MHD Stagnation Point Flow of Williamson Fluid over a Stretching Cylinder with Variable Thermal Conductivity and Homogeneous/Heterogeneous Reaction

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1 Commun. Theor. Phys. 67 (2017) Vol. 67, No. 6, June 1, 2017 MHD Stagnation Point Flow of Williamson Fluid over a Stretching Cylinder with Variable Thermal Conductivity and Homogeneous/Heterogeneous Reaction M. Bilal, M. Sagheer, S. Hussain, and Y. Mehmood Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan (Received November 21, 2016; revised manuscript received March 16, 2017) Abstract The present study reveals the effect of homogeneous/hetereogeneous reaction on stagnation point flow of Williamson fluid in the presence of magnetohydrodynamics and heat generation/absorption coefficient over a stretching cylinder. Further the effects of variable thermal conductivity and thermal stratification are also considered. The governing partial differential equations are converted to ordinary differential equations with the help of similarity transformation. The system of coupled non-linear ordinary differential equations is then solved by shooting technique. MATLAB shooting code is validated by comparison with the previously published work in limiting case. Results are further strengthened when the present results are compared with MATLAB built-in function bvp4c. Effects of prominent parameters are deliberated graphically for the velocity, temperature and concentration profiles. Skin-friction coefficient and Nusselt number for the different parameters are investigated with the help of tables. PACS numbers: e, b, ad DOI: / /67/6/688 Key words: stagnation point flow, homogeneous/heterogeneous reaction, Williamson fluid, stretching cylinder, variable thermal conductivity, thermal stratification Nomenclature A, B chemical species T temperature of fluid a, b concentration of A, B T w wall temperature B 0 magnetic field strength T ambient temperature C p specific heat U w stretching velocity along x-axis C f skin friction coefficient U 0, V 0 constants D A diffusion species coeff. of A (u, w) velocity component D B diffusion species coeff. of B w e ambient fluid velocity f non-dimensional velocity α variable thermal diffusivity l s heterogeneous reaction rate constant α thermal conductivity L 1 strength of homogeneous reaction β heat generation coefficient L s strength of heterogeneous reaction γ curvature parameter l characteristic length η similarity variable l 1 homogeneous reaction rate constant θ dimensionless temperature m, n dimensionless concentration λ Weissenberg number M magnetic parameter ψ stream function Nu Nusselt number ε thermal conductivity parameter P r Prandtl number δ rate of mass diffusion coefficient P 0 stretching ratio parameter σ electric conductivity Q heat generation coefficient ν kinematic viscosity q w heat flux ρ density of fluid r, R radius σ Steffan Boltzman constant Re Reynolds number τ w wall shear stress s, t dimensional constants Γ Williamson parameter Sc Schmidt number ϕ thermal stratification parameter 1 Introduction Heat transfer analysis over a stretching cylinder is one of the current research topic among researchers due to its extensive applications in many engineering processes. In Corresponding author, me.bilal.786@gmail.com c 2017 Chinese Physical Society and IOP Publishing Ltd available literature, usually the thermal conductivity is taken as constant. Heat is transferred due to difference in temperature. If there is a large temperature difference then assumption of constant thermal conductivity will

2 No. 6 Communications in Theoretical Physics 689 lead to a noticeable error. Thus to minimize this type of error, it is necessary to consider a temperature dependent variable thermal conductivity within the thermal boundary layer region. Hussain et al. [1] analyzed the effect of MHD Jeffrey fluid flow on heat transfer with variable thermal conductivity. They used the inclined magnetic field on peristaltic flow and obtained solution analytically using perturbation method. Lin et al. [2] presented the numerical solution of unsteady pseudo-plastic nanoliquid in a thin film flow over a linearly stretching surface. They considered the effect of viscous dissipation and variable thermal conductivity along with four different types of nanoparticles. Influence of variable thermal condutivity on an exponentially stretching sheet for third grade fluid using inclined magnetic field was addressed by Hayat et al. [3] In another article, Hayat et al. [4] documented the behaviour of mixed convection and variable thermal conductivity on viscoelastic nanofluid over a stretching cylinder with heat source/sink. Si et al. [5] explored numerically the laminar film condensation of non-newtonian pseudo-plastic fluid on an isothermal vertical plate with variable thermal conductivity. Malik et al. [6] made an attempt, which deals with the heat transfer analysis of Williamson fluid over a stretching cylinder with heat generation/absorption and variable thermal conductivity. They conclude that increase in the thermal conductivity parameter enhances the temperature. References [7 10] include some recent work related to variable thermal conductivity. An extensive literature can be found regarding non- Newtonian fluid flows over a stretching cylinder because of their industrial and engineering applications such as metallurgical processes, extraction of petroleum, pipe industry, and many others. Several models have been proposed to analyze the pseudo-plastic fluids (e.g. Cross model, Carreau model, Power law model, Ellis model and Williamson fluid model). Every model has its own characteristics. In Williamson fluid model, a variable viscosity of larger range can be considered. Williamson [11] introduced a model that describes the pseudo-plastic materials and fluid flow experimentally. Malik and Salahuddin [12] used the Williamson fluid model over a stretching cylinder with the effect of magnetohydrodynamics. Solutions are obtained by shooting method for the fluid flow. Malik et al. [13] discussed the flow and heat transfer of Williamson fluid over a stretching cylinder with homogeneous/heterogeneous reaction using Keller Box method. In another article, Salahuddin et al. [14] have numerically investigated the MHD flow of Williamson fluid with Cattaneo Christov heat flux model over a non-linear stretching surface. Explicit finite difference method is employed for the numerical solutions. Hayat et al. [15] offered an overview of the literature about MHD boundary layer flow of Williamson fluid with Ohmic dissipation and thermal radiation. They noticed the decreasing influence of Weissenberg number on velocity profile. A list of references concerning Williamson fluid can be found in Refs. [16 18]. A phase is a distinct, uniform state of a system that has no observable boundary, which may divide the system into components. Chemical reactions are broadly classified into homogenous reactions (occurring in single phase) and heterogeneous reactions (involving multiple phases). A reaction is often aided with a catalyst, which enhances the reaction rate by providing a substitute path for reaction having lower activation energy. In recent years, scientists are very much concerned in creating effective and efficient processes that include an amalgamation of both homogenous and heterogeneous reactions. The researchers are currently much concerned with the study of the complex interactions of these reactions. Hayat et al. [19] explored Jeffrey fluid flow with homogeneous/heterogeneous reaction using Cattaneo Christov heat flux model. They conclude that concentration of molecules increases with the increment of Schmidt number. On another article Hayat et al. [20] presented an analytical solution of Oldroyd-B fluid with MHD and homogeneous/heterogenenous reaction using Cattaneo Christov heat flux model. Reddy et al. [21] analyzed the effect of homogeneous/heterogenenous reaction on MHD and non-linear thermal radiation between rotating plates. They employed shooting technique to solve the problem. Homogeneous/heterogenenous reaction in magneto-nanofluid in a permeable shrinking surface was addressed by Mansur et al. [22] Kameswaran et al. [23] explored homogeneous/heterogenenous reaction in a viscous nanofluid flow past a stretching sheet analytically when the auto catalyst and diffusion coefficients of the reactants are equal. In all the above mentioned articles, thermal conductivity is considered as a constant. Very less work is available with variable thermal conductivity specially in stretching cylinder. Our interest in the present article is to analyze the work of Malik et al. [13] with the assumption of variable thermal conductivity, magnetohydrodynamics and heat generation/absorption. Williamson fluid is considered over a stretching cylinder with homogeneous/heterogeneous reaction. Solutions for flow and heat transfer are computed numerically by shooting method [24 28] integrated with Runge Kutta method of order four. Numerical solutions are further supported with MATLAB built-in function bvp4c. Graphical results of velocity, temperature and concentration are presented and also numerical results of skin-friction coefficient and local Nusselt number are tabulated for emerging parameters. 2 Mathematical Formulation We have considered MHD stagnation point flow of Williamson fluid under the combined effect of homogeneous/heterogeneous reaction and heat generation/absorption over a linearly stretching cylinder along z-axis. Mangetic field of strength B 0 is applied normal to the flow as shown in Fig. 1. Induced magnetic field

3 690 Communications in Theoretical Physics Vol. 67 is neglected because of the supposition of small Reynolds number. Further a variable thermal conductivity and the effect of thermal stratification at boundary layer flow are also incorporated. Fig. 1 Geometry of the Problem. It is also clear that temperature at the sheet T w is greater than the temperature far away from the sheet T. There is an isothermal cubic auto catalytic (homogeneous) reaction on boundary layer flow however first order reaction (heterogeneous) is taken on catalyst surface. These are represented by A + 2B 3B, rate = l 1 ab 2, (1) A B, rate = l s a, (2) where l 1, l s are rate constants and a, b denote concentrations of the chemical species A, B respectively. Further considering that there is no change in temperature for both the reactions. Employing boundary layer approximations and taking in account all considerations mentioned above, the governing equations of the system, are (ru) r + (rw) z = 0, (3) u w r + w w z = w dw [ e 2 e dz + υ w r w r r + 2Γ w 2 w r u T r + w T z = 1 r u a r + w a z = D A u b r + w b z = D B r 2 + Γ ( w 2r r ( αr T r r ( 2 a r a r r ( 2 b r b r r subject to the boundary conditions ) 2 ] σb2 0 ) + Q(T T ) ρ (w w e), (4), (5) ρc p ) l 1 ab 2, (6) ) + l 1 ab 2, (7) w w (z) = U 0z, l u = 0, T = T w = T 0 + sz l w = w e V 0z l, D A a r = l sa, D B b r = l sa at r = R,, T T = T 0 + tz l, a a, b b as r. (8) Using the following transformations U0 ( r 2 R 2 ) νu0 η =, ψ = Rzf(η), w = U 0z f (η), νl 2R l l νu0 R u = l r f(η), θ(η) = T T, m(η) = a, n(η) = b. (9) T w T 0 a 0 a 0 In Eq. (5), α is variable thermal conductivity, which is defined as α = α (1 + εθ(η)), (10) satisfaction of Eq. (3) is evident, however Eqs. (4), (5), (6), and (7) lead to the following non-dimensional form (1 + 2ηγ)f + ff f 2 + 2γf (1 + 2ηγ)1/2 γλf 2 + λ(1 + 2ηγ) 3/2 f f + P 2 0 M(f P 0 ) = 0, (11) (1 + εθ)((1 + 2ηγ)θ + γθ ) + P r(fθ f (θ + ϕ)) + ε(1 + 2ηγ)θ 2 + P rβθ = 0, (12) (1 + 2ηγ)m + 2γm + Scfm ScL 1 mn 2 = 0, (13) δ(1 + 2ηγ)n + 2γδn + Scfn + ScL 1 mn 2 = 0. (14) The transformed boundary conditions are: f(0) = 0, f (0) = 1, m (0) = L s m(0), θ (0) = 1 ϕ, δn (0) = L s n(0) at η = 0, f ( ) P 0, m( ) 1, θ( ) 0, n( ) 1, as η. (15) Different dimensionless parameters appearing in Eqs. (11) (15), are defined as δ = D B, P r = ν, γ = D A α Sc = ν 3/2 ΓU0 z, λ = D, A 2νl 3/2 νl U 0 R 2, L 1 = la2 l 1 U 0, L s = l s D A νl U 0, M = σb2 0l U 0 ρ, P 0 = V 0 U 0, ϕ = t s, β = Ql ρc p U 0. (16)

4 No. 6 Communications in Theoretical Physics 691 Assumption that diffusion coefficient of chemical species A and B are of analogous magnitude leads us to suppose that D A and D B are identical provided δ = 1. Thus, we have m(η) + n(η) = 1. (17) Using Eq. (17), Eqs. (13) and (14) take the form (1+2ηγ)m +2γm +Scfm ScL s m(1 m) 2 = 0, (18) and the corresponding boundary conditions for Eq. (18) are m (0) = L s m(0), m( ) 1. (19) Skin friction coefficient and Local Nusselt number in dimensional form are C f = 2τ w zq w ρww 2, Nu = α(t w T 0 ), (20) where τ w and q w are the shear stress and surface heat flux given by ( w τ w = µ r + Γ ( w ) 2 ) 2 r, r=r ( T ) q w = α r. (21) r=r The dimensionless form of skin friction coefficient and local Nusselt number are represented as C f Re 2 where Re = U 0 z 2 /νl. = f (0) + λ 2 f 2 (0), NuRe 1/2 = θ (0), (22) 3 Solution Methodology Non-linear system of ordinary differential equations obtained from partial differential equations, is solved numerically by shooting method with integration scheme of Runga Kutta method of order four. To solve the system (11), (12) and (18) with respective boundary conditions, first we have to convert it into a system of first order ordinary differential equations. Afterwards, the missing conditions ι 1, ι 2 and ι 3 are supposed initially and then refined iteratively by Newtons iterative scheme subject to the tolerance of For numerical computations, the largest value of η has been taken as η = 8 instead of η because values of η greater than η = 8 do not make a considerable difference. The results are further strengthened by using bvp4c, a built-in MATLAB function. For first order initial value problem we have denoted f by y 1, θ by y 4 and m by y 6 to have the following equations. y 1 = y 2 y 1 (0) = 0, y 2 = y 3 y 2 (0) = 1, y 3 = (y2 2 y 1 y 3 2γy 3 3/2(1 + 2ηγ) 1/2 γλy 2 3 P M(y 2 P 0 ) (1 + 2ηγ) + λ(1 + 2ηγ) 3/2 y 3 y 3 (0) = ι 1, y 4 = y 5 y 4 (0) = 1 ϕ, y 5 = [P r(y 1y 5 y 2 (y 4 + ϕ)) + (1 + εy 4 )γy 5 + ε(1 + 2ηγ)y P rβy 4 ] (1 + εy 4 )(1 + 2ηγ) y 5 (0) = ι 2, y 6 = y 7 y 6 (0) = ι 3, y 7 = ScLy 6(1 y 6 ) 2 Scy 1 y 7 2γy ηγ Table 1 Comparison of skin friction coefficient with the previously published work for different values of P 0 when γ = 0, λ = 0, M = 0, θ = 0, m = 0 for f (0). A Ref. [28] Ref. [29] Ref. [30] Present This system of first order initial value problem with the supposition of missing initial conditions is solved with Runge Kutta method of order four. Newton iterative scheme helps us to refine the initial guesses until the following stoping criteria is met y 7 (0) = Ly 6 (0). (23) max{ y 2 (8) P 0, y 4 (8) 0, y 6 (8) 1 } < ϵ. (24) Numerical solutions are obtained and for the purpose of verification of the code, a comparison with some published work is presented in Table 1, where a very good agreement in the results can be seen. 4 Results and Discussions Numerical solutions for the derived system of equations with associated boundary conditions are obtained by using shooting method. The results are verified by MATLAB built-in function bvp4c. The numerical results of skin friction coefficient and Local Nusselt number for various values of different parameters are enumerated in Table 2. It is evident from the table that friction factor increases with the increasing values of magnetic parame-

5 692 Communications in Theoretical Physics Vol. 67 ter M and curvature parameter γ. A reverse behaviour is noticed in case of dimensionless Weissenberg number λ and stretching ratio parameter P 0 as both have decreasing tendency for the skin friction coefficient. A rise in values of curvature parameter γ, magnetic parameter M and Weissenberg number λ, depreciates the local Nusselt number while raising the stretching ratio parameter P 0 enhances the rate of heat transfer in fluid. Table 2 Numerical values of Skin friction coefficient and local Nusselt number for different values of parameters when ε = 0.2, P r = 0.72, β = 0.1, and ϕ = 0.3. C f Re/2 NuRe 1/2 γ λ P 0 M Shooting bvp4c Shooting bvp4c Table 3 Numerical values of local Nusselt number for different values of parameters when γ = 0.3, λ = 0.2 and P 0 = 0.2, M = 0.6. NuRe 1/2 ϵ P r ϕ β Shooting bvp4c In Table 3, the effect of thermal conductivity parameter ε, Prandtl number P r, thermal stratification parameter ϕ and heat generation/absorption parameter β on heat transfer rate are presented. It is observed from the table that increase in thermal conductivity will reduce the heat transfer rate and hence the temperature increases. On the other hand Nusselt number enhances with the enhancement of Prandtl number P r, thermal stratification parameter ϕ, and heat generation/absorption parameter β. Fig. 2 Influence of γ on f. The effects of curvature parameter γ, dimensionless Weissenberg number W e, magnetic parameter M, and stretching ratio parameter P 0 on velocity profile are displayed through Figs In Fig. 2, values of velocity function f (η) and the boundary layer thickness increase by increasing γ. Increase in γ implies reduction in radius of cylinder and hence the speed of the fluid got fasten. Figure 3 elucidates the behaviour of Weissenberg number on velocity profile. Velocity reduces slightly when λ is raised. As we know that Weissenberg number λ is the ratio of relaxation time to specific process time. Increase in relaxation time will increase the resistance of the fluid and resultantly velocity reduces. In Fig. 4, influence of magnetic parameter M on velocity profile is displayed,

6 No. 6 Communications in Theoretical Physics 693 which shows the reduction in speed of the fluid with increasing M. It happens due to the retarding force known as Lorenth force, which is a resistive force. Influence of stretching ratio parameter P 0 on velocity profile is displayed in Fig. 5. As the values of P 0 increase, the velocity and the boundary layer thickness increase. Fig. 6 Influence of β on θ. Fig. 3 Influence of λ on f. Fig. 7 Influence of ϵ on θ. Fig. 4 Influence of M on f. Fig. 8 Influence of γ on θ. Fig. 5 Influence of P 0 on f. Figures 6 12 show the variations of different parameters on temperature profile. Figure 6 demonstrates the effect of heat generation/absorption coefficient. As expected temperature increases with the increment of heat generation. Same effect is seen for the thermal conductivity parameter ε. As increase in thermal conductivity will definitely raise the temperature of the fluid and this can be confirmed from Fig. 7. Influence of curvature parameter γ on temperature profile is depicted in Fig. 8. As increase in curvature parameter will reduce the radius of the cylinder, which causes less resistance to the fluid flow but due to the smaller diameter the resistive force between the fluid

7 694 Communications in Theoretical Physics Vol. 67 and wall of the cylinder is enhanced and consequently the temperature increases near the wall. Fig. 10. As we know that P 0 is the ratio of far away velocity to the surface velocity and it is also obvious that surface velocity is always greater than ambient velocity, so increase in stretching ratio implies reduction in temperature distribution. Fig. 9 Influence of M on θ. Fig. 12 Influence of P r on θ. Fig. 10 Influence of P 0 on θ. Fig. 13 Influence of γ on g(η). Fig. 11 Influence of ϕ on θ. Figure 9 is plotted to see the impact of magnetic parameter M on dimensionless temperature. Magnetic parameter produces a resistive force in the fluid motion due to Lorentz force. This frictional force enhances the temperature of the fluid. Variation of velocity stretching ratio parameter P 0 on temperature profile is illustrated in Fig. 14 Influence of L 1 on g(η). Figure 11 is plotted to portray the behaviour of thermal stratification parameter ϕ on temperature profile. En-

8 No. 6 Communications in Theoretical Physics 695 hancement in thermal stratification yields higher the density of the fluid in lower region and it increases on upper region. So the temperature differences deliberately reduces between the cylinder and ambient fluid and this effect clearly elaborated through Fig. 11. Figure 12 indicates that the temperature of the fluid decreases for increasing values of P r. It is contemplated that raising the Prandtl number decreases the thermal diffusivity and hence the thermal boundary layer, which eventually enhances the rate of heat transfer. Influence of Schmidt number Sc on mass distribution is shown in Fig. 16. Accession of mass concentration is noted for raising Schmidt number. Schmidt number is actually the ratio of momentum diffusivity to mass diffusivity. Increase in Schmidt number will enhance the momentum diffusivity, which resultantly raises the mass distribution. Figure 17 is plotted to see the effect of Weissenberg number λ on temperature distribution. A minor increase in temperature distribution is observed for increasing Weissenberg number. Fig. 15 Influence of L s on g(η). Fig. 17 Influence of λ on θ(η). 5 Concluding Remarks In the present article, we have numerically investigated the effect of stagnation point flow of Williamson fluid over a stretching cylinder with the effect of homogenous/heterogenous reaction, magnetohydrodynamics and variable thermal conductivity. Shooting technique is summoned to address the system of equations with high nonlinearity. The main findings of the present investigation are summarized as below. Fig. 16 Influence of Sc on g(η). Figures are plotted to analyze the behaviour of concentration profile subject to variation of different parameters. Figure 13 shows the increasing effect of curvature parameter on concentration. Increasing curvature certainly reduces the radius of cylinder and hence more fluid is expected to flow, which will enhance the mass transfer. Figures 14 and 15 portray the effect of strength of homogeneous/heterogeneous reaction on concentration profile. It is observed that both L 1 and L S have decreasing behaviour but the boundary layer thickness is increased. Homogenous and heterogenous reactions show opposing behaviour on concentration distribution. Temperature field is growing function of thermal conductivity and heat generation coefficient. Prandtl number and thermal stratification have the similar impact on local Nusselt number. Increasing the values of magnetic parameter boost the temperature and its allied boundary layer thickness. Mass distribution is increasing function of Schmidt number.

9 696 Communications in Theoretical Physics Vol. 67 References [1] Q. Hussain, S. Asghar, T. Hayat, and A. Alsaedi, Appl. Math. Mech. Engl. Ed. 36 (2015) 499. [2] Y. Lin, L. Zheng, and G. Chen, Power Technol. 274 (2015) 324. [3] T. Hayat, Anum Shafiq, A. Alsaedi, and S. Asghar, AIP Adv. 5 (2015) [4] T. Hayat, M. Waqas, S. A. Shehzad, and A. Alsaedi, Int. J. Num. Met. Heat Fluid Flow 26 (2016) 214. [5] X. Si, X. Zhu, L. Zheng, X. Zhang, and P. Lin, Int. J. Heat Mass Transfer. 92 (2016) 979. [6] M. Y. Malik, M. Bibi, F. Khan, and T. Salahuddin, AIP Adv. 6 (2016) [7] M. G. Reddy, J. Eng. Phys. Thermophy. 88 (2015) 240. [8] M. Ghalambaza, A. Behseresht, J. Behseresht, and A. Chamkha, Adv. Powder Tech. 26 (2015) 224. [9] G. J. Li, J. Ma, and B. W. Li, J. Heat Transfer. 137 (2015) [10] I. L. Animasaun and N. Sandeep, Power Technol. 301 (2016) 858. [11] R. V. Williamson, Ind. Engr. Chem. Res. 11 (1929) [12] M. Y. Malik and T. Salahuddin, Int. J. Nonlinear Sci. Num. Sim. 16 (2015) 161. [13] M. Y. Malik, T. Salahuddin, A. Hussain, S. Bilal, and M. Awais, AIP Adv. 5 (2015) [14] T. Salahuddin, M. Y. Malik, A. Hussain, S. Bilal, and M. Awais, J. Mag. Mag. Mat. 5:doi.org/ /j.jmmm (2015). [15] T. Hayat, A. Shafiq, and A. Alsaedi, Alex. Engr J. 55 (2016) [16] B. C. Prasannakumara, B. J. Gireesha, R. S. R. Gorla, and M. R. Krishnamurthy, J. Aerosp. Eng :1-10 (2016). [17] T. Hayat, S. Bibi, M. Rafiq, A. Alsaedi, and F. M. Abbasi, J. Mag. Mag. Mat. page doi.org/ /j. jmmm (2015). [18] M. M. Bhatti and M. M. Rashidi, J. Mol. Liq. doi: /j.molliq , (2016). [19] T. Hayat, S. Qayyum, M. Imtiaz, and A. Alsaedi, PLOs ONE 11 (2016) e [20] T. Hayat, M. Imtiaz, A. Alsaedi, and S. Almezal, J. Mag. Mag. Mat. 401 (2016) 296. [21] J. V. R. Reddy, V. Sugunamma, and N. Sandeep, Int. J. Engr. Res. Africa 20 (2016) 130. [22] S. Mansur, A. Ishak, and I. Pop, J. App. Fluid Mech. 9 (2016) [23] P. K. Kameswaran, S. Shaw, P. Sibanda, and P. V. S. N. Murthy, Int. J. Heat Mass Tran. 57 (2013) 465. [24] M. R. Eid, J. Mol. Liq. 220 (2016) 718. [25] W. Ibrahim, Prop. Power Res. 5 (2016) 164. [26] O. D. Makinde, T. Iskander, F. Mabood, W. A. Khan, and M. S. Tshehla, J. Mol. Liq. 221 (2016) 778. [27] H. Rosali, A. Ishak, R. Nazar, and I. Pop, Prop. Power Res. 5 (2016) 118. [28] F. Mabood, S. M. Ibrahim, M. M. Rashidi, M. S. Shadloo, and G. Lorenzini, Int. J. Heat Mass Tran. 93 (2016) 674. [29] T. R. Mahapatra and A. S. Gupta, Heat Mass Trans. 38 (2002) 517. [30] R. Nazar, N. Amin, D. Filip, and I. Pop, Int. J. Eng. Sci. 42 (2004) [31] M. Y. Malik and T. Salahuddin, Int. J. Nonlinear Sci. Num. Sim. 16 (2015) 161.

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