INFLUENCE OF VISCOUS DISSIPATION AND HEAT SOURCE ON MHD NANOFLUID FLOW AND HEAT TRANSFER TOWARDS A NONLINEAR PERMEABLE STRETCHING SHEET

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1 INFLUENCE OF VISCOUS DISSIPATION AND HEAT SOURCE ON MHD NANOFLUID FLOW AND HEAT TRANSFER TOWARDS A NONLINEAR PERMEABLE STRETCHING SHEET SRINIVASASAI PANUGANTI, MUDDA RAMESH, GNANESWARA REDDY MACHIREDDY 3, PADMA POLARAPU 4,,3 Department o Mathematics, ANU Ongole Campus, Ongole, A.P, India Department o Mathematics,S.R.Govt.Arts&Science college,kothagudem,khammam,telangana- 57 p.srinivasasai@gmail.com,mudda_ramesh@yahoo.com,mgrmaths@gmail.com,padmapolarapu@gmail.com ABSTRACT A numerical analysis has been presented on ree convective boundary layer lo o a nanoluid along a non-linear permeable stretching sheet in the existence o viscous dissipation and internal heat generation. We have included the Bronian motion and thermophoresis eects. A suitable similarity transmutation is used to transorm the governing equations into non-linear ordinary dierential equations. A second order inite dierence model knon as Keller-Box method is used to solve numerically the governing boundary layer equations. The present mathematical values have been compared ith the previously published values and good agreement is ound. The impacts o various emerging parameters on velocity proiles, temperature and concentration proiles are presented graphically and mathematical results o reduced Nusselt number, skin raction coeicient and the Sherood numbers are presented in tabular orm and graphs. An increase in the viscous dissipation parameter enhances the temperature and declines the concentration. The heat generation parameter enhances the skin riction coeicient. The eect o increasing the values o chemical reaction parameter is to decrease the concentration proiles. We made a conclusion that the rate o mass transer enhances or rise o the chemical reaction parameter, KEYWORDS: Nanoluid, Flo and heat transer, Nonlinear permeable stretching sheet. B applied magnetic ield M magnetic parameter C concentration o the nanoluid Nb bronian motion parameter C skin riction coeicient Nt thermophoresis parameter C x reduced skin riction coeicient Nr thermal radiation parameter C p speciic heat at constant pressure Nu nusselt number C W concentration o the nanoluid along the Nu x reduced nusselt number stretching sheet Pr prandtl number C ambient concentration q r radiative heat lux D B bronian diusion coeicient q m mass lux D T thermophoresis coeicient q heat lux E C viscous dissipation parameter R thermal radiation parameter dimensionless stream unction Re x reynolds number g acceleration due to gravity R permeability parameter, Gr local thermal Grasho number Sh sherood number Gm local mass Grasho number Sh x reduced sherood number k thermal conductivity o the luid T luid temperature k e mean absorption coeicient T W temperature o the nanoluid along the stretching k p permeability o the porous medium sheet Le leis number

2 T u U W U v x y α η υ β β c β T temperature o the luid ar aay rom the sheet velocity in x direction velocity o the stretching surace uniorm velocity velocity in y direction horizontal distance vertical distance thermal diusivity parameter similarity variable kinematic viscosity thermal expansion coeicient coeicient o volumetric concentration Expansion coeicient o volumetric thermal expansion ρ density o the base luid μ dynamic viscosity ψ stream unction σ s stephan- Boltzmann constant τ ratio beteen the eective heat capacity o the nanoparticles and the luid τ W shear stress θ dimensionless temperature variable φ dimensionless rescaled nanoparticle volume raction λ buoyancy parameter δ solutal buoyancy parameter γ heat source/sink parameter n, n constants m stretching parameter. INTRODUCTION A luid hich contains nanometer sized particles, called nanoparticles and a base luid is called a nanoluid. The nanoparticles used in nanoluids are generally made o Carbides, Metals, Oxides, Nitrides or nonmetals (Carbon nanotubes, Graphite) and the base conductive luid is a luid, like ater, ethylene glycol, toluene and oil. Studies related to nanoluids gave much interest in recent time due to their excessive enhanced perormance properties, particularly ith respect to the transer o heat. Nanoluids have ne properties that make them potentially useul in many advantages in heat transer, including uel cells, microelectronics, pharmaceutical processes and hybrid-poered engines. A great scientiic interest is created in Nanoparticles since it is an eectively bridge beteen bulk materials and atomic or molecular structures. The technique o mixing nanoparticles and the base luid as irst proposed by Choi (995). Later Das et al. (3) shoed experimentally a to-to our-old raise in thermal conductivity enrichment or ater based nanoluids containing Al O 3 or Copper nanoparticles over a small temperature range 5C. A satisactory report or the abnormal increase o the thermal conductivity as given by Buongiorno (6). Buongiorno and Hu (5), investigated on the nanoluid coolants in advanced nuclear systems. Mixed convection boundary layer lo through a vertical lat plate illed ith nanoluids and enclosed in a porous medium as inspected by Ahmad and Pop (). Hady et al. () inspected the impacts o radiation on the lo and heat transer o a nanoluid past a non-linearly stretching sheet. Olanreaju and Olanreaju et al. () examined the impacts o radiation on the boundary layer lo o nanoluids past a moving surace. Hamid et al. () analyzed the impacts o radiation on Marangoni boundary layer lo over a lat plate in a nanoluid. EI-Aziz (9) studied radiation eect on the lo and heat transer over an unsteady stretching surace. Singh et al. () investigated the thermal radiation and magnetic ield impacts on an unsteady stretching permeable sheet in the existence o ree stream velocity. Gbadeyan et al. () studied the boundary layer lo o a nanoluid over a stretching sheet ith convective boundary conditions in the existence o magnetic ield and thermal radiation. Manna et al. () studied the inluence o radiation on unsteady MHD ree convective lo over an oscillating vertical porous plate. Magneto hydrodynamics (MHD) is the study o the interrelation o conducting luids ith electromagnetic incident. The lo o an electrically conducting luid in the existence o a magnetic

3 ield is o importance in dierent areas o engineering and technology such as MHD poer generation, MHD pumps, MHD lo meters, etc. The study o MHD boundary layer lo on a continuous stretching sheet has pulled attention during the last e decades because o its various appliances in industrial manuacturing techniques. In particular the metallurgical processes such as tinning o copper ires, draing and annealing involve cooling o continuous strips or ilaments by draing them through a quiescent luid. Deormation and lo o materials require energy. The mechanical energy is dissipated, i.e. during the luid lo it is converted into internal energy (heat) o the material. The increase o internal energy expresses itsel on temperature rise. Viscous dissipation alters the temperature distributions by playing a role like an energy source, hich leads to inluence heat transer rates. The merit o the eect o viscous dissipation depends on hether the plate is being cooled or heated. Viscous dissipation is o interest o many applications. Signiicant temperature increments are detected in polymer processing los such as injection molding or extrusion at high rates. Aerodynamic heating in the thin boundary layer over high speed aircrat increases the temperature o the skin. In a completely dierent purpose, the dissipation unction is utilized to deine the viscosity o dilute suspensions (Einstein 96, 9): viscous dissipation or a luid ith appended particles is equated to the viscous dissipation in a pure Netonian luid both exist in the same lo. It is no ell accepted act that the terms magneto hydrodynamics (MHD) heat generation and thermal radiation broadly appear in various engineering processes. MHD is signiicant in the control o boundary layer lo and metallurgical procedures. The thermal radiation and heat generation possessions may results in high temperature actors processing operations. Ingredients may be intelligently arranged thereore ith judicious implementation o radiative heating to generate the required characteristics. This recurrently takes place in plasma studies, agriculture, engineering and petroleum industries. The investigation o heat generation or absorption in moving luids is essential in problems dealing ith chemical reactions and these concerned ith disassociating luids. Heat generation is also important in the content o exothermic or endothermic chemical reactions. The heat and mass transer problems ith chemical reaction are o useul in many procedures, and thereore have acquired a considerable amount o attention in recent years. In order to transorm cheaper ra materials to high value products, the ra materials are made to go through chemical reaction in all industrial chemical procedures. Such chemical transormations exist in a reactor. The reactor plays an important act o bringing reactants into close contact and providing an proper environment or adequate time and alloing the dismissal o inished products. Thus e are especially interested in cases in hich diusion and chemical reaction roughly the same speed. In processes, like drying, vaporization at the surace o a ater body, energy transer in a et cooling toer and the lo in desert cooler, groves o ruit trees, electric poer generation; the heat and mass transmission take place simultaneously. In many chemical engineering procedures, a chemical reaction beteen a oreign mass and the luid take place. These procedures take place in various industrial applications, such as the polymer construction, the manuacturing o ceramics or glassare, ood processing. Most chemical reactions contain the breaking and ormation o chemical bonds. It requires energy to break a chemical bond but energy is discharged hen chemical bonds are ormed.

4 Kuznetsov and Nield () have investigated the impact o nanoparticles on the natural convection boundary layer lo toards a vertical plate integrating the Bronian motion and Thermophoresis treating both the temperature and the nanoparticles raction as stable along the all. Further, Nield and Kuznetsov (9) have studied the problem proposed by Cheng and Minkoycz (977) about the natural convection over a vertical plate in porous medium saturated by a nanoluid integrating the Bronian motion and thermophoresis. All these researchers studied linear stretching sheet in the nanoluid, but a numerical research as done by Rana and Bhargava () ho studied the steady laminar boundary luid lo o nanoluid past a nonlinear stretching lat surace ith the combined impacts o Bronian motion and thermoporesis. Fazlul Kader Murshed et al. (5) extended the ork o Rana and Bhargava by taking thermal radiation on boundary layer lo o a nanoluid toards a nonlinearly permeable stretching sheet. Also recently Nadeem and Lee () investigated analytically the problem o steady boundary layer lo o nanoluid over an exponentially `idening surace including the eects Bronian motion parameter and thermophoresis parameter. Investigations ere done on dual solutions o radiative MHD nanoluid lo past an exponentially unolding sheet ith heat generation/absorption (5). Khan and Pop () investigated the boundary layer lo o a nanoluid past a stretching sheet. Makinde and Olanreaju () examined the buoyancy impacts on thermal boundary layer past a erecting plate ith a convective boundary condition. Gnanesara Reddy (4) examined the thermal radiation and chemical reaction impacts on MHD mixed convective boundary layer slip lo in a porous medium ith ohmic heating and heat source. Gnanesara Reddy (4) examined the impacts o viscous dissipation, thermal radiation and hall current on MHD convection lo over a unold vertical lat plate. It as ound that the velocity, cross lo velocity and temperature distribution raises as the radiation parameter or viscous dissipation parameter increases. Gnanesara Reddy (4) investigated the impacts o Thermophoresis, viscous dissipation and joule heating on steady MHD lo over an inclined radiative isothermal porous surace ith variable thermal conductivity. Gnanesara Reddy et al. (5) studied the impacts o viscous dissipation and heat source on unsteady MHD lo over a stretching sheet. But so ar, no contribution is made ith the impacts o viscous dissipation, permeability heat source and chemical reaction parameter on heat and mass transer o MHD ree convective boundary layer lo toards a nonlinear stretching sheet and hence the present ork is carried out due to its abundant applications. So the aim o present paper is to study the simultaneous impacts o viscous dissipation, internal heat generation, chemical reaction on magneto hydro dynamic lo and heat transer o nanoluids through porous medium over a non-linear stretching sheet. The governing boundary layer equations are reduced to a combination o nonlinear ordinary dierential equations using similarity transmutations. By using Keller box method the nonlinear ordinary dierential equations are settled mathematically. A parametric study is conducted to illustrate the impacts o various governing parameters on the velocity proiles, temperature distribution, concentration proiles, local Nusselt number, Skin riction coeicient and Sherood number are discussed in detail.. FORMULATION OF THE PROBLEM We consider a steady, laminar, incompressible, to-dimensional boundary layer lo o a viscous 3

5 ater based nanoluids over a stretching sheet coinciding ith the plane y = and the lo being restricted to y >. The lo is created in vie o nonlinear stretching o the sheet caused by the simultaneous application o to equal and opposite orces along the x -axis. Fixing the origin, the sheet is then stretched ith a velocity U (x) = U x m here U is the uniorm velocity. m ( m ) is a constant parameter. The luid is considered to be a gray, absorbing emanating radiation but non scattering medium. The lo is subjected to a uniorm transverse magnetic ield o strength B = B hich is applied in the positive y-direction, normal to the surace. The luid is assumed to be slightly conducting, so that the magnetic Reynolds number is supposed to be small so that the induced magnetic ield is negligible in comparison to the applied magnetic ield. It is supposed that the porous medium is homogeneous and present everyhere in local thermodynamic equilibrium. The remaining properties o the luid and the porous medium are assumed to be constant. Both the base luid and the nanoparticles are supposed to be in thermal equilibrium and no slip occurs beteen them. The Hall Eect is also negligible. The pressure gradient and external orces are neglected. A schematic model o the problem under consideration is depicted in Fig.. Under the above assumptions, and employing the Oberbeck- Boussinesq approximation, the emerging equations o the lo ield in presence o viscous dissipation, heat source and porous medium can be ritten in the dimensional orm as u u x y p p u u u v x y x y T ( CC ) p C C T T u g B u k p T T T C D T x y y y T x u v T DB T q Q u r C y c c y p p p () T T 3 C C C DT T u v DB k ( ) r C C x y y T y here u and v are the velocities in the x and y-directions, respectively, g is the acceleration due to gravity, μ-the viscosity, ρ -the density o the base luid, ρ-the density o the nanoparticle, β T -the coeicient o volumetric thermal expansion, β C -the coeicient o volumetric concentration expansion, T-the temperature o the nanoluid, C-the concentration o the nanoluid, T and C -the temperature and concentration along the stretching sheet, T and C -the ambient temperature and concentration, D B - the Bronian diusion coeicient, D T -the thermophoresis coeicient, B -the magnetic induction, q r -radiative heat lux, k-the thermal conductivity, (ρc) p -the heat capacitance o the nanoparticles, (ρc) -the heat capacitance o the base luid, k p -the permeability o the porous medium, Q-the volumetric rate o heat generation/absorption, α = k (ρc) is the thermal diusivity parameter and τ = (ρc) p (ρc) is the ratio beteen the eective heat capacity o the nanoparticles material and heat capacity o the luid. The associated boundary conditions are (4) 4

6 p u, v, T T, C C as y m u U s x, v,t T, C C at y 5 By using the Rosseland approximation (99), the radiative heat lux q r is given by q r 4 4 s T 3k y e here σ s is the Stephen Boltzmann constant and k e is the mean absorption coeicient. It should be noted that by using the Rosseland approximation, the present analysis is limited to optically thick luids. I the temperature dierences ithin the lo are suiciently small, then equation (6) can be linearised by expanding T 4 into the Taylor series about T, hich ater neglecting higher order terms takes the orm T 4T T 3T (7) Then the radiation term in equation (3) takes the orm 3 4 st T e qr 6 y 3k y Invoking equation (8), equation (3) gets modiied as T T k 6 T u v x y C C 3k p e p 3 s D T T T y 9 T C DT T Q u B y y T x c c y p p Let e introduce a stream unction ψ(x, y) in the lo ield such that u = ψ y (8) (6), v = ( ψ). It is x obvious that the equation o continuity () is satisied. Assuming that the external pressure on the plate, in the direction having diluted nanoparticles, to be constant, the similarity transormations are taken as m U x T T C C,,, m T T C C m m U x k k 4 Gr e y, Nr, R, 3, 3 4 T 3Nr Re Gm DB 3,Pr, Le,, Nb C C, Re D ρ x B s C gnt T D u x x ρ T x T Re Nt T T, Re, Gr, ρρ p ρ Gm ρ gn C C B k m M R m Re U ( m ) x m,,, x m U x Q k m Ec c T T, m U x, m U x r m p c x x U x 5

7 Using the similarity transormations, the governing equations () (4) are transormed to the nonlinear dierential equations, ''' '' m ' M R m m () '' R Nb Pr Nt ' '' Ec () φ + Leφ + Nt Nb θ Leγ φ = (3) here λ- the buoyancy parameter, δ-the solutal buoyancy parameter, Pr-the Prandtl number, Le-the Leis number, υ-the kinematic viscosity o the luid, Nb-the Bronian motion parameter, Nt-the thermophoresis parameter, Re x -the local Reynolds number based on the stretching velocity, Gr-the local thermal Grasho number, Gm-the local concentration Grasho number, M-the magnetic parameter, R -the permeability parameter, R-the radiation parameter, Ec-the Eckert number, γthe heat source parameter γ = k r (m+)u xm -the chemical reaction parameter and, θ, φ are the dimensionless stream unctions, temperature, rescaled nanoparticle volume raction respectively. Here, β T and β C are proportional to x 3, that is β T = nx 3 and β C = n x 3, here n and n are the constants o proportionality (Makinde and Olanreaju, (). The corresponding boundary conditions are,,, at ',, as (4) No, e are interested to study the quantities o practical interest, the skin riction coeicient C and the local Nusselt number Nu, Sherood number S. The parameters respectively, characterize the surace drag, all heat transer rate and mass transer rate. These quantities are deined by: C u U U y y Re m x '' (5) qx Nu k T T y T T T y Re m q / x (6) 6

8 qmx Sh k C C y C C C y S Re m 3. NUMERICAL SOLUTION Since equations () (3) are highly non-linear, it is diicult to ind the closed orm solutions. Keller Box method (9, 974) is used to solve these equations ith the boundary conditions (4). The numerical values o local Nusselt number, Skin riction coeicient and local Sherood number are presented in tabular orm. 4. RESULTS AND DISCUSSION To get a physical insight into the lo problem, comprehensive numerical computations are conducted or various values o emerging parameters that describe the lo characteristics and the results are illustrated graphically. The value o Pr is taken as.7 hich corresponds to air. The values o Sc are chosen such that they represent Helium (.3), ater vapor (.6) and Ammonia (.78). The other parameters are chosen arbitrarily. The present results are compared ith that o Khan and Pop (), Anar et al. () and got an excellent agreement. Keller Box method is used to solve the resulting nonlinear ordinary dierential equations () (3) subject to the boundary conditions (4). Velocity, temperature and concentration proiles ere obtained and e applied the results to calculate the skin riction coeicient, the local Nusselt number and the local Sherood number in equations (5), (6) and (7). The accuracy o the method depends on the choice o the appropriate initial guesses. We have taken the olloing initial guesses :,, 8 e e e The choices o the above suitable initial guesses depend on the convergence criteria and the all shear stress () is commonly used as a convergence criterion because in the boundary layer lo calculations the greatest error appears in the all shear stress parameter as it is mentioned in the book by Cebeci and Bradsha (988). Thus, e used this convergence criterion in the present study. A uniorm grid o size.4 is chosen to satisy the convergence criterion o 7 in our study, hich gives about six decimal places accurate to most o the prescribed quantities. From the process o numerical computation, the numerical values o the local Nusselt number, Skin riction coeicients and Sherood number are represented in tabular orm. The impacts o various emerging parameters on the Skin riction coeicient (), Nusslet number θ () and Sherood number φ () are shon in Table. It is observed that as M, R, Nt, m or δ increaes all the three Skin riction coeicient, Nusslet number and Sherood number declines. As Le or λ increases then the Nusselt number, Skin riction coeicient and Sherood number rises. We came o he conclusion that the thermal radiation parameter R decreases both the Nusselt number, Skin riction coeicient and enhances the Sherood number. An increase in the Eckert number Ec or Nb leads to a raise in the Skin riction coeicient and Sherood number. An increase in the Eckert number Ec or Nb leads to a all in the Nusslet number. When the values o Pr or γ increase, both the Nusslet number and Sherood number rise hereas the Skin riction coeicient q m / x (7) 7

9 decreases. The numerical results are discussed or the values o the physical parameters graphically and in tabular orm. The present results are compared ith that o Khan and Pop () and Anar et al. () in the absence o permeability, viscous dissipation and heat source and ound that there is an excellent agreement (Table ). Table Numerical values o the Skin riction coeicient, Nusselt number and the Sherood number or various physical parameters. M R Ec Q R Pr Nb Nt Le m δ γ λ () θ () φ () Table Comparison o Reduced Nusselt number, Reduced Sherood number ith γ = Nb Nt Pr Le λ δ M Anar & Khan et Poornima & Present results al. () Reddy (3) θ () φ () θ () φ () θ () φ () Table 3 Computations o velocity and temperature proiles or dierent values o δ. δ (η) θ(η) (η) θ(η) (η) θ(η) η = η = η = Table 4 Computations o concentration proiles φ(η) or dierent values o M, R, λ and γ. φ(η) φ(η) φ(η) φ(η) M R λ γ η

10 The eects o various governing parameters upon the nature o the lo and transport are discussed in detail. The numerical results are depicted graphically in Figs. 3. Unless otherise mentioned, the deault values are Pr =.7, Le =, λ =, m =, M =, Nt = Nb = Ec =., R =, R =., γ =. γ =. and δ = or our subsequent results. The velocity proiles or various values o emerging parameters are elucidated in Figures 9. From these Figures it is observed that the luid velocity is highest at the stretching porous sheet and decreases exponentially to its ree stream zero value satisying the ar ield boundary conditions. It is observed that the eect o increasing the values o Prandtl number, the stretching parameter m, magnetic parameter, heat source parameter and solutal buoyancy parameter is to decrease velocity proiles as illustrated in Figures -6, hereas a reverse trend is observed ith increasing the values o buoyancy parameter, radiation parameter and the permeability parameter, hich is clear rom Figs.7-9. We deine the Prandtl number Pr as the ratio o momentum diusivity to thermal diusivity. It is noticed that an increase in the Prandtl number makes the luid to be more viscous, hich leads to a decrease in the velocity (Fig. ). When the magnetic parameter increases, the velocity proiles decrease. This is because, the application o the magnetic ield ithin the boundary layer produces a Lorentz orce hich opposes the lo and decelerates the luid motion hich is clear rom Fig. 4. The inluence o the solutal buoyancy parameter, on velocity (η) is shon in Figs. 6 and Table 3. It is noticed that a rise in solutal buoyancy parameter causes a decrease in velocity. The positive (+ve) buoyancy orce acts like a supportive pressure gradient and hence accelerates the luid in the boundary layer. This results in higher velocity as λ increases. The radiation parameter R deines the relative contribution o conduction heat transer to thermal radiation transer. It is clear rom Fig. 8 that an increase in the radiation parameter results in increasing velocity ithin the boundary layer. As shon in Fig. 9 the eect o increasing the value o porous permeability is to raise the value o the velocity component in the boundary layer due to the act that drag is reduced by increasing the value o porous permeability on the luid lo hich results in raised velocity. The temperature proiles or various values o governing parameters are elucidated in Figures. From these Figures e made a conclusion that the luid temperature is highest at the stretching porous sheet and decreases exponentially to its ree stream zero value satisying the ar ield boundary conditions. We have observed that the eect o increasing the values o Prandtl number, the buoyancy parameter and the Permeability parameter is to decrease temperatue as illustrated in Figures -, hereas an opposite trend is observed ith increasing the values o the stretching parameter m, radiation parameter, thermophoresis parameter, Bronian motion parameter, Eckert number, magnetic parameter, solutal buoyancy parameter and heat source parameter hich is clear rom Figs.3-. Enhancing the Prandtl number results in a reduction in thermal diusivity. We have noticed that the Fig. temperature decreases ith an increase in the Prandtl number. This results in reducing the thermal boundary layer thickness. The reason is that smaller values o Pr are equivalent to rising the thermal conductivities, and thereore heat is able to diuse aay rom the heated stretching sheet more immediately than or higher values o Pr. Hence or higher values o Prandtl numbers, the boundary layer thickness decreases so the rate o heat transer is increased. The radiation parameter R is responsible to thickening the thermal boundary layer. This enables the 9

11 luid to release the heat energy rom the lo region and causes the system to cool. This is true because increasing the Rosseland approximation results in an increase in temperature (99). From Fig. 4 it is noticed that the radiation eect enhances temperature o the luids and the absence o radiation deines small temperatures. Thereore, radiation enhances the buoyancy orce. It is seen that the increase o the radiation parameter leads to decrease the boundary layer thickness and thereby decrease in the value o the heat transer in the presence o thermal and solutal buoyancy orce. Due to the larger values o Nt the thermophoretic orces are produced. These orces have the tendency to leave the nanoparticles in the reverse direction o temperature gradient (i.e., rom hot to cool) hich causes a non-uniorm nanoparticle distribution. Consequently, the increasing values o Nt corresponds to an increase in the temperature. The Bronian motion is stronger in case o smaller nanoparticles hich corresponds to the greater Nb and converse is the situation or smaller values o Nb. In thermal conduction nanoparticle s motion plays a pivotal role. Due to the increased chaotic motion o the nanoparticles (i.e., or larger b) the kinetic energy o the particle is enhanced hich as a result enhances the temperature o the nanoluid. Hence increasing Nb irmly rises temperature values throughout the regime as illustrated in Fig. 6.An increase in viscous dissipation parameter Ec increases the temperature because the heat energy is stored in the liquid due to the rictional heating. The viscous dissipation parameter Ec expresses the relationship beteen the kinetic energy in the lo and the enthalphy *3+. It expresses the conversion o kinetic energy into internal energy by ork done against the viscous luid stresses. Larger viscous dissipative heat causes a rise in the temperature. This behavior is evident rom Fig. 7. The positive value o viscous dissipation parameter Ec cools the sheet i.e., loss o heat rom the sheet to the luid. The inluence o the solutal buoyancy parameter, on temperature θ(η) is shon in Fig. 9 and Table 3. It is observed that the temperature increases as the solutal buoyancy parameter increases. Fig. has been plotted to depict the impact o heat source parameter on temperature proiles. From this Figure e observe that temperature increases ith increase in the heat source parameter because hen heat is released, the buoyancy orce increases the temperature. The concentration proiles are illustrated in Figures 9 or various values o emerging parameters. From these Figures it is observed that the luid concentration is highest at the stretching porous sheet and reduces exponentially to its ree stream zero value satisying the ar ield boundary conditions. It is observed that the eect o increasing the values o thermophoresis parameter, the buoyancy parameter, the viscous dissipation parameter, Leis number, heat source parameter radiation parameter and chemical reaction parameter is to decrease concentration proiles as illustrated in Figures -7, hereas a reverse trend is observed ith increasing the values o Bronian motion parameter and magnetic parameter hich is clear rom Figs The impact o the thermal buoyancy parameter on the concentration ield is shon in Fig. and Table 4. It is noticed that the concentration decreases as the thermal buoyancy parameter increases. It is observed that the concentration decreases as the thermal buoyancy parameter increases. There ould be a signiicant reduction in the concentration boundary layer thickness hen Le is increased. This phenomenon occurs due to Leis number eects hich raises the concentration gradient at the surace, and as a result rises the local Sherood number as elucidated in Fig. 4. Table 4 and Fig. 5

12 sho the eect o the radiation parameter on the concentration. It is noticed that the concentration reduces as the radiation parameter enhances Table 4 and Fig. 6 sho the eect o heat source parameter on the concentration proiles. It is observed that the concentration declines as γ increase. The eect o increasing the values o chemical reaction motion parameter is to decrease the concentration proiles as shon in Fig. 7. The eect o increasing the values o Bronian motion parameter is to increase the concentration proiles as shon in Fig. 8. The eect o the magnetic parameter on the concentration ield is illustrated in Fig. 9 and Table 4. It is shon that a rise in the magnetic parameter causes an increase in the concentration. The eects o the magnetic parameter and R on skin riction coeicient are shon in Fig. 3. This Figure indicates an increase in the values o skin riction coeicient ith the increase in the values o magnetic parameter. The olloing conclusions ere dran.. Velocity decelerates ith an increase in the Prandtl number, solutal buoyancy parameter Magnetic parameter and stretching sheet parameter. But an opposite trend is observed ith an increase in the thermal buoyancy and radiation and permeability parameters.. It is ound that temperature decreases ith an increase in the Prandtl number, thermal buoyancy parameter, and permeability parameter, here as an reverse trend is observed ith an increase in the Eckert number, solutal buoyancy parameter, stretching sheet parameter, radiation parameter, Nb, Nt, γ. 3. Concentration decreases ith an increase in Ec, R, Nt, γ or γ here as an opposite trend is observed ith increase in M, Nb, or λ. 4. Skin riction coeicient increases ith an increase in the Ec or Nb or Le or λ and decreases ith an increase in M,R, R, Pr, Nt, m, δ, Pr, γ or γ. 5. Nusselt number or Heat transer rate rises ith a rise in the Pr, Le, γ or λ and it alls ith an increase in the M, R, R, Nt, m or δ, Ec, Nb or γ. 6. Mass transer rate increases ith an increase in the R, Ec, Pr, Nb, Le, γ or γ or λ and decreases ith in increases in the M, R, Nt, m or δ. Fig.. A schematic model o the problem under consideration

13 ' () ' () ' () ' () ' () ' () ' () ' () Pr =.7,., Fig.. Variations o velocity proiles or dierent values o Pr Fig. 6. Velocity Proiles or dierent values o δ..8 =.,., m =,, 3.4 =.,.5, Fig. 3. Variations o velocity proiles or dierent values o m M =.,.5, Fig. 4. Variations o velocity proiles or dierent values o M Fig. 7. Variations o velocity proiles or dierent values o λ Fig. 8. Variations o velocity proiles or dierent values o R..8.6 R =.,.4, R =,, =.,., Fig. 5. Velocity Proiles or dierent Values o γ. Fig. 9. Variations o velocity proiles or dierent values o R.

14 ().8.6 Pr =.7,., R =.,.4,.6 ().4 () Fig.. Variations o temperature proiles or dierent values o Pr Fig. 4. Variations o temperature proiles or dierent values o R..8.8 Nt =.,.,.3 ().6.4 =.,.5,. () Fig.. Variations o temperature proiles or dierent values o λ Fig. 5. Variations o temperature proiles or dierent values o Nt R =,, 3 ().6 Nb =.,., Fig.. Variations o temperature proiles or dierent values o R..8 m =.,., Fig. 6. Variations o temperature proiles or dierent values o Nb. ().6.4. () Ec =.,.,., Fig. 3.Variations o temperature proiles or dierent values o m Fig. 7. Variations o temperature proiles or dierent values o Ec.

15 .8.8 ().6.4 M =.,.5,. ().6.4 =.,.5, Fig. 8. Temperature Proiles or dierent values o M. () =.,., Fig. 9. Temperature Proiles or dierent values δ. Fig.. Concentration proiles o dierent values o λ. () Ec =.,.,., () =.,.,.3 Fig. 3. Variations o concentration proiles or dierent values o Ec Fig.. Temperature Proiles or dierent values o γ. () Nt =.,., Fig.. Variations o concentration proiles or dierent values o Nt. () Fig. 4. Concentration Proiles or dierent values o Le. () Le =, 5, R =.,.5, Fig. 5.Concentration Proiles or dierent values o R. 3

16 .8 dierent values o Nb..6.8 ().4 =.,.,.,.3.6. ().4 M =.5,., Fig. 6. Concentration Proiles or dierent values o γ Fig. 9. Concentration Proiles or distinct values o M. ().6.4. =.,., M =.5 M =. M = Fig. 7. Concentration Proiles or distinct values o γ. () Nb =.,.,.5 " () Fig. 3. Variation o the skin riction coeicient ith M or dierent values o R. R Fig. 8. Variations o concentration proiles or ACKNOWLEDGEMENTS The authors ish to express sincere thanks to the honorable reerees or their valuable comments and constructive suggestions to improve the quality o the paper. REFERENCES []. Ahmad, S. and I. Pop (). Mixed convection boundary layer lo rom a vertical lat plate embedded in a porous medium illed ith Nanoluids. International Communications in Heat and Mass Transer. 37, []. Anar, M.I., I. Khan, Sharidan S. and M.Z. Salleh (). Conjugate eects o heat and mass transer o nanoluids over a nonlinear stretching sheet. International Journal o Physical Sciences. 7(6), [3]. Aziz, E.l. (9). Radiation eect on the lo and heat transer over an unsteady stretching sheet. Int. Commu. Heat and Mass Trans. 36, [4]. Brester, M.Q, (99). Thermal radiatiative transer and properties. John Wiley. Ne Work. [5]. Buongiorno, J. (6). Convective transport in nanoluids. ASME J. Heat Transer, 8,

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