Rana, P, Shukla, N, Beg, OA, Kadir, A and Singh, B

Transcription

1 Unsteady electroagnetic radiative nanofluid stagnation point flow fro a stretching sheet with cheically reactive nanoparticles, Stefan blowing effect and entropy generation Rana, P, Shukla, N, Beg, OA, Kadir, A and Singh, B Title Authors Type URL Unsteady electroagnetic radiative nanofluid stagnation point flow fro a stretching sheet with cheically reactive nanoparticles, Stefan blowing effect and entropy generation Rana, P, Shukla, N, Beg, OA, Kadir, A and Singh, B Article Published Date 018 This version is available at: USIR is a digital collection of the research output of the University of Salford. Where copyright perits, full text aterial held in the repository is ade freely available online and can be read, downloaded and copied for non coercial private study or research purposes. Please check the anuscript for any further copyright restrictions. For ore ioration, including our policy and subission procedure, please contact the Repository Tea at: usir@salford.ac.uk.

2 1 PROC IMECHE-PART N- JOURNAL OF NANOMATERIALS, NANOENGINEERING AND NANOSYSTEMS PUBLISHERS: SAGE ACCEPTED FEBRUARY 7 TH 018 eissn: ISSN: IN PRESS UNSTEADY ELECTROMAGNETIC RADIATIVE NANOFLUID STAGNATION-POINT FLOW FROM A STRETCHING SHEET WITH CHEMICALLY REACTIVE NANOPARTICLES, STEFAN BLOWING EFFECT AND ENTROPY GENERATION Puneet Rana 1*, Nisha Shukla 1, O. Anwar Bég, A. Kadir 3 and Bani Singh 1 1 Departent of Matheatics, Jaypee Institute of Ioration Technology, A-10, Sector-6, Noida-01307, Uttar Pradesh, India. Fluid Mechanics, Propulsion and Nanosystes, Aeronautical and Mechanical Engineering, School of Coputing, Science & Engineering, University of Salford, Newton Building, M54WT, UK. 3 Corrosion and Materials, Aeronautical and Mechanical Engineering, School of Coputing, Science & Engineering, University of Salford, Newton Building, M54WT, UK. Abstract: The present article investigates the cobined iluence of nonlinear radiation, Stefan blowing and cheical reactions on unsteady EMHD stagnation point flow of a nanofluid fro a horizontal stretching sheet. Both electrical and agnetic body forces are considered. In addition, the effects of velocity slip, theral slip and ass slip are considered at the boundaries. An analytical ethod naed as hootopy analysis ethod is applied to solve the nondiensional syste of nonlinear partial differential equations which are obtained by applying siilarity transforations on governing equations. The effects of eerging paraeters including Stefan blowing paraeter, electric paraeter, agnetic paraeter etc. on the iportant physical quantities are presented graphically. Additionally, an entropy generation analysis is provided in this article for theral optiization. The flow is observed to be accelerated both with increasing agnetic field and electrical field. Entropy generation nuber is arkedly enhanced with greater agnetic field, electrical field and Reynolds nuber, whereas it is reduced with increasing cheical reaction paraeter. Keywords: Stefan blowing; Nonlinear radiation; Nanofluid; EMHD; Entropy; Cheical reaction; Electrical field; Magnetic field; Hootopy solutions; Unsteady flow.

3 Noenclature B Magnetic Field Strength (T) Nb Brownian Motion Paraeter (--) 0 C Nanoparticles Concentration (--) Nt Therophoresis Paraeter (--) D Brownian Diffusion ( /s) Pr Prandtl nuber (--) B D Therophoresis Diffusion ( /s) q Ebedding Paraeter (--) T E Electric Paraeter (--) R Radiation Paraeter (--) E Electric Field Strength (N/C) Re Reynolds Nuber (--) 0 Ec Eckert Nuber (--) s Stefan Blowing Paraeter (--) G Diensionless Strea function (--) S Diensionless Concentration(--) k Theral Conductivity (W/-K) Sc Schidt Nuber (--) K Cheical Reaction Paraeter (--) t Tie(s) M Magnetic Paraeter (--) Tw t r T Teperature Ratio (--) N Velocity Slip Paraeter (--) u 1 Velocity (/s) along x -axis N Theral Slip Paraeter (--) v Velocity (/s) along y -axis N Mass Slip Paraeter (--) x, y Siilarity Variables (--) 3 Greek Sybols Theral Diffusivity ( /s) Diensionless Teperature Difference Stagnation Paraeter (--) c Heat Capacity (J/K 3 )

4 3 Dynaic Viscosity (Ns/ ) Diensionless Teperature (--) Electrical Conductivity (S/) Diffusive Constant (--) Strea Function ( /s) 1 Diensionless Velocity slip paraeter Kineatic Viscosity ( /s) Diensionless Theral slip paraeter Non-Diensional Tie (--) 3 Diensionless Mass slip paraeter Ratio of Heat Capacities (--) 1 Subscripts Abient condition Nanofluid w Condition on surface p Nanoparticles 1. INTRODUCTION Stefan blowing (wall injection) finds substantial applications in industrial systes such as drying and purifying processes where boundaries are perforated. The blowing effect occurs due to the ass transfer of olecules or nanoparticles fro one location to another. Mass transfer is also fundaental to absorption, evaporation, cobustion, distillation and aterials synthesis. The concept of blowing effect is provided by the Stefan proble which is an application of ass transfer [1] of species. Stefan blowing proble states that there exist a relation between the rate of ass transfer and flow field at the wall, because ass transfer is dependent upon the flow field and flow field is generated by ass blowing at wall. Fang and Jing [] have considered the Stefan blowing effects to investigate the ass and heat transfer of a viscous fluid over a linearly stretching sheet and observed that velocity, teperature and concentration are increasing

5 4 functions of blowing paraeter. Uddin et al. [3] have provided a nuerical study of bioconvection nanofluid flow over a plate incorporating the effects of Stefan blowing, velocity, theral and ass slips at the wall. In addition, Uddin et al. [4] investigated the second order velocity slip and Stefan blowing effects for bio-nanofluid flow with passively boundary conditions. Nanofluids [5 8] constitute based fluids containing nano-sized etallic particles. The presence of etallic nano-particles enhances theral conductivity properties of such fluids. Shahzad et al. [9] have presented the analytical study of the agnetohydrodynaic flow of Cu based nanoparticles. They have observed that the heat transfer enhances near the surface due to the presence of Cu-nanoparticles. Sheikholeslai and Bhatti [10] have applied finite eleent ethod to investigate the iluence of Coulob force on the force convective nanofluid flow.and observed an enhanceent in Nusselt nuber due to electric field. A broad literature is available regarding the different types of study on nanofluid [11 18] due to a vast aount of applications of it in industries and engineering field. In high-teperature aterials processing, theral radiation is significant. Generally two odelling approaches are eployed for siulating radiative heat transfer effects, naely linear and nonlinear odels. Nonlinear radiation is valid for both high and low teperature difference but linear radiation is valid only for low teperature difference. Thus, to provide a ore general and physically realistic siulation, in the present work, we consider the nonlinear theral radiation in the present article. Nuerous studies of radiative flows have been counicated. Khan et al. [19] have investigated the nonlinear radiation effect on a agnetohydrodynaic nanofluid flow fro a sheet and have shown that the rate of heat transfer is increased for a shrinking sheet whereas it is decreased for a stretching sheet. Bhatti and Rashidi [0] have

6 5 exained the cobined iluence of theral radiation and diffusion on nanofluid flow over a stretching sheet. Sheikholeslai [1] has investigated the effect of Lorentz force and radiation on a Fe O H O nanofluid and achieved and enhanceent in rate of heat transfer due to an 3 4 enhanceent in radiation paraeter. Rashidi et al. [] have considered the ipact of theral radiation with aligned agnetic field on a Cu and Al O -water nanofluid flow over a stretching 3 sheet. Further studies include [3,4] for different coigurations and with ultiple body force effects. Materials processing systes often involve cheical reactions, which ay be destructive or constructive in nature and can iluence significantly heat and ass diffusion phenoena. Generally boundary layer flow odels utilize first order cheical reaction effects and assue the reaction to be destructive. Cheical reactions are instruental in transforing aterial constitution. This phenoenon also arises in cheical engineering industries, electrocheistry, hydrolysis, electro-plating and cobustion processes (furnaces, fires, jet propulsion etc). Interesting studies of reactive flows include Mishra et al. [5] on agnetic viscoelastic fluids, Mohaed [6] on two-phase nanofluids, Venkateswarlu and Narayana [7] on radiative rotating nanofluid flows, Bég et al.[8] on dissipative radiative hydroagnetic double diffusion transport and Matin and Pop [9] on porous nanofluid flows. Further investigations include [30 33] in which cheical reaction has been shown to have a significant iluence on heat and/or ass transfer characteristics. The second law of therodynaics provides a way to quantify the level of disorder of a therodynaically syste. This study is known as entropy generation analysis which is very useful in optiizing theral engineering systes to operate at high working efficiency. The

7 6 ethod of entropy generation iniization was originally proposed by Bejan [34] in Many researchers have investigated the effects of physical paraeters on entropy generation in various theral flow regies. Tshehla and Makinde [35] have calculated the rate of entropy generation in steady flow of a liquid with variable viscosity for two concentric cylindrical pipes. Das et al.[36] have elaborated on entropy generation in an unsteady MHD flow of nanofluid fro a stretching sheet, observing that etallic nanoparticles generate a large aount of entropy. Rehan et al. [37] have considered the steady Jeffery nanofluid flow over a stretching sheet, indicating that entropy generation nuber is an increasing function of therophoresis paraeter, Eckert nuber and Brinkan nuber. Qing et al. [38] have studied the entropy generation analysis on a agnetohydrodynaic flow of Casson nanofluid with the effects of cheical reactions and nonlinear theral radiation over a stretching surface. Bhatti et al. [39] have applied successive linearization ethod to investigate the entropy generation analysis for non-newtonian nanofluid over stretching surface. The preceding studies have not considered entropy generation for the cobined unsteady electro-agnetic nanofluid stagnation flow fro a stretching sheet with the siultaneous effects of Stefan blowing, cheical reaction, nonlinear theral radiation, velocity slip, theral slip and ass slip. Both electrical and agnetic body forces are incorporated in the atheatical odel [40]. The syste of governing equations has been transfored into a non-diensional syste of nonlinear partial differential equations by applying siilarity transforations. The nondiensional boundary value proble is thereafter solved with the hootopy analysis ethod (HAM) eploying power-series expansions. Hootopy analysis ethod is discovered by Liao[41 43] which is an excellent analytical technique to solve the nonlinear ordinary and partial differential equations. Rehan et al. [44] have copared the nuerical results obtained via

8 7 shooting technique with HAM results of Casson fluid flow over an exponentially stretching sheet. In this article, we eploy this technique to solve the syste of nonlinear partial differential equations [45] which a non-diensional for of governing equations. The present proble has applications in theral anageent [46] of high heat flux electronics such as heat pipes, energy conversion devices and bioedical research. The article is structured as follows. Section 1 is the introductory part. Section covers the proble forulation. Section 3 entails the entropy generation analysis. Section 4 elaborates the analytical hootopy solutions and convergence characteristics. A brief discussion of results is presented in section 5. Section 6 suarizes the conclusions. The current work is relevant to electroagnetic nano-aterials processing.. MATHEMATICAL MODEL In this proble, we analyze the coposite iluence of Stefan blowing, destructive first order cheical reaction and nonlinear radiation on two-diensional tie dependent EMHD (Electroagneto-hydrodynaic) stagnation point flow[47] of an electrically-conducting incopressible nanofluid fro a stretching sheet. Fig.1 represents the geoetry of the proble in which the point O is a stagnation point. The horizontal x -axis is located along the sheet and the vertical y - axis is fixed in a direction perpendicular to the sheet. At the surface of sheet, the effects of velocity, theral and ass slips are also considered. Such phenoena are known to arise in aterials processing systes. Under these considerations, the boundary layer equations for conservation of ass, oentu, energy and nanoparticle (species) concentrations are defined ay be shown to assue the for [1,48,49]:

9 8 u v y 0, (1) u u u u du u B u v u ( u u ), t y t dx y 0 EB 0 0 () T T T T u C T DT T 1 qr u v 1 DB t x y y ( c) y y y T y ( c) y, (3) C C C C DT T u v DB K( C C ), t y y T y (4) The following boundary conditions are iposed at the sheet (wall) and in the free strea[3] : at y 0, w ( ) u u u x N1, y v DB C, 1Cw y T T Tw N, y C C Cw N3, y as y, u u, T T, C C, (5) where u (/sec), v (/sec) are the velocities along the x () and y () axes respectively and t (s) represents the tie. The velocity of the sheet is u w ( x) bx where b is the sheet (wall) stretching paraeter and free strea velocity is defined asu ax. The ter ( /s) represents the kineatic viscosity whereas (Ns/ ) represents the dynaic viscosity of nanofluid. In addition, the ter (kg/ 3 ) is density of nanofluid, (S/) is electrical conductivity of nanofluid, k is theral conductivity of nanofluid, ( c) (J/K- 3 ) is heat capacity of nanofluid and k ( /s) ( c) is theral diffusivity of nanofluid. B 0 (T) and

10 9 E 0 (N/C) define agnetic and electric field strength respectively. The ters D B ( /s) and D T ( /s) signify Brownian and therophoresis diffusion respectively and K (s -1 ) represents cheical reaction coefficient. T (K) and T (K) are nanofluid and abient teperatures, C and C are nanoparticles volue fraction and abient volue fraction and the ter ( c) 1 ( c) p represents the ratio of heat capacities (J/K- 3 ) of nanoparticles to nanofluid whereas N 1, N and N3 are velocity slip, theral slip and ass slip paraeters respectively. Tw and Cw are respectively nanofluid teperature and nanoparticles volue fraction at surface. The radiative heat flux q r (W/ ) is defined as[50]: q r 4 4Stefan-Boltzann constant ( 1) T, (6) 3 Rosseland ean absorption coefficient ( k ) 1 y which is obtained by Rosseland approxiations. This approxiation is valid for optically-thick fluids which can absorb or eit radiation at their boundaries. Proceeding with the analysis, the following siilarity transforations are applied to convert Eqns. (1)-(5) into non-diensional for [51,5]: a x y y, a (, ) y x G x y, u a x G( x, y), v a yg( x, y), T T ( xy, ) T T w C C, S( x, y) C C w, y 1 e, at, (7) where x is siilarity variable, G( x, y ), ( xy, ) and S( x, y) are non-diensional strea function, teperature and nanoparticles concentration, respectively and represents the non-diensional

11 10 tie. The ter represents the diensional strea function which satisfies Eqn. (1) and is defined by the Cauchy-Rieann equations, u, v y x. The non-diensional fors of Eqns. ()-(5) can be written as: x x y 3 G G G x G G G y 1 G ( y 1) y( y 1) M y 1 E 0, 3 (8) 1 S x G Nb Nt y( y 1) ( y 1) yg Ec Pr x x x x y 3 3 ( t 1) 3 3( 1) r t r 4 x x x x R 0, 3Pr 3( tr 1) (9) S Nt S S x S Sc yg y( y 1) ( y 1) ys 0, Nb y (10) at x 0, s S(0, y) G(0, y), Sc G(0, y) G(0, y), 1 (0, y) (0, y) 1 S(0, y), S(0, y) 1 3, as x, G( x, y) 1, ( xy, ) 0, S( x, y) 0, (11) where the non-diensional physical paraeters are defined as:

12 11 M a B 0 is agnetic paraeter, E E 0 is electric field paraeter, Bu 0 ( c) Pr k is 1 DB( Cw C ) Prandtl nuber, Nb 1 DT( Tw T ) is Brownian otion paraeter, Nt is T therophoresis paraeter, uw Ec c ( Tw T ) is local Eckert nuber, Sc is Schidt D B nuber, 3 4 1T R T is radiation paraeter, w tr k k 1 T is teperature ratio, ax Re is local Reynolds nuber, K a ( Cw C ) is cheical reaction paraeter, s is Stefan blowing (1 C ) y w a a paraeter, 1 N1 is velocity slip paraeter, N y y is theral slip paraeter, a 3 N3 is ass slip paraeter and is the stagnation paraeter. y In this study, the iportant engineering quantities are the skin friction coefficient, the local Nusselt nuber and Sherwood nuber. These quantities evaluate the transport phenoena at the wall (sheet) and are defined respectively as: Skin friction coefficient: C w, ax (1) where w represents wall stress, which is defined as: w u y y 0. (13)

13 1 Using eqs. (7), (1) and (13), we obtain y Cf C G ''(0, y). (14) Re Local Nusselt nuber[19,1,53]: x Nu qw qr, (15) y 0 k ( T T ) w where qw represents heat flux at wall and is defined as: 0 q k T h j, (16) w p p y Here hp cp ( Tw T ) is enthalpy and jp is the su of the Brownian and therophoresis diffusion ters, defined as. T jp p DBC DT. T (17) Using eqs. (6), (7), (15), (16) and (17), we obtain y 4R 3 Nur Nu '(0, y) 1 Nt t '(0, ) Re 3 r NbS y. (18) Sherwood Nuber: Sh q x, DC B (19) where q represents ass flux at wall and defined as: q jp. (0) p

14 13 Using Eqns. (6), (17), (19) and (0), we obtain y Nt Shr Sh S '(0, y) '(0, y) Re Nb. (1) 3. ENTROPY GENERATION ANALYSIS The voluetric rate of entropy generation, due to the effects of heat transfer, nonlinear radiation, viscous dissipation, diffusion and electroagnetic field is defined as [36,54]: S G 16 1 T u Rg DB C 1 3 k k1 y T y C y k T g B C T u u T y y T T R D B E B u, () where R g (J ol -1 K -1 ) is gas constant. The characteristic rate of entropy generation is: S c w k ( T T ). (3) xt The non-diensional entropy generation nuber is obtained by applying siilarity transforations on the ratio of SG and S C which is defined as: S Re G 4R Pr. Ec G G G 1 S S Ns 1 ym 1 E, Sc y 3 (4) where Rg DC is diffusive constant and k Tw T T is non-diensional teperature difference.

15 14 4. SOLUTIONS WITH HOMOTOPY ANALYSIS METHOD (HAM) To solve the non-diensional syste of nonlinear partial differential Eqns. (7)-(9) with boundary conditions (10), the hootopy analysis ethod has been applied. On the basis of suggestions of Liao[4], we have selected the following initial guesses, linear operators and auxiliary functions: The initial guesses: 1 s G0 ( x, y) 1 e x 1 Sc e e ( xy, ), S0( x, y), (5) 1 1, 0 3 which are satisfied the boundary conditions (10). The linear operators: L G G 3 G G x x 3, L, L x x S S S S, (6) x x which are satisfied the conditions: x x LG ( C1 Ce C3e x ) 0, L ( C4C5e ) 0, LS ( C6C7e ) 0. (7) The auxiliary functions: HG 1, H 1, H 1. S The zero th order equation: (1 q) L [ ( x, y, q) G ( x, y)] qh H ( x, y) N [ ( x, y, q), ( x, y, q), ( x, y, q)], (8) G G 0 G G G G S (1 q) L [ ( x, y, q) ( x, y)] qh H ( x, y) N [ ( x, y, q), ( x, y, q), ( x, y, q)], (9) 0 S

16 15 (1 q) L [ ( x, y, q) S ( x, y)] qh H ( x, y) N [ ( x, y, q), ( x, y, q), ( x, y, q)], (30) S S 0 S S S S S associated with: s S (0, y) G (0, y) 0, Sc G 1 G(0, y) (0, y) 0, (0, y) (0, y) S (0, y) 1 0, S (0, y) 3 1 0, as x, G ( xy, ) 1, ( xy, ) 0, ( xy, ) 0. (31) S where q [0,1] and h, G h and hs are the auxiliary paraeters. The convergence of Eqns. (8)- (31) are dependent on the values of these auxiliary paraeters. The ter NG, N and NS are defined as: 3 G G G G x G G NG y 1 ( 1) ( 1) 1 3 G y y y M y E x x y x, (3) 1 S x G N Nb Nt y( y 1) ( y 1) y G Ec Pr x x x x y 4R 3 3 ( t 1) 3 3( 1) r t r 3Pr 3( t r 1), (33) Nt x N Sc y y( y 1) ( y 1) y Nb y S S S S S G S. (34)

17 16 On differentiating Eqns. (8)-(30) ties and dividing by!, then with subsequent substitution of q 0, we obtain: G G 1 1 G G L ( G ( x, y) G ( x, y)) h H R ( x, y), (35) 1 1 L ( ( x, y) ( x, y)) h H R ( x, y), (36) S S 1 1 S S L ( S ( x, y) S ( x, y)) h H R ( x, y), (37) with boundary conditions: at x 0, G s S ( x, y) G ( x, y) 0, ( x, y) G( x, y) (, ) 1 0, (, ) xy Sc xy 0, S S (, ) (, ) x y x y 3 0, as x, G ( x, y) 0, ( x, y) 0 and S ( x, y) 0. (38) G In the above Eqns., the ter R ( x, y ), R ( x, y) and R ( x, y) are defined as: S R G 1 ( x, y) 1! q 1 NG 1 q0 3 1 G 1 Gi 1 G 1 G i1 y G 3 1 x i0 G x G y( y 1) ( 1) y 1 1 y G 1 1 (1 ) (1 ), (39) M y M y E 1 R (, ) 1! q 1 N 1 q0 = 1 Pr i Si 1 i i Nb Nt i0 i0

18 i Gi G 1i 1 x 1 4R 1 ygi Ec y( y 1) ( y 1) tr 1 i0 i0 y 3Pr, 3 3 3t 1 1 i 1i 1 i 1i 1 k ik1 ik1 1i ji j ji j k r i i0 j0 k0 i0 j0 k0 i0 1 1 i 1 i i 1i 1 i i j 1 i 3( tr 1) ji j j i0 i0 j0 x i0 j0 x x, (40) 1 N S Nt S S x S R x y Sc y G y y y 1 1 S S (, ) ( 1) ( 1) 1 1! q x Nb x 0 i 0 x y x q ys 1, (41) where G 1! G ( x, y, q) q q0, 1! ( x, y, q) q q0 and S 1! S ( x, y, q) q q0. (4) The convergence of the following series of solutions is dependent on the appropriate choice of auxiliary paraeters h G, h and h S : G( x, y, q) G ( x, y) G ( x, y), (43) 0 1 ( x, y, q) ( x, y) ( x, y), (44) 0 1 S( x, y, q) S ( x, y) S ( x, y). (45) 0 1

19 18 To solve the above eqs., we have applied the sybolic software Maple-18 and obtained the ters G ( x, y ), ( xy, ) and S ( x, y ) in the following for: G ( x, y) c c x c e G, (46) 1 3 PS ( x, y) c c e, (47) 4 5 PS S ( x, y) c c e S, (48) 6 7 PS where the suffix PS represents the particular solution. The all ci ( i1..7) are calculated with the help of the boundary conditions (38). 4.1 Convergence of HAM solutions: The convergences of series (43)-(45) are dependent on the appropriate values of auxiliary paraeters h G, h and h S which can be obtained by sketching h-curves[55]. We have sketched the h-curves with G''(0, y ), '(0, y) and S'(0, y) for different values of y up to the 8 th order of approxiations and obtained a horizontal line in the ranges hg [ 0.048, 0], h [ 0.048,0] and h [ 0.0, 0] which are displayed in Fig.. Table-1 represents the order of convergences S of G''(0, y ), '(0, y) and S'(0, y) for the values of auxiliary paraeters, h 0.04, h 0.01 and h which indicates that the results are convergent up to four decial S places at 0 th order of approxiations. Hence, these are the appropriate values of auxiliary G paraeters hg, h and h S. In Table-, we present a coparison of the values of { S'(0, y)} obtained by hootopy analysis ethod and shooting ethod for steady non-radiative flow which presents a good agreeent between both. Table-3 shows the coparison of the present values of

20 19 G''(0, y) with previous published values i.e. Wang[56] and Abbas et al. [57] for M = 0, E = 0, s = 0, 1 0 and different values of. Excellent correlation is achieved coiring the validity of the HAM solutions. 5. DISCUSSION OF RESULTS In this section, we present a brief discussion of iluence of physical paraeters on velocity G '( x, y ), teperature ( xy, ), concentration S( x, y ), skin friction coefficient Cf, local Nusselt nuber Nur, Sherwood nuber Shr and entropy generation nuber Ns via graphs and tables. The default values of leading paraeters are taken as: M 0.1, E 0.1, Nt 0.1, Nb = 0.1, Pr 5, Ec = 0.3, R = 0.1, t 1.5, 0.5, 0.1, s 0.5, 1 3, Re 1, r 0.5, 0.1. The sybolic software Maple 18 running on personal coputer (core i5) is used for finding HAM solution. The nuerical HAM values of Nusselt nuber and Sherwood nuber are presented in Table-4 with variation in the values of Ec, Nt,, & 3 and fixed values of other paraeters. This table shows that these both quantities are decreased with an increase in the value of Eckert nuber and theral slip paraeter but increased with therophoresis and stagnation paraeter and also represents that Nusselt nuber increases as the values of ass slip paraeter increases but Sherwood nuber decreases with an increase in the value of this paraeter. The profiles of velocity G '( x, y ), teperature ( xy, ) and concentration S( x, y ) in opposition to x are presented in Fig 3 in which Fig 3(a) and 3(b) are sketched to elucidate the effects of agnetic field paraeter M and electric field paraeter E on velocity. It is evident that G '( x, y) is an increasing function of both of these paraeters. The behavior of M can be

21 0 B understood by the ter 0 ( u u ) in the priitive oentu conservation eqn. (). Since, in our study, the boundary layer velocity u is less than the external free strea velocity u, B therefore the ter 0 ( u u ) becoe negative. Furtherore in the transfored oentu conservation eqn. (8), the ter G M y 1 E, both agnetic and electrical body force ters becoe effectively positive. These body forces therefore assist oentu developent and lead to flow acceleration. Thus velocity increases with an increase in the value of agnetic paraeter, M. Fig 3(c) depicts teperature profile versus transfored transverse coordinate, x for different values of radiation paraeter R. It is apparent that ( xy, ) increases with an increase in the value of R. This paraeter is defined as R 3 4 T 1 k k 1 and features in the augented theral diffusion ter in the heat conservation eqn. (9). It defines the relative contribution of theral radiation heat transfer to theral conduction heat transfer. When R <1 theral conduction doinates. When R = 1 both theral conduction and theral radiation contributions are equal. For R >1 theral radiation doinates over theral conduction. In the present siulations, we coine attention to the last of these three cases i.e. 0< R <1 wherein theral radiative flux is substantial. Fig. 3 (c) clearly reveals that there energizing of the flow with increasing R values. This enhances theral diffusion and therefore elevates teperatures and also theral boundary layer thickness. Siilar observations have been reported by Shehzad et al. [58] and Venkateswarlu et al. [7]. Fig.3 (d) depicts the concentration profile vs. x for different values of cheical reaction paraeter which shows that concentration is a decreasing function

22 1 of. We consider the destructive type of hoogenous cheical reaction. Increasing the cheical reaction paraeter produces a decrease in velocity and therefore also oentu boundary layer thickness is therefore increased substantially with greater cheical reaction effect. However concentration distributions decrease when the cheical reaction increases. Physically, for a destructive case, with stronger cheical reaction, greater destruction of the original nano-particle species takes place. This, in turn, suppresses olecular diffusion of the reaining species which leads to a fall in concentration agnitudes and a corresponding depletion in concentration boundary layer thickness. Figs. 4(a) -4(c) illustrate the iluence of agnetic paraeter M, electric paraeter E, Stefan blowing paraeter s, diensionless tie, velocity slip paraeter 1 and stagnation paraeter on skin friction coefficient Cf. Cf increases with an increase in the values of M since agnetic body force accelerates the flow whereas it decreases with electric field E. However the converse response is coputed with s, 1 and. In this figure, we have coputed the effects of blowing paraeter on skin friction for the values of s 1, 0,1. Here s 1indicates suction at the sheet surface, s 0 iplies a solid wall (no lateral ass flux) and s 1indicates injection at the wall. We have observed that as the value of blowing paraeter (ass transfer) increases, skin friction at the surface decreases i.e. strong lateral ass flux into the boundary layer decreases the skin friction at the sheet surface. Fig.5 depicts the cobined effects of s and on local Nusselt nuber and Sherwood nuber; evidently both quantities are decreasing function of s but increasing function of.with greater progression of tie therefore heat and ass transfer at the sheet (wall) is enhanced and wall

23 suction (s=-1) induces a siilar iluence. Wall injection (s=1) however depresses heat and ass transfer rates at the wall. Fig.6 describes the effects of radiation paraeter on Nur and Shr and shows that as the value of radiation paraeter increases, the Nusselt nuber increases but Sherwood nuber decreases. Clearly with greater teperatures generated at higher values of radiation paraeter the heat transferred to the wall is boosted. Conversely wall ass transfer rate is depressed owing to an increase in concentration of nano-particles in the boundary layer. Fig. 7. shows that Nur and Shr are both increased with cheical reaction paraeter. The destruction in nano-particle species with stronger cheical reaction effect (higher values), leads to a depletion in concentration values in the boundary layer. This encourages the transfer of species to the wall and elevates Sherwood nuber. The species diffusion to the wall also assist theral diffusion and elevates wall heat transfer rates. The second law of therodynaics states that generally entropy of a syste always increases. This stateent is coired by Fig 8 (a)-(c) which depict the effects of agnetic paraeter M, electric paraeter E and Reynolds nuber Re on entropy generation nuber Ns. In all three graphs 8a-c, there is a non-trivial increase in entropy generation nuber Ns. Magnetic field, electrical field and inertial effect (Reynolds nuber) therefore all encourage entropy generation in the nanofluid regie. However Fig. 8(d) indicates that entropy generation nuber is decreased with an increent in cheical reaction paraeter. When the teperature of any substance is iniized, this inhibits olecular otion and therefore the entropy of the syste decreases.

24 3 Fig. 9(a)-(c). visualize the cobined effects of paraeters (M, E), (Nt, Nb) and (Re, R), in threediensional plots of Ns. Inspection of the Fig.9(a) reveals that Ns increases with an increase in the value of therophoresis paraeter Nt whereas the contrary behavior is generated with increasing Brownian otion paraeter Nb. Fig. 9(b) shows the cobined effects of Reynolds nuber and radiation paraeter. With increasing Reynolds nuber, the entropy generation nuber also increases which is consistent with the results of Fig. 8(c).There is a weak odification of Ns due to radiation paraeter. Furtherore it is apparent that Fig. 9(c) concurs with the results of Fig. 8(a) and 8(b). 6. CONCLUDING REMARKS An analytical study of the collective iluence of Stefan blowing and cheical reaction on unsteady nonlinear radiative EMHD stagnation-point flow of nanofluid fro a stretching sheet has been presented with slip effects. Hootopy analysis ethod (HAM) solutions have been derived for the transfored, nonlinear partial differential boundary value proble. The principal conclusions of the present siulations ay be suarized as: Velocity is an increasing function of agnetic and electric paraeter. Teperature increases with an increase in the value of radiation paraeter whereas concentration decreases with an increase in the value of cheical reaction paraeter. Skin friction coefficient increases with an increase in the value of M and whereas it deonstrates the opposite response with s,, E and 1. Local Nusselt nuber decreases with an increase in the value of Stefan blowing paraeter s whereas it increases with R,

25 4 and. A siilar response is coputed for Sherwood nuber with s, and ; however the converse trend is observed with R. Entropy generation nuber is an increasing function of M, E, Re, and Nt whereas it deonstrates the contrary behaviour with and Nb. REFERENCES 1. Spalding DB. Mass transfer in lainar flow. Proc R Soc Lond Ser Math Phys Sci 1954; 1: Fang T and Jing W. Flow, heat and species transfer over a stretching plate considering coupled Stefan blowing effects fro species transfer. Coun Nonlinear Sci Nuer Siul 014; 19: Uddin MJ Kabir MN and Bég OA. Coputational investigation of Stefan blowing and ultiple-slip effects on buoyancy-driven bioconvection nanofluid flow with icroorganiss. Int J Heat Mass Transf 016; 95: Uddin MJ Alginahi Y Bég OA and Kabir MN. Nuerical solutions for gyrotactic bioconvection in nanofluid-saturated porous edia with Stefan blowing and ultiple slip effects. Coput Math Appl 016; 7: Choi SUS and Eastan JA. Enhancing theral conductivity of fluids with nanoparticles. Argonne National Lab., IL (United States); biblio/19655/ (accessed 8 Mar017). 6. Buongiorno J. Convective Transport in Nanofluids. ASME J Heat Transf 006; 18: Sheikholeslai M. Magnetohydrodynaic nanofluid forced convection in a porous lid driven cubic cavity using Lattice Boltzann ethod. J Mol Liq 017; 31: Sheikholeslai M. Iluence of agnetic field on nanofluid free convection in an open porous cavity by eans of Lattice Boltzann ethod. J Mol Liq 017; 34: Shahzad F Haq RU and Al-Mdallal QM. Water driven Cu nanoparticles between two concentric ducts with oscillatory pressure gradient. J Mol Liq 016; 4: Sheikholeslai M and Bhatti MM. Active ethod for nanofluid heat transfer enhanceent by eans of EHD. Int J Heat Mass Transf 017; 109:

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30 9 Table-1 Order of Convergence of HAM results: Order of convergence of the values of G''(0, y ), '(0, y) and S'(0, y) for the fixed values of paraeters 0.5, Nt = Nb = 0.1, Sc = 3, Ec = 0.5, M = 0.1, Pr = 5, E = 0.1, s = 0.5, R = 0.1, tr 1.5, 0.1, 1 3 5, y = 0.5 Order ''(0, ) G y '(0, y) S'(0, y) Table- Validation of HAM Results: Coparison of HAM values of S'(0, y) with the values obtained by shooting ethod (liiting case) for the different values of Stefan blowing paraeter s and fixed values of other paraeters Nt = Nb = 0.1, Sc = 5, Ec = 0.3, M = 0.1, Pr = 5, E = 0.1, R = 0, 0.1, 1 3 5, y = 1, 0.5 s S'(0, y) HAM Shooting

31 30 Table-3: Coparison of present values of G''(0, y ) to the published results for particular case: Present values Wang [56] Abbas et al.[51] Table 4: HAM values of Nusselt and Sherwood nuber: The nuerical values of Nusselt and Sherwood nuber obtained by hootopy analysis ethod on the fifteen order of approxiations for the different values of Ec, Nt,, & 3 and fixed values of paraeters Nt = Nb = 0.1, Sc = 10, M = 0.1, Pr = 5, E = 0.1, R = 0.1, 1 3, y = 0.5, 0.5. (, 3) Ec Nt (, ) (, 3) (3,3) Nur Shr Nur Shr Nur Shr

32 31 u ( x) y T u ( x) u ( ) w x T w u ( ) w x x Fig.1 Physical structure of proble.

33 3 Fig. h-curve with G''(0, y ), '(0, y) and S'(0, y) for different values of y on 8 th order of approxiations.

34 33 Fig.3 Effects of physical paraeters on velocity G '( x, y ), teperature ( xy, ) and concentration S( x, y ).

35 34 Fig.4 Cobined effects of (E, M ), (s, ) and ( 1, ) on skin friction coefficient.

36 35 Fig. 5 Cobined effect of s and on Nusselt and Sherwood nuber. Fig.6 Cobined effect of radiation paraeter R and on Nusselt and Sherwood nuber.

37 Fig. 7 Effect of cheical reaction paraeter on Nusselt and Sherwood nuber. 36

38 Fig. 8 Effect of physical paraeters M, E, Re and on entropy generation nuber Ns. 37

39 Fig.9 Cobined effects of (Nt, Nb), (Re, R) and (E, M) on entropy generation nuber Ns. 38

40 39 List of Caption of Figures Fig 1. Physical structure of proble. Fig. h-curve with G''(0, y ), '(0, y) and S'(0, y) for different values of y on 8 th order of approxiations. Fig.3 Effects of physical paraeters on velocity G '( x, y ), teperature ( xy, ) and concentration S( x, y ). Fig.4 Cobined effects of (E, M ), (s, ) and ( 1, ) on skin friction coefficient. Fig. 5 Cobined effect of s and on Nusselt and Sherwood nuber. Fig.6 Cobined effect of radiation paraeter R and on Nusselt and Sherwood nuber. Fig. 7 Effect of cheical reaction paraeter on Nusselt and Sherwood nuber. Fig. 8 Effect of physical paraeters M, E, Re and on entropy generation nuber Ns. Fig.9 Cobined effects of (Nt, Nb), (Re, R) and (E, M) on entropy generation nuber Ns.