat et al. [14] elaborated the concept of mixed convection stagnation point flow of Casson fluid under the effect of convective boundary conditions. Flow through a porous linearly stretching surface of MHD three-dimensional Casson fluid was analyzed by Nadeem et al. [15]. Hayat et al. [16] studied the variable thermal radiation effect over a three-dimensional stretched flow of Jeffery fluid. Nadeem et al. [17] examined the magneto-hydrodynamic boundary layer flow of a Casson fluid past an exponentially shrinking surface. Haq et al. [18] investigated the effect of MHD and 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 a nonlinearly stretching surface with suction/injection effects. Pramanik [20] presented the concept of Casson fluid flow and 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 surface. The effect of a convective boundary condition on the two-dimensional boundary layer flow of a nanofluid over a 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/shrinking surface with Brownian motion and thermophoresis effects. Nadeem and Lee [24] discussed the effects of Brownian motion parameter and thermophoresis parameter on the steady boundary layer flow of a nanofluid past an exponential stretching 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 Brownian motion parameter and thermophoresis parameter on MHD mixed convective boundary layer flow of a nanofluid over a porous exponentially stretching sheet. The researchers [27,28] studied the heat and mass transfer characteristics of non-Newtonian fluid through stretching sheet. Makinde et al. [29] elaborated the influence of thermophoresis and radiation on heat transfer of MHD flow with varying viscosity past a heated plate immersed in a porous medium. Khan et al. [30] and Khan et al. [31] discussed the mixed convective heat and mass transfer of power law nanofluid and third grade nanofluid past on a heated vertical surface and heated stretching surface.

In this study, we analyzed the three-dimensional magnetohydrodynamic Newtonian and non-Newtonian fluid flow, heat and mass transfer over a stretching surface in the presence of thermophoresis and Brownian motion is investigated. 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 friction factor, 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 considered that the sheet is stretched along xy-plane while fluid is placed along the z-axis. It is assumed that induced magnetic field is negligible. Thermophoresis and Brownian motion effects are taken into account. Here, we assumed that the sheet has stretched with the linear velocities u = ax and v = by along the xy-plane, respectively, with constants a and b (see Fig. 1). Moreover, it is considered that a constant magnetic field is applying normal to the fluid flow. The heat and mass transfer process is taken in to account. The rheological equation of state for an isotropic flow of a Casson

C. Sulochana et al. / Journal of the Nigerian Mathematical Society % illll) III—I

u — ax

Fig. 1. Physical model and coordinate system.

fluid can be expressed as follows:

2yß b + 2 ( ß b +

en, n > n

eu, n < n

In the above equation n = eijeij and eij denotes (i, j)th component of the deformation rate, n be the product of the component of deformation rate itself, nc be critical value of this product based on the non-Newtonian model, ^B be the 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:

d u dv dw — + — + — = 0,

d x d y d z

= V 1 +

d u d u d u

u--+ v--+ w —

d x dy d z

dv dv dv

u--+ v--+ w —

d x dy dz

d T d T d T k d2 T u--+ v--+ w- =--TT

d X dy d z p Cp d z

= V 1 +

n 1 d 2 u a Bo2

ßj ' dz2 P

1Ï d 2 v a Bo2

ß) P

v - 1 +

d C d T DT Db--+ —

d z d z T„o

ßj ki

d C d C d C d2 C DT u--+ v--+ w— = +

where u, v and w are the velocity components in the x, y and z-direction, respectively, fi = ^2nc/pz is the Casson fluid parameter, Bq is the magnetic induction, T is the temperature, v be the viscosity, k1 is the permeability, p is the density of the fluid, k is the thermal conductivity of the fluid, t is the ratio of the heat capacitances, DB and DT are the 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,

d T d C

-k — = hf(Tf - T), - Db — = hs (Cs - C ) at z = 0

d z u0

as z ^ œ}

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In the above expression, Uw and Vw are stretching velocities in x and y directions, respectively. h f is the convective heat transfer coefficient, hs is the convective mass transfer coefficient and Tf, Cf are the convective fluid temperature and concentration below the moving sheet.

Introducing the following similarity transformations

u = axf ' (n),

0(n) =

v = byg'(n), w =-Vav( f (n)+ cg(n))

t / \ C - Cœ / -¡~

Ф (n) = ^-, П = zJa/v

Tf Tœ C f Cœ

Eq. (2) is automatically satisfied, and Eqs. (3)-(6) becomes

1 + ^ ) f''' - ( f' )2 + ( f + cg) f" - ( M + Kl 1 + - I )f' = 0

1 + ^ ) g''' - (g' )2 + ( f +

'' ' M + K ( 1 + e I ) g' = 0,

—0'' + ( f + cg)0' + Nbe'ф' + Nt (9')2 = 0,

// / Nt ..

ф '' + Le ( f + cg) ф '+— 9 '' = 0.

The transformed boundary conditions are

f = 0, g = 0, f' = 1, g' = c,

0' = -Bi1 (1 - 9(0)), ф' = -Bi2 (1 - ф (0)) at n = 0

f ' = 0, g' ^ 0,

ф ^ 0 as n ^ œ.

(10) (11) (12)

In the above expression M = ^pO0 is a magnetic parameter, K = -0ц is the porosity parameter, Pr = is Prandtl number, R = is the radiation parameter, Nt = T Dt Vf Tœ) is the Thermophoresis Parameter,

= TDb (Cf Cœ) is the Brownian motion parameter. Bi1 = (Vv/a), Bi2 = Dh^ (Vv/a) are the Biot numbers

and c = b/a is the stretching parameter.

Expression for skin friction coefficient Cf on the surface along the x and y-directions, which are denoted by Cfx and Cfy, respectively are as follows:

PUw PUw

where Twx, and rwy are the wall shear stress along x and y-directions, respectively.

Re1'1 Cfx = (l + /"(0), Re]/2Cfy = (l + ^ g"(0)

where Rex = ux (x)x/v 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

Rex 1/2 Nu = -0'(0), Rex 1/2Sh = -ф'(0).

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 system of nonlinear ordinary differential equation (10)-(13) with the boundary conditions (14) and (15) forms highly

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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 adopted bvp4c technique in Matlab package. We consider f = fi, f' = /2, f" = /3, g = /4, g' = /5, g" = fó, 0 = f7, 0' = /8, 0 = /9, 0' = fio. Eqs. (10)-(13) are transformed into systems of first order differential equations as follows:

f ' = f2 f2 = f3

f5 = f6 = f7 = f8 = f9 =

(f2)2 - (fl + Cf4) f'' ^ M + K Í 1 +

(f5)2 - (fl + Cf4) f6 ^ M + K 1 +

- Pr ((f1 + Cf4)f8 + Nbf8f10 + Nt (f8)2) f10

fío = - Le ( f1 + cf4 ) f10 + Nb P^ ( f1 + cf4 ) f8 + Nbf8 f10 + Nt ( f8 .

Subject to the following initial conditions

f1 (0) = f4(0) = 0, f2(0) = 1, /5(0) = A,... etc..

Here, we assumed some unspecified initial conditions in Eq. (20), Eq. (19) is then integrated numerically as an initial valued 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 fluids to illustrate the effects of different governing parameters viz., the Magnetic field parameter M, Brownian motion parameter Nb, Thermophoresis parameter Nt, Biot numbers Bi1, Bi2, stretching ratio parameter c, on the nature of the flow, heat and mass transfer.

The non-dimensional velocities f' (n) and g' (n), temperature^ (n), and concentration ^ (n) for different values of magneticfield parameter M for both Newtonian and Casson fluids are shown in Figs. 2-5. It is evident that the

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

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

velocities f (n) and g' (n) declines with rise in magnetic parameter M. But reverse results has been observed with an increase in the magneticfield parameter. The magnetic field opposes the rate of transportation. Generally, an increase in the magneticfield results a strong reduction in the dimensionless velocity. This is due to the fact that the magneticfield introduces a retarding body force known as Lorentz force. This happens because of the interaction of the magneticfield and electric fields for the motion of an electrically conducting fluid, and the stronger Lorentz force produces much more 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) boundary layer thickness become thicker for stronger magnetic field.

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

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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 the value of Nt. It is interesting to mention here that the influence of thermophoresis is significantly high in Newtonian fluid while compared with the non-Newtonian fluid. Figs. 8 and 9 depict to analyze the temperature and concentration profiles for various values of Brownian motion parameter Nb. It is noticed that the dimensionless temperature в (n) increases with an increase in the Brownian motion parameter. But we noticed an opposite results in the concentration profiles 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 the concentration distribution is driven by the temperature field, one anticipates that a higher Biot number would promote

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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 higher concentration 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 both Newtonian and Casson fluids. It is seen from the figures that the usual decay occurs to the horizontal velocity profiles f' (n) for various values of stretching parameter c, while vertical velocity is enhanced with the hike in the stretching parameter c. The stretching parameter c is the ratio of horizontal stretching to the vertical stretching. Figs. 15 and 16 reveal the influences of stretching ratio parameter c on the temperature and concentration profiles. It is observed that the stretching ratio parameter has quite opposite effect on the temperature and concentration profiles, a rise in stretching ratio parameter c, decays the temperature and concentration profiles.

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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 excellent agreement of the present results with the published results. This proves the validity of the present results along with the accuracy of the numerical technique we used in this study. Tables 2 and 3 shows the influence of the non-dimensional governing parameters on friction factors along with the local Nusselt and Sherwood numbers for Newtonian and non-Newtonian fluids respectively. It is evident that with an increase in the magnetic field parameter we have noticed depreciation in the friction factor along with the heat and mass transfer rate. We noticed the similar type of result in friction factor case with an increase in the stretching ratio parameter. But an increase in the stretching ratio parameter helps to enhance the heat and mass transfer rate. A raise in the values of Biot numbers, thermophoresis and Brownian motion parameters does not shown significant influence on friction factor. But we noticed mixed response in heat and

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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 and mass transfer rate.

5. Conclusions

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

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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 + ^ f''(0) for different values of M, K, ft when c = 0.5. M K ft Nadeem et al. [27] Mahanta and Shaw [28] Present results

0 0 to 1.0932 1.093257 1.093252

0 0 5 1.1974 1.197426 1.197425

0 0.5 1 1.8361 1.836083 1.836082

10 0 to 3.3420 3.342023 3.342020

10 0 5 3.6610 3.660730 3.660730

10 0.5 1 4.8310 4.830598 4.830596

ARTICLE 1 N P R ESS

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

Table 2

Variation in physical quantities for Newtonian fluid.

M Nt Nb Bi1 Bi2 c ß Cfx Cfy -e ' (0) -4> ' (0)

1.0 -1.829575 -1.829575 0.301751 0.244650

2.0 -2.083805 -2.083805 0.288260 0.236367

3.0 -2.310564 -2.310564 0.277004 0.229586

0.5 -1.829575 -1.829575 0.293770 0.213208

1.0 -1.829575 -1.829575 0.274299 0.144515

1.5 -1.829575 -1.829575 0.255629 0.089302

1.0 -1.829575 -1.829575 0.285647 0.272275

2.0 -1.829575 -1.829575 0.253739 0.286051

3.0 -1.829575 -1.829575 0.222739 0.290593

0.4 -1.829575 -1.829575 0.229439 0.269031

0.6 -1.829575 -1.829575 0.282101 0.260999

0.8 -1.829575 -1.829575 0.318272 0.255507

0.4 -1.829575 -1.829575 0.320008 0.221406

0.6 -1.829575 -1.829575 0.316782 0.284745

0.8 -1.829575 -1.829575 0.314360 0.332272

0.5 -1.489532 -0.695215 0.261896 0.223924

1.0 -1.564022 -1.564022 0.318272 0.255507

1.5 -1.682498 -2.647314 0.379145 0.289266

1 -1.731420 -1.892121 0.320008 0.221406

2 -1.544321 -1.744322 0.271241 0.294873

3 -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.

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Blue: Newtonian Fluid Red : Non-Newtonian Fluid

u 0 1 2 3 4 5 V 6 7

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

Table 3

Variation in physical quantities for Casson fluid.

M Nt Nb Bi1 Bi2 c Cfx Cfy -e'(0)

1.0 -1.056839 -1.056839 0.351302 0.276392

2.0 -1.203687 -1.203687 0.342317 0.270575

3.0 -1.334672 -1.334672 0.334282 0.265384

0.5 -1.056839 -1.056839 0.343652 0.249566

1.0 -1.056839 -1.056839 0.324731 0.190895

1.5 -1.056839 -1.056839 0.306227 0.143374

1.0 -1.056839 -1.056839 0.336545 0.299985

2.0 -1.056839 -1.056839 0.306797 0.311765

3.0 -1.056839 -1.056839 0.277110 0.315664

0.4 -1.056839 -1.056839 0.248208 0.292515

0.6 -1.056839 -1.056839 0.311373 0.284995

0.8 -1.056839 -1.056839 0.356310 0.279683

0.4 -1.056839 -1.056839 0.358055 0.240507

0.6 -1.056839 -1.056839 0.354790 0.313754

0.8 -1.056839 -1.056839 0.352273 0.370115

0.5 -0.928995 -0.433610 0.298763 0.246840

1.0 -0.975414 -0.975414 0.356310 0.279683

1.5 -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 comments and suggestions. Third author acknowledge the UGC for financial support under the UGC Dr. D.S. Kothari Fellowship Scheme.

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