Chemical Engineering Research Bulletin 21(2019) 36-45 Available online at http://banglajol.info/index.php/CERB

NUMERICAL SOLUTION OF MHD NANOFLUID OVER A STRETCHING SURFACE WITH CHEMICAL REACTION AND VISCOUS DISSIPATION G. Narender1*, Santoshi Misra2 and K. Govardhan3

1Department of H&S(Mathematics), CVR College of Engineering, Hyderabad, T.S., India 2Departmet of Mathematics, St. Ann's College for Women, Hyderabad, T.S., India 3Departmetof Mathematics, GITAM University, Hyderabad Campus, T.S., India

Received: 28 March 2019 received in revised form17 September 2019

Abstract: The main objective of this paper is to focus on a numerical study of chemical reaction and viscous dissipation effects on the steady state boundary layer flow of MHD nanofluid past the horizontally stretching sheet with the existence of nanoparticles. A proper similarity transformation is utilized to convert the boundary layer equations into the nonlinear and coupled ordinary differential equations. These ODEs are sorted out numerically by applying the shooting mechanism. Graphical representations are also included to explain the effect of evolving parameters against the above- mentioned distributions. Significance of different physical parameters on dimensionless velocity, temperature and concentration are elaborated through graphs and tables. For increasing values of Eckert number, the temperature profile increases whereas the chemical reaction parameter increases, the boundary layer thickness decreases.

Keywords: MHD Nanofluid, Chemical reaction parameter; Stretching sheet; Eckert number; Velocity slip Parameter.

DOI: https://doi.org/10.3329/cerb.v21i1.47370

Introduction layer flow of nanofluid past a stretched sheet. In future, advancement in nano-technology is The study of flow and heat transfer generated by expected for making unbelievable changes in our means of stretching medium has plenty of lives. A very big number of researchers are significance in numerous industrialized working in this area due to its great use in the developments, (e.g., in the process of rubber and engineering and its linked areas. In the process of plastic sheets manufacturing, upgrading the solid air cleaning, development of microelectronics, materials like crystal, turning fibers etc.). The most safety of nuclear reactors etc., heat and mass widely used coolant liquid among them is water. In transfer of thermophoretic magnetohydrodynamic above cases, flow and heat transfer investigation is flow consumes prospective uses. Choi [6] was the of major importance because final product quality first who introduced the idea of “nanofluids" and be determined to bulk level based on coefficient of presented the report on the heat transfer properties skin friction and heat transfer surface rate. of nano-fluids. The thorough exposure on Numerous investigators talked over different traits thermophoretic flow was examined by Derjaguin of stretching flow problem. Some of them are and Yalamov [7]. Heat and mass transfer of MHD Crane [1], Chaim [2], Liao and Pop [3], Khan and thermophoretic stream above plane surface was Sanjayanand [4] and T. Fang et al. [5]. also studied by Issac and Chamkha [8]. Thermophoresis effect on aerosol particles was Currently, in the fields of engineering and fluid investigated by Tsai [9]. science, heat transfer and boundary layer flow of nanofluid are the thrust areas of research. Many In fluid temperature, no doubt, viscous dissipation researchers examined the convective boundary produces a considerable ascend. This would happen

*Corresponding author email: [email protected] ©Bangladesh Uni. of Engg. & Tech

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because of change in kinetic motion of fluid into studies related to magnetohydrodynamic have thermal energy. Viscous dissipation is unavoidable been discussed in Refs. [22,23,24]. in case of flow field in high gravitational field. Viscous flow past a nonlinearly stretching sheet In this article, the findings of M. G. Reddy [25] was deliberated by Vajravelu [10]. For external have been reproduced and extended by considering natural convention flow over a stretching medium, the viscous dissipation and chemical reaction. The the effect of viscous dissipation was also studied by acquired arrangement of partial differential Mollendroff and Gebhart [11], whereas the impact equations is transformed into non-linear and of Joule heating and viscous dissipation on the coupled ordinary differential equations by using a forced convection flow with thermal radiation was similarity transformation. With the help of presented by Duwairi [12]. shooting technique, numerical solution of the system of ordinary differential equations is The chemical reaction with the diffusion of species achieved. for the boundary layer fluid have numerous applications in atmosphere pollution, water, fluids 2. Problem Formulation relevant to atmosphere and many other problems Consider the numerical investigation of MHD of chemical engineering. For boundary layer boundary layer flow of anin compressible laminar flow of reactive chemically species with nanofluid. The flow is two-dimensional, steady, the diffusion which are used by a body over the laminar, viscous flow of an electrically conducting surface considered by Chamber and Young [13]. electrically conducting fluid towards a stretching For non-Newtonian fluids and their solution for surface with chemically reactive species the species of diffusion with chemical reactive in a undergoing chemical reaction is considered. The flow over a stretching sheet with porous medium schematic diagram of the system under reported by Akyildiz et al. [14]. Cortell [15] also investigation is shown in the Figure 1. discuss the two types of viscoelastic fluid over a The plate has been stretched with velocity ( ) = porous stretching sheet with the chemically , ( > 0) along direction. In addition, fluid reactive species. Hiemenz flow through porous 𝑢𝑢𝑤𝑤 𝑥𝑥 is flowing in the presence of magnetic field.The media considered by Chamka and Khaled [16] magnetic𝑎𝑎𝑎𝑎 𝑎𝑎 field is supposed𝑥𝑥 − to be applied along the with the presence of magnetic field. Heat transfer direction.The temperature at surface is , with steady condition considered by Sriramalu et , represent fluid velocity, nanoparticle al. [17] for incompressible viscous fluid with 𝑦𝑦 − 𝑇𝑇𝑓𝑓 concentration at surface respectively. porous type species over a stretching surface. 𝑢𝑢𝑤𝑤 𝐶𝐶𝑤𝑤 Khan et al. [18] discussed MHD viscoelastic fluid, The following systems of equations are transfer of mass and heat over a permeable incorporated for mathematical model [25]. shows stretching surface with stress work and energy that in the presence of magnetic field over the dissipation. The fluid on stretching surface close surface, the governing equations of conservation with stagnation-point discussed by Tripathy et al. of momentum, energy, mass and nanoparticle [19]. Seddeek and Salem [20] observed that the fraction, under the boundary layer approximation, mass and heat transfer distribution on stretching are as follows: type surface with thermal diffusivity and variable Continuity equation: viscosity. M. Ali et al. [21] studied the effects of magnetohydrodynamic on heat transfer and + = 0 (1) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 convective mass flow. They concluded that mass 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 flux of particle is as small as the stream velocity Momentum equation: 1 2( ) and temperature profile are not influenced by + = 0 ( ) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜎𝜎𝐵𝐵 𝑥𝑥 2 2 thermophysical phenomenon experienced by 𝜕𝜕𝑥𝑥 𝜕𝜕𝜕𝜕 𝜌𝜌𝑓𝑓 𝜕𝜕𝜕𝜕 𝜌𝜌𝑓𝑓 +𝑢𝑢 2𝑣𝑣+ 2 − − 𝑢𝑢 (2) relatively small number of particles. Some further 𝜕𝜕 𝑢𝑢 𝜕𝜕 𝑢𝑢 𝜈𝜈 �𝜕𝜕𝑥𝑥 𝜕𝜕𝑦𝑦 �

Chemical Engineering Research Bulletin 21(2019) 36-45 38

Figure 1: Geometry of the flow under consideration.

2 2 2 1 0 ( ) + = ( ) + 2 + 2 = + , = , 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜎𝜎𝐵𝐵 𝑥𝑥 𝜕𝜕 𝑣𝑣 𝜕𝜕 𝑣𝑣 𝜕𝜕𝜕𝜕 𝑢𝑢 𝜕𝜕𝜕𝜕 𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜌𝜌𝑓𝑓 𝜕𝜕𝜕𝜕 − 𝜌𝜌𝑓𝑓 𝑣𝑣 𝜈𝜈 �𝜕𝜕𝑥𝑥 𝜕𝜕𝑦𝑦 � 𝑤𝑤 = ,𝑤𝑤 𝑢𝑢 𝑢𝑢 𝐿𝐿ℎ𝜕𝜕𝜕𝜕 𝑣𝑣 𝑣𝑣 ⎫ (3) 𝜕𝜕𝜕𝜕 = at = 0 ⎪ −𝑘𝑘 𝜕𝜕𝜕𝜕 �𝑇𝑇𝑓𝑓 − 𝑇𝑇� = 0, = 0, , as ∞ Energy equation: 𝑤𝑤 ∞ ∞ ⎬ 𝐶𝐶 𝐶𝐶 𝑦𝑦 ⎪ (6) 2 2 2 2 𝑢𝑢 𝑣𝑣 𝑇𝑇 → 𝑇𝑇 𝐶𝐶 → 𝐶𝐶 𝑦𝑦 → ⎭ The radiative heat flux , by using the Rosseland + = + + 2 + 2 + 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜇𝜇 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑇𝑇 approximation for radiation, can be written as 2 2 𝑟𝑟 𝑢𝑢 𝜕𝜕𝜕𝜕 𝑣𝑣 𝜕𝜕𝜕𝜕 𝐶𝐶𝑝𝑝 ��𝜕𝜕𝜕𝜕 � �𝜕𝜕𝜕𝜕 � � 𝛼𝛼 �𝜕𝜕𝑥𝑥 𝜕𝜕𝑦𝑦 � 4 4 𝑞𝑞 τ + + + = (7) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝐷𝐷∞𝑇𝑇 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝛼𝛼 𝜕𝜕𝑞𝑞𝑟𝑟 3 ∗ − 𝜎𝜎 𝜕𝜕𝑇𝑇 �𝐷𝐷𝐵𝐵 �𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 � 𝑇𝑇 ��𝜕𝜕𝜕𝜕 � �𝜕𝜕𝜕𝜕 � �� − 𝜌𝜌𝑓𝑓 � 𝜕𝜕𝜕𝜕 � ∗ 𝑞𝑞where𝑟𝑟 𝑘𝑘 and𝜕𝜕𝜕𝜕 stand for the Stefan-Boltzmann (4) constant ∗ and coefficient∗ of mean absorption. 𝜎𝜎 𝑘𝑘 4 Concentration equation: Expansion of about ∞ by making use of Taylor's series is: 2 2 2 2 3 𝑇𝑇 𝑇𝑇 2 4 4 4 ∞ 1 12 ∞ 2 + = 2 + 2 + 2 + 2 = + ( ) + ( ) + ∞ ∞ 1! ∞ 2! ∞ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝐶𝐶 𝜕𝜕 𝐶𝐶 𝐷𝐷𝑇𝑇 𝜕𝜕 𝐶𝐶 𝜕𝜕 𝐶𝐶 𝑇𝑇 𝑇𝑇 24 ∞ 3 24 4 0𝜕𝜕(𝜕𝜕 𝜕𝜕∞𝜕𝜕) 𝐵𝐵 𝜕𝜕𝑥𝑥 𝜕𝜕𝑦𝑦 𝑇𝑇 𝜕𝜕 𝑥𝑥 𝜕𝜕 𝑦𝑦 (5) ( ) + ( ) 𝑢𝑢 𝑣𝑣 𝐷𝐷 � � � � − 𝑇𝑇3! 𝑇𝑇 ∞ 𝑇𝑇4−! 𝑇𝑇 ∞ 𝑇𝑇 − 𝑇𝑇 (8) 𝑇𝑇 4 4 where𝑘𝑘 𝐶𝐶 − 𝐶𝐶 and are the components of velocity Disreg𝑇𝑇 arding− 𝑇𝑇 the higher𝑇𝑇 − 𝑇𝑇 order terms, = ∞ + 3 respectively in the and directions, is the fluid 4 ∞ ( ∞) 𝑢𝑢 𝑣𝑣 4 𝑇𝑇 𝑇𝑇 pressure, is the density of base fluid, is the 3 𝑇𝑇 𝑇𝑇=−4𝑇𝑇 (9) 𝑥𝑥 𝑦𝑦 𝑝𝑝 𝜕𝜕𝑇𝑇 𝜕𝜕𝜕𝜕 density of the𝑓𝑓 particles, is the kinematic viscosity𝑝𝑝 𝜌𝜌 𝜌𝜌 ⇒Using𝜕𝜕𝜕𝜕 (9) in𝑇𝑇 (7)𝜕𝜕𝜕𝜕 and the differentiate w.r.t. , we get of the base fluid, is electrical conductivity, is 16 3 2 𝜈𝜈 = ∞ (10) density, is temperature, is thermal diffusivity, 3 ∗ 2 𝑦𝑦 𝜕𝜕𝑞𝑞𝑟𝑟 𝜎𝜎 𝑇𝑇 𝜕𝜕 𝑇𝑇 ( ) 𝜎𝜎 𝜌𝜌 ∗ = is the ratio of nanoparticle heat capacity We𝜕𝜕𝜕𝜕 use− similarity𝜅𝜅 𝜕𝜕𝑦𝑦 transformation to solve Eqs. (1 – ( ) 𝑇𝑇 𝛼𝛼 𝜌𝜌𝜌𝜌 𝑝𝑝 5) 𝜏𝜏and the𝜌𝜌𝜌𝜌 𝑓𝑓base fluid heat capacity, is Brownian = ( ), ( ) = , 𝑇𝑇∞−𝑇𝑇𝑚𝑚 diffusion coefficient, is thermophoretic𝐵𝐵 𝐷𝐷 ∞ 𝑇𝑇 −𝑇𝑇𝑚𝑚 𝜓𝜓( )√=𝑎𝑎𝑎𝑎 𝑥𝑥𝑥𝑥 𝜂𝜂, 𝜃𝜃=𝜂𝜂 , (11) diffusion coefficient, is dynamic𝑇𝑇 viscosity, is 𝐷𝐷 𝐶𝐶−𝐶𝐶 ∞ 𝑎𝑎 specific heat at constant pressure. The boundary the velocity𝑤𝑤 component of stream function which is 𝜇𝜇 𝑐𝑐 𝛽𝛽 𝜂𝜂 𝐶𝐶 −𝐶𝐶 𝜂𝜂 𝑦𝑦�𝜈𝜈 conditions for Eqs. (1 – 5) are defined as

39 Chemical Engineering Research Bulletin 21(2019) 36-45

= , = (12) ℎ = (18) (( ∞) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑥𝑥𝑞𝑞𝑚𝑚 𝑥𝑥 𝐷𝐷𝐵𝐵 𝐶𝐶𝑤𝑤 −𝐶𝐶 𝑢𝑢So, we𝜕𝜕𝜕𝜕 have𝑣𝑣 − 𝜕𝜕𝜕𝜕 𝑆𝑆where, is the shear stress, is the wall mass = ′( ), = ( ) (13) flux from the surface, and is the heat flux at the 𝜏𝜏𝑤𝑤 𝑞𝑞𝑚𝑚 wall surface, given by: 𝑤𝑤 where𝑢𝑢 𝑎𝑎 𝑎𝑎prime𝑓𝑓 𝜂𝜂 shows𝑢𝑢 differentiation−√𝑎𝑎𝑎𝑎𝑓𝑓 𝜂𝜂 with respect to 𝑞𝑞 . = , = (19) The equation continuity (1) is satisfied identically. 𝜕𝜕𝜕𝜕 =0 𝜕𝜕𝜕𝜕 =0 𝜂𝜂 𝑤𝑤 𝜕𝜕𝜕𝜕 𝑚𝑚 𝐵𝐵 𝜕𝜕𝜕𝜕 The governing Eqs. (2) – (5) are reduced into the Using𝑞𝑞 −the𝑘𝑘 �dimensionless�𝑦𝑦 𝑞𝑞 variables,−𝐷𝐷 � �we𝑦𝑦 get following nonlinear ODEs. ℎ = 1 + ′(0), ′(0), ′′′ ′′ ′ 2 ′ + ( ) = 0 (14) 𝑁𝑁𝑁𝑁𝑥𝑥 𝑆𝑆 𝑥𝑥 �𝑅𝑅𝑥𝑥 − 𝑅𝑅 𝜃𝜃 �𝑅𝑅𝑥𝑥 − 𝛽𝛽 4 ′′ 𝑓𝑓1 + 𝑓𝑓𝑓𝑓 −′′ +𝑓𝑓 Pr−[ 𝑀𝑀′𝑓𝑓+ ′ ′ + ( ′)2 + = (0) (20) 3 ( ′′)2] = 0 (15) 𝑓𝑓 𝑥𝑥 � 𝑅𝑅� 𝜃𝜃 ⁡𝑓𝑓𝜃𝜃 𝑁𝑁𝑁𝑁𝜃𝜃 𝛽𝛽 𝑁𝑁𝑁𝑁 𝜃𝜃 𝐶𝐶where�𝑅𝑅 −𝑓𝑓 denotes the Reynolds number and is 𝐸𝐸𝐸𝐸 𝑓𝑓 expressed as: + + = 0 (16) 𝑅𝑅𝑅𝑅𝑥𝑥 ′′ ′ 𝑁𝑁𝑁𝑁 ′′ ( ) 𝛽𝛽 𝐿𝐿𝐿𝐿𝐿𝐿𝛽𝛽 𝑁𝑁𝑁𝑁 𝜃𝜃 − 𝐿𝐿𝐿𝐿𝛾𝛾𝛽𝛽 = . with the boundary conditions 𝑥𝑥𝑈𝑈𝑤𝑤 𝑥𝑥 𝑥𝑥 (0) = , ′(0) = 1 + A ′′(0), 3.𝑅𝑅𝑅𝑅 Numerical𝑣𝑣 Treatment ′(0) = 1 (0) , = 0, (0) = 1 𝑓𝑓 𝑆𝑆 𝑓𝑓 𝑓𝑓 In this section, the scheme for the numerical at = 0, ⎫ (17) 𝜃𝜃 −𝐵𝐵𝐵𝐵� − 𝜃𝜃 � 𝛽𝛽 solution of the system of three coupled ODEs (14) ′(∞) 0, (∞) 0, (∞) 0 ⎪ 𝜂𝜂 – (16) with BCs (17) will be discussed. Because of as ∞ ⎬ 𝑓𝑓 → 𝜃𝜃 → 𝛽𝛽 → ⎪ its efficiency, the shooting technique has been A, , Pr The dimensionless𝜂𝜂 → constants , ,⎭ , , performed to apply. Firstly, it is noticed that , , , , represent the magnetic parameter, 𝑀𝑀 𝑆𝑆 𝑅𝑅 𝑁𝑁𝑁𝑁 heurist infinity for the independent variable chosen a suction parameter,the velocity slip parameter, the 𝑁𝑁𝑁𝑁 𝐸𝐸𝐸𝐸 𝐿𝐿𝐿𝐿 𝛾𝛾 𝐵𝐵𝐵𝐵 as . radiation parameter, the Prandtl number, the The comment on the choice on the , for 𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 Brownian motion parameter, the thermophoresis solving is presented at the end of the section. parameter, the Eckert number, the Lewis number, 𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 Equation (14) is solved with ′′(0) = , assumed the chemical rate parameter, the Biot number number using the initial conditions respectively, which are defined as, 𝑓𝑓 𝑟𝑟 2 (0) = , (0) = 1 + × , (0) = (21) 0 = , = , = , ′ ′′ 𝜎𝜎𝐵𝐵 𝑣𝑣𝑤𝑤 𝑎𝑎 𝑓𝑓 is iteratively𝑆𝑆 𝑓𝑓 found using𝜆𝜆 𝑟𝑟 𝑓𝑓Newton’s𝑟𝑟 method using 𝑀𝑀 3 𝑆𝑆 − 𝐴𝐴 𝐿𝐿 �� � ′( ) 𝜌𝜌4𝜌𝜌 ∞ √𝑎𝑎𝑎𝑎 𝜈𝜈 ′ = , = , 𝑟𝑟 ( ) = which is obtain by solving, 3 ∗ 𝜕𝜕𝑓𝑓 𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑇𝑇 𝜎𝜎 𝜈𝜈 𝑚𝑚𝑚𝑚𝑚𝑚 𝑅𝑅 ∗ ( 𝑃𝑃𝑃𝑃 ) 𝐹𝐹′′′ 𝜂𝜂= 2 ′ ′ 𝜕𝜕′′𝜕𝜕 ′′ ′ + ′ (22) = 𝑘𝑘 𝑘𝑘 ∞ 𝛼𝛼, 𝐵𝐵 𝑤𝑤 𝜏𝜏𝐷𝐷 𝐶𝐶 − 𝐶𝐶 with𝐹𝐹 (𝑓𝑓0𝐹𝐹) =−0𝑓𝑓,𝐹𝐹 (−0)𝑓𝑓=𝐹𝐹 and𝑀𝑀𝑓𝑓 (0) = 1 (23) 𝑁𝑁𝑁𝑁 2 = 𝜈𝜈 ∞ , = , ′ ′′ 𝐹𝐹 𝐹𝐹 𝜆𝜆 𝐹𝐹 𝑇𝑇 𝑓𝑓 ∞ 𝑤𝑤 ∞ After finding ( ) we solved the equations (15) 𝜏𝜏𝐷𝐷 �𝑇𝑇 − 𝑇𝑇 � 𝑢𝑢 1 2 0 ( ∞) ℎ( ) and (16) with the initial conditions. 𝑁𝑁𝑁𝑁= , = 𝐸𝐸𝐸𝐸 , =𝑓𝑓 𝑓𝑓 . 𝜈𝜈𝑇𝑇 𝜌𝜌 �𝑇𝑇 − 𝑇𝑇⁄ � 𝑓𝑓 𝜂𝜂 𝜈𝜈 𝑘𝑘 𝑈𝑈 𝐶𝐶𝑤𝑤 −𝐶𝐶 𝜈𝜈⁄𝑎𝑎 y2 𝐿𝐿𝐿𝐿 𝐷𝐷𝐵𝐵 𝛾𝛾 𝜈𝜈 𝐵𝐵𝐵𝐵 𝑘𝑘 ′ y 2 ′′2 Nusselt number , skin friction coefficient , 1 Pr y2+Nb y2y4+Nty2+Ec ′ 4R and Sherwood number ℎ , are expressed as: y2 1+ 𝑁𝑁𝑁𝑁𝑥𝑥 𝐶𝐶𝑓𝑓 = − �𝑓𝑓 3 𝑓𝑓 � (24) y′ y 𝑥𝑥 3 ⎛ � 4 � ⎞ 𝑆𝑆 ⎛ ′ ⎞ = , = 2 , y Nb 1 (( ∞) ⎜ 4⎟ ⎜ Le y y + Le y ⎟ 𝑥𝑥𝑞𝑞𝑤𝑤 𝜏𝜏𝑤𝑤 ⎜ 4 Nt 2 γ 3 ⎟ 𝑁𝑁𝑁𝑁𝑥𝑥 𝑘𝑘 𝑇𝑇𝑤𝑤 −𝑇𝑇 𝐶𝐶𝑓𝑓 𝜌𝜌𝑈𝑈𝑤𝑤 𝑎𝑎𝑎𝑎𝑎𝑎 ⎝ ⎠ ⎝ − 𝑓𝑓 − ⎠

Chemical Engineering Research Bulletin 21(2019) 36-45 40

Associated boundary conditions in Eq. (17) can be 3.1. Code Validation written as For the validity of our results, the skin friction

y1(0) p1 factor, the local Nusselt number and Sherwood y (0) (p1 1) number have been compared with those already 2 = (25) y3(0) 1 published in literature as shown in Tables 3.1 and 𝐵𝐵𝐵𝐵 p − ⎛y4(0)⎞ � 2 � 3.2. From tables the results achieved by the present code are found convincingly very closed to the ′ ⎝Here 1⎠= (0) 2(0) = (0). published results [25].

𝑝𝑝 𝜃𝜃 𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝 𝛽𝛽 " 1, 2 are to be found satisfying end conditions Table 3.1. Comparison of results of (0)for various 1 0, 3 0 as . Adams Moultan fourth values of , and 𝑝𝑝 𝑝𝑝 −"(𝑓𝑓 ) order method (with the corresponding predictor) is 𝑦𝑦 → 𝑦𝑦 → 𝜂𝜂 → ∞ 𝑀𝑀 𝑆𝑆 𝐴𝐴M. G. Reddy [25] Present value used to solve the initial value problem. Assumed −𝒇𝒇 𝟎𝟎 0.5𝑴𝑴 0.2𝑺𝑺 0.1𝑨𝑨 1.13992 1.136756 values of 1and 2 are corrected using Newton 1.0 0.2 0.1 1.28430 1.282955 method. Derivatives of (∞, 1, 2) and 0.5 0.5 0.1 1.26830 1.265184 𝑝𝑝 𝑝𝑝 (∞, 1, 2) with respect to any 0.5 0.2 0.5 0.800331 0.8066351 𝜃𝜃 𝑝𝑝 𝑝𝑝 parameter ( 1 2 ) are found by solving the 𝜙𝜙 𝑝𝑝 𝑝𝑝 ′(0) ′(0) equation which are obtained by differentiating Table 3.2. Comparison of results of and 𝑝𝑝 𝑝𝑝 𝑜𝑜𝑜𝑜 𝑝𝑝 ′( ) ′( ) system (21). = for all = 1,2,3,4. These M. G. M.−𝜃𝜃 G. −𝛽𝛽 Present Present 𝜕𝜕𝑦𝑦𝑖𝑖 Reddy− 𝜽𝜽 𝟎𝟎 Reddy− 𝜷𝜷 𝟎𝟎 equations are 𝑌𝑌𝑖𝑖 𝜕𝜕𝜕𝜕 𝑖𝑖 value value 𝑵𝑵𝑵𝑵 𝑵𝑵𝑵𝑵 [25] [25] 0.08406 0.0822515 ′ = (2), 0.1 0.1 2.96584 2.9588820 1 76 300 0.08139 0.0792839 𝑌𝑌′ 𝑌𝑌 0.5 0.1 3.00253 3.0008590 2 = 4 [ ( ) (2) + ( (2) (4) 44 800 1+ −𝑃𝑃𝑃𝑃3 0.08384 0.0819882 𝑅𝑅𝑅𝑅 0.1 0.5 2.79651 2.7606710 𝑌𝑌2 4�)+2 � 𝐹𝐹𝐹𝐹2 2𝐼𝐼, 𝑌𝑌 𝑁𝑁𝑁𝑁 𝑦𝑦 𝑌𝑌 − 97 600 0.08110 0.0789362 𝑌𝑌′ 𝑦𝑦 𝑁𝑁𝑡𝑡𝑦𝑦 𝑌𝑌 0.5 0.5 2.97637 2.9675340 3 = (4), 44 500

𝑌𝑌 𝑌𝑌 ′ ′ 4. Graphical Results 4 = ( ) (4) 2 + (3) 𝑁𝑁𝑁𝑁 The objective is to inspect governing parameters 𝑌𝑌 −𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐼𝐼 𝑦𝑦 − 𝑌𝑌 𝐿𝐿𝑒𝑒𝑒𝑒𝑒𝑒 This system is solved𝑁𝑁𝑁𝑁 with three different sets of on the velocity, temperature and concentration initial conditions (0) = 0 for all = 1,2,3,4. distribution in this section. In every one of these Newton’s method is estimations, we have considered = 2, = 2, = 𝑦𝑦𝑖𝑖 𝑖𝑖 2, ======0.5, = 0.1 and 1 𝑃𝑃𝑃𝑃 𝑀𝑀 𝑅𝑅 1 1 = 5. 1 1 1 2 1 𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 𝑆𝑆 𝐴𝐴 𝐸𝐸𝐸𝐸 𝛾𝛾 𝐵𝐵𝐵𝐵 = 𝜕𝜕𝑦𝑦 𝜕𝜕𝑦𝑦 − (26) 2 𝑁𝑁𝑁𝑁𝑁𝑁 2 𝑂𝑂𝑂𝑂𝑂𝑂 3 3 3 𝐿𝐿𝐿𝐿 𝑝𝑝 𝑝𝑝 𝜕𝜕𝑝𝑝 𝜕𝜕𝑝𝑝 𝑦𝑦 𝜕𝜕𝑦𝑦1 𝜕𝜕𝑦𝑦2 =∞ �𝑝𝑝 � �𝑝𝑝 � − � � �𝑦𝑦 � 𝜕𝜕𝑝𝑝 𝜕𝜕𝑝𝑝 𝜂𝜂 It may be noticed that the choice of initial guess of 1, 2 is very crucial. Once we obtain solution for a set of physical parameters, a single parameter changed𝑝𝑝 𝑝𝑝 slightly to achieve convergence of Newton’s method. The choice of = 6 was more than enough for end condition. The 𝑚𝑚𝑚𝑚𝑚𝑚 convergence criteria are chosen to 𝜂𝜂be successive value agree up to 3 significant digits.

Figure 2. Dimensionless Velocity

𝑣𝑣𝑣𝑣 𝐴𝐴

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4.1 Impact of slip parameter ( ). The reason beyond this electrically conducting fluid produces a resistive force known as Lorentz The effect of slip parameter𝐴𝐴 on the force, which opposes the flow and tends to make dimensionless velocity profile ′( ) is presented in 𝐴𝐴 the fluid motion slowdown in the boundary layer Figure2. Increasing the values of the slip parameter and therefore reduces the profile of temperature 𝑓𝑓 𝜂𝜂 reduces the velocity field and boundary ( ) increases with the increase in magnetic thickness as depicted in Figure 2. parameter. 𝐴𝐴 𝜃𝜃 𝜂𝜂 4.2 Impact of Magnetic parameter ( ).

Figure 3 shows the behavior of 𝑀𝑀 on velocity profile it is noticed the effects of magnetic field are to reduce the velocity profile. Because𝑀𝑀 of the application of transverse magnetic field in an electrically conducting fluid, a resistive force like a drag force is produced, which is Lorentz force.

Figure 5. Dimensionless Concentration

4.3 Impact of suction parameter ( ). 𝑣𝑣𝑣𝑣 𝑀𝑀

The impact of the section parameter𝑆𝑆 on the dimensionless velocity profile ′( ) ispresented in 𝑆𝑆 Figure 6. Velocity profile diminishes and 𝑓𝑓 𝜂𝜂 accompanied with boundary layer width increases for gradually growing values of the suction Figure 3. Dimensionless Velocity parameter .Figure 7 displays the variation of The presence of Lorentz force retards 𝑣𝑣the𝑣𝑣 𝑀𝑀 force on temperature with suction parameter. As the values 𝑆𝑆 the velocity field. Figure. 4 and Figure. 5 shows of suction parameter increase, the temperature the variations in energy profile ( ) and graph is decreasing. Moreover, the thermal 𝑆𝑆 dimensionless concentration profile ( ) for boundary layer thickness and surface temperature 𝜃𝜃 𝜂𝜂 different estimations of Magnetic Parameter . It is also decreasing. Figure 8 shows that by 𝛽𝛽 𝜂𝜂 is analyzed that the temperature profile ( ), increasing suction parameter , concentration 𝑀𝑀 concentration profile ( ) and thermal boundary profile decreases. By an increasing in the suction, 𝜃𝜃 𝜂𝜂 𝑆𝑆 layer thickens are increasing functions of Magnetic concentration profile as wellas boundary layer Parameter. 𝛽𝛽 𝜂𝜂 thickness decrease.

Figure 4. Dimensionless Temperature Figure 6. Dimensionless Temperature

𝑣𝑣𝑣𝑣 𝑀𝑀 𝑣𝑣𝑣𝑣 𝑆𝑆

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that temperature profile enhances as the Biot number is increased gradually. Physically, the Biot number defines the ratio between resistance rate of heat transfer inside the body to the resistance at the body surface. The reason behind is that convective heat exchange at the surface will raise the boundary layer thickness therefore the nanofluid with convective boundary condition is more effective model as compared to the constant surface temperature state. Figure 7. Dimensionless Temperature

𝑣𝑣𝑣𝑣 𝑆𝑆

Figure 9. Dimensionless Temperature Figure 8. Dimensionless Temperature 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃 4.4 Impact of Prandtl number ( ). 𝑣𝑣𝑣𝑣 𝑆𝑆

Figure 9 presents that an elevation𝑃𝑃𝑃𝑃 in Prandtl number shows a reduction in the temperature profile ( ). Obviously, greater Prandtl number 𝑃𝑃𝑃𝑃 has weaker thermal diffusivity due to which 𝜃𝜃 𝜂𝜂 low range temperature. This indicates resection in energy𝑃𝑃𝑃𝑃 exchange ability and finally it causes a reduction in thermal boundary surface. 4.5 Impact of Radiation parameter ( ). Figure 10. Dimensionless Temperature 𝑣𝑣𝑣𝑣 𝑅𝑅 The influence of radiation parameter on profile𝑅𝑅 of temperature distribution is displayed in Figure 10. From the figure it is observed that by increasing the radiation parameter, temperature profile decreases significantly. It is because the increasing values of radiation parameter lead to decrease the thickness of the boundary layer and enhance the heat transfer rate with chemical effect on the melting surface.

4.6 Impact of Convective Heating ( ). Figure 11. Dimensionless Temperature 𝑣𝑣𝑣𝑣 𝐵𝐵𝐵𝐵 Figures11is presented to visualize the effect𝐵𝐵𝐵𝐵 of the on temperature distribution. This figure defines

𝐵𝐵𝐵𝐵

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4.7 Impact of Brownian motion and the chemical dissipation and therefore results in thermophoresis on unitless energy profile decrease in the profile of concentration. The most significant influence is that chemical reaction tends Figure 12 shows the impact of Brownian motion to increase the overshoot in the concentration and the thermophoresis parameters on the profiles and their associated boundary layer. temperature distributions ( ) in the thermal boundary layer. Thermophoresis is a component that drives small materials 𝜃𝜃away𝜂𝜂 from hot layer to the cooler end. It is noticed that as thermophoresis parameter increases the thermal boundary layer thickness increases and the temperature gradient at the surface decrease (in absolute value) as both and increase. 𝑁𝑁𝑁𝑁 4.8 𝑁𝑁𝑁𝑁 Impact of Lewis number ( )

Lewis number is decreasing functi𝑳𝑳𝑳𝑳 on of ( ). Figure 13. Dimensionless Concentration Increase Lewis number may decrease Brownian diffusion coefficient𝐿𝐿𝐿𝐿 because Lewis number 𝛽𝛽is 𝜂𝜂the 𝑣𝑣𝑣𝑣 𝐿𝐿𝐿𝐿 ratio of momentum diffusivity to Brownian diffusion coefficient, shown in Figure 13.

4.9 Impact of Eckert number ( ). Figure 14 displays the influence of on the 𝐸𝐸𝐸𝐸 energy profile. Energy profile increases when the Eckert number is increased. Due to friction,𝐸𝐸𝐸𝐸 the heat energy is kept in owing to accelerating values of , which results in the enhancement of the temperature profile. Figure14. Dimensionless Temperature 𝐸𝐸𝐸𝐸 𝑣𝑣𝑣𝑣 𝐸𝐸𝐸𝐸

Figure12. Dimensionless Temperature and Figure 15. Dimensionless Concentration 𝑣𝑣𝑣𝑣 𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 4.10. Impact of chemical reaction parameter 5. Conclusion 𝑣𝑣𝑣𝑣 𝛾𝛾 ( ) From the above discussion, we can make the Figure𝜸𝜸 15explains the influence of the chemical following conclusions. reaction parameter on the profile of concentration. Higher values of yield an increment in the It is noted that increasing values of chemical energy profile and Concentration profile whereas reaction parameter concentration as well as the an opposite effect 𝑀𝑀has noticed for the velocity thickness of concentration decrease. It is because profile. Because of the presence of Lorentz force the chemical reaction in this system results in

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retards the force on the velocity field and his force Reduced Nusselt number has the tendency to slow down the fluid motion. Prandtl number 𝑁𝑁𝑁𝑁𝑁𝑁 By increasing in the suction parameter , velocity pressure 𝑃𝑃𝑃𝑃 ( 3 1) profile increases whereas temperature profile heat capacity of the fluid 𝑝𝑝 − − 𝑆𝑆 𝑓𝑓 effective heat capacity of the nanoparticle decreases with an increase in suction while due to 3 1 𝐽𝐽𝑚𝑚 𝐾𝐾 𝑐𝑐 material ( ) increasing blowing, it increases. The thickness of 2 𝑐𝑐𝑝𝑝 radiative heat− flux− ( ) 𝐽𝐽𝑚𝑚 𝐾𝐾 concentration boundary layers reduces due to wall mass flux − 𝑟𝑟 increasing in the suction parameter. Increase of 𝑞𝑞 wall heat flux 𝑘𝑘𝑘𝑘𝑚𝑚 𝑚𝑚 Prandtl number and Radiation parameter 𝑞𝑞 local Reynolds number 𝑤𝑤 causes decrease in temperature profile. Due to 𝑞𝑞ℎ reduced Sherwood number 𝑃𝑃𝑃𝑃 𝑥𝑥 increase in thermal Radiation parameter more heat 𝑅𝑅ℎ𝑅𝑅 local Sherwood number 𝑆𝑆 𝑟𝑟 𝑅𝑅to fluid produces that enhance the energy filed and fluid temperature ( ) 𝑆𝑆 𝑥𝑥 temperature at the stretching sheet ( ) momentum boundary layer. Temperature increase 𝐾𝐾 𝑇𝑇 ambient temperature ( ) by enlarging thermophoresis parameter and 𝑤𝑤∞ 𝑇𝑇 𝐾𝐾 1 , velocity components along and axis ( . ) Brownian motion parameter . Physically, heated 𝑇𝑇 𝐾𝐾 1 𝑁𝑁𝑁𝑁 velocity of the stretching sheet ( . ) − particles come away from high temperature as 𝑢𝑢 𝑣𝑣 𝑥𝑥 𝑦𝑦 𝑚𝑚 𝑠𝑠 𝑁𝑁𝑁𝑁 Cartesian coordinates ( axis is aligned− along the compared to low-temperature so the temperature of ,𝑤𝑤 𝑚𝑚 𝑠𝑠 𝑢𝑢 stretching surface and axis is normal to it) ( ) fluid increases. It is reported that for enhancing 2𝑥𝑥 1 𝑥𝑥 𝑦𝑦 thermal diffusivity ( ) 𝑦𝑦 𝐿𝐿 values of the Biot number temperature profile dimensionless nanoparticle− volume fraction 𝛼𝛼 𝑚𝑚 𝑠𝑠 enhanced. Physically, the Biot number is basically similarity variable 𝛽𝛽 2 1 the ratio between resistances of the heat transfer in stream function( ) 𝜂𝜂 the body to the resistance at body surface. This is dimensionless temperatur− e 𝑚𝑚 𝑠𝑠3 because convective heat exchange along the 𝜓𝜓 fluid density ( ) 𝜃𝜃 3 surface will enhance the momentum boundary nanoparticle mass density− ( ) 𝜌𝜌𝑓𝑓 𝑘𝑘𝑘𝑘𝑘𝑘 electrical conductivity of the fluid− layer. For larger values of Lewis number and 𝜌𝜌𝑝𝑝 𝑘𝑘𝑘𝑘𝑘𝑘 chemical reaction parameter, concentration field parameter defined by ratio between the effective 𝐿𝐿𝐿𝐿 𝜎𝜎 heat capacity of the nanoparticle material and ( ) shows decreasing behavior. Temperature heat capacity of the fluid. = ( ) ( ) . profile will increase due to increases in the Eckert 𝜏𝜏 number.𝛽𝛽 𝜂𝜂 𝜏𝜏 𝜌𝜌𝜌𝜌 𝑝𝑝 ⁄ 𝜌𝜌𝜌𝜌 𝑓𝑓 References Acknowledgment [1] L. J. Crane. Flow past a stretching plate. The suggestions of reviewer regarding an earlier Zeitschrift für Angewandte Mathematik und Physik ZAMP, 21(4):645-647, 1970. version are appreciated. [2] T. C. Chaim. Hydromagnetic flow over a surface Nomenclature stretching with a power-law velocity. Int. J. 2 Engineering Science, 33(3):429-435, 1995. Magnetic field strength ( ) Constant ( 1) − [3] S. J. Liao and I. Pop. Explicit analytic solution for 𝐵𝐵 𝑊𝑊𝑊𝑊𝑚𝑚 Brownian diffusion− coefficient similarity boundary layer equations. Int. J. Heat 𝑎𝑎 Thermophoretic𝑠𝑠 diffusion coefficient 𝐵𝐵 and Mass Transfer, 47(1):75-85, 2004. 𝐷𝐷 Thermal conductivity ( 1 1) 𝑇𝑇 𝐷𝐷 2 4 [4] S. K. Khan and E. Sanjayanand. Viscoelastic Stefan-Boltzmann constant(− − ) 𝜅𝜅 𝑊𝑊𝑚𝑚 𝐾𝐾 boundary layer flow and heat transfer over an ∗ Mean absorption − − 𝜎𝜎 𝑘𝑘𝑘𝑘𝑚𝑚 𝐾𝐾 exponential stretching sheet. Int. J. Heat and Mass ∗ Eckert number Transfer, 48(8):1534-1542, 2005. 𝑘𝑘 Lewis number 𝐸𝐸𝐸𝐸 Magnetic parameter [5] T. Fang, F. L. Chiafon, and J. Zhang. The 𝐿𝐿𝐿𝐿 Brownian motion parameter boundary layers of an unsteady incompressible 𝑀𝑀 Thermophoresis parameter stagnation-point flow with mass transfer. Int. J. 𝑁𝑁𝑁𝑁 Nusselt number Non-Linear Mechanics, 46(7):942-948, 2011. 𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁

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[6] S. Choi. Enhancing thermal conductivity of fluids stretching sheet. J. Energy, Heat Mass Transfer, with nanoparticles in developments and 23:483-495,2001. applications of non-Newtonian flows, Am. Soc. [18] S.K. Khan, A.M. Subhas, and R.M. Sonth. Mech. Eng., 99-105 1995. Viscoelastic MHD flow, heat and mass transfer [7] B. V. Derjaguin and Y. Yalamov. Theory of over a porous stretching sheet with dissipation thermophoresis of large aerosol particles. J. energy and stress work. Heat Mass Transfer, Colloid science, 20(6):555-570, 1965. 40:47-57, 2003. [8] J. Chamkha and C. Issac. Effects of heat [19] R. S. Tripathy, G.C. Dash, S.R. Mishra, and S. generation/absorption and thermophoresis on Baag. Chemical reaction effect hydromagnetic flow with heat and mass transfer onMHDfreeconvectivesurfaceoveramovingvertical over a at surface. Int. J. Numerical Methods for planethroughporousmedium. Alexandria Heat & Fluid Flow, 10(4):432-449, 2000. Engineering Journal, 54(3):673-679, 2015. [9] R. Tsai. A simple approach for evaluating the [20] M.A. Seddeek and A.M. Salem. Laminar mixed effect of wall suction and thermophoresis on convection adjacent to vertical continuously aerosol particle deposition from a laminar flow stretching sheet and variable viscosity and variable over a at plate. Int. C. Heat and Mass Transfer, thermal diffusivity. Heat Mass Transf., 41:1048- 26(2):249-257, 1999. 1055, 2005. [10] K. Vajravelu. Viscous flow over a nonlinearly [21] M. Ali, M. A. Alim, R. Nasrin, M. S. Alam. stretching sheet. Applied mathematics and Numerical analysis of heat and mass transfer along computation, 124(3):281-288, 2001. a stretching wedge surface. Journal of Naval Architecture and Marine Engineering,1813-8235, [11] B. Gebhart and J. Mollendorf. Viscous dissipation December 2017. in external natural convection flows. J. Fluid mechanics, 38(01):97-107, 1969. [22] M. Ali, M. A. Alim, R. Nasrin, M. S. Alam, M. J. Haque Munshi. Similarity Solution of Unsteady [12] H. M. Duwairi. Viscous and Joule heating effects MHD Boundary Layer Flow and Heat Transfer on forced convection flow from radiate isothermal past a Moving Wedge in a Nanofluid using the porous surfaces. Int. J. Numerical Methods for Buongiorno Model, Procedia Engineering, 194, Heat & Fluid Flow, 15(5):429-440, 2005. 407 – 413, 2017. [13] P. L. Chamber and J. D. Young. On diffusion of a [23] M. Ali, M. A. Alim, R. Nasrin, and M. S. Alam, chemically reactive species in a laminar boundary Study the effect of chemical reaction and variable layer flow. Phys. Fluids, 1:48-54, 1958. viscosity on free convection MHD radiating flow [14] F. T. Akyildiz, H. Bellout, and K. Vajravelu. over an inclined plate bounded by porous medium, Diffusion of a chemically reactive species in a AIP Conference Proceedings, 1754, 040009, 2016. porous medium over a stretching sheet. J. Math. [24] M. S. Alam, M. Ali, M. A. Alim, and M. J. Haque Anal. Appl, 320:322-339, 2006. Munshi. Unsteady boundary layer nanofluid flow [15] R. Cortell. Flow and heat transfer of a fluid and heat transfer along a porous stretching surface through a porous medium over a stretching surface with magnetic field, AIP Conference with internal heat generation/absorption and Proceedings, 1851, 020023, 2017. suction/blowing. Fluid Dyn. Res, 37:231-245, [25] M. G. Reddy. Influence of Magneto hydrodynamic 2005. and Thermal Radiation Boundary Layer Flow of a [16] A. J. Chamka and A.R.A.Khaled. Similarity Nanofluid Past a Stretching Sheet. J. Sci. Res. 6 solution for hydromagnetic mixed convection and (2): 257-272, 2014. mass transfer heimntz flow through porous media. In. J. Numer. Methods Heat Fluid Flow, 10:94- 115, 2000. [17] A. Sriramalu, N. Kishan, and A.J. Anand. Steady flow and heat transfer of a viscous incompressible fluid flow through porous medium over a