Radiative Magnetohydrodynamics Casson Nanofluid Flow and Heat and Mass Transfer past on Nonlinear Stretching Surface

The magnetohydrodynamics (MHD) stagnation point Casson nanofluid flow towards stretching surface with velocity slip and convective boundary condition has been investigated in this article. Effects of thermal radiation, viscous dissipation, heat source and chemical reaction have also been incorporated. Using appropriate similarity transformation Partial Differential Equations (PDEs) are converted into Ordinary Differential Equations (ODEs) and shooting technique along with Adams–Moulton method of order four has been used to obtain the numerical results. Different physical parameters effects on velocity, temperature and concentration of nanofluid flow have been presented graphically and discussed in detail. Numerical values of the skin friction coefficient, Nusselt number and Sherwood number are also and discussed.


Introduction
Fluid is a phase of matter that deforms or flows under an applied external force. Fluid exists in the form of liquids, gases or plasma [1]. It is a substance with vanishing shear modulus or, in more simple words, substance which cannot resist any applied shear force. Fluid is the basic need of everyday life and because of its importance in many natural processes, scientists in different part of world are trying to explore various facts regarding the flow of fluid. Fluid dynamics is the sub-branch of fluid mechanics in which we study the fluid flow, also by analyzing the cause of flow. And how forces influence the fluid flow. It provides methods for understanding the evolutions of stars, ocean, current, tectonics plate, as well as the blood circulation [2]. Few important applications of fluid flows include wind turbines, oil pipelines, rocket engine and air-conditioning systems [3]. Archimedes was the first mathematician who formulated the Archimedes principle about the static of fluid and is considered to be the basics of fluid mechanics. The proper study of fluid mechanics start from early fifteen centuries. Fluid can be further classified into Newtonian or non-Newtonian fluid, depending on the relationship between two physical quantities i.e., stress and strain.
The mixture of nanoparticles with dimension less than 100nm and the conventional low thermal conductivity fluid is known as nanofluid. The word nanofluid was first introduced by Choi [4] that presented a new class of fluid. The thermal conductivity of nanofluid can be increased by using nanoparticle of gold, copper, silver etc., into the base fluid. The factor that lead to an increase in the thermal conductivity of nanofluids was studied by Buongiorno [5]. He observed that, both thermophoresis effect and the Brownian motion causes a change in thermal conductivity of the fluid. Naramgari and Soluchana [6] analyzed the effect of thermal radiation on MHD nanofluid over a stretching surface. Abolbashari et al. [7] investigated the transfer of energy and heat in the steady laminar Casson nanofluid flow using both slip velocity and surface boundary condition. Ghadikolaei et al. [8] investigated the influence of different physical parameters such as chemical reaction, thermal radiation, suction, Joule heating, heat generation and absorption in the MHD flow of Casson nanofluid using a porous nonlinear sheet. Magnetohydrodynamics is that branch of mechanics which we deals with the study of conduction fluid flow in the presence of an external magnetic field. It has wide range of applications in various fields of science such as, metallurgical science, mental working process, aerodynamics, fluid dynamics, and many others engineering disciplines for example, ceramic and biomedical engineering etc. [9]. The 2D mixed convection MHD boundary layer stagnation point flow in the existence of thermal radiation using a vertical plate which was filled with nanofluid has been demonstrated by Eftekhari and Moradi [10]. Kumar et al. [11] analyzed the impact of the transfer of heat in MHD Casson nanofluid using nonlinear surface. Aman et al. [12] investigated the flow of 2D incompressible viscous fluid using a shrinking surface in the existence of an external magnetic field. Stagnation point always exist on the flow field surface such that close to this point the fluid come to at rest. There for, the Stagnation point can be defined as, the point in the flow field where the fluid velocity become zero. The study of the flow of nanofluid close to the stagnation point has many practical applications, some of them are listed as, cooling of electronic devices by fan, solar receiver, the cooling of nuclear reactor at the time of emergency shutdown, and several hydrodynamic processes [13]. Due to these important applications of stagnation point flow has attracted a great attention of scientific community. Hiemenz [14] was the first mathematician who first time proposed the 2D stagnation point flow. Eckert [15] got the accurate solution by extending Hiemenz problem by adding the energy equation. Mahapatra and Gupta [16], In view of that Ishak et al. [17], and Hayat et al. [18] investigated the impact of the transfer of heat on stagnation point over a porous plate. Non-Newtonian fluid has wide range of application such as oil recovery, filtration, polymer engineering, ceramics production and petroleum production. There is exists a sub-class of non-Newtonian fluids called Casson fluid. This fluid has large viscosity which tends to infinity at zero rate of shear i.e., if the magnitude of shear stress is much weaker than the magnitude of applied stress it behaves like a solid. On the other hand, if shear stress become greater than the yield stress the fluid start flowing. Casson [19] first time developed the Casson model for various suspension of cylindrical particles. Soup, fruit juice, jelly, honey and tomatoes sauce are general example of Casson fluid. Benazir et al. [20] have analyzed the unsteady MHD Casson fluid flow over a at plate and vertical 3 cone over porous medium along with double dispersion effects. The laminar convective boundary layer non-Newtonian Casson fluid flow thermally fixed over a stretching sheet have been analyzed by Animasaun et al. [21] Afikuzzaman et al. [22] have investigated the unsteady flow of MHD fluid with hall current through parallel plates and considering the magnetic field perpendicular to the plates. The heat transfer with MHD Casson fluid flow towards a linear stretching sheet with temperature distribution over the sheet has been analyzed by Govardhan et al. [23]. In this article, we provide a detail review of Ibrahim et al. [24] study and the study is extended by considering various other effects such as Soret effect, Dufour effect and inclined magnetic field.

Problem Formulation
We have considered a 2D steady incompressible MHD stagnation point flow of a Casson nanofluid over a stretching sheet. The sheet is placed in the plane B is a constant. It is also assumed that the magnetic Reynolds number is small, and the induced magnetic field is negligible. A convective heating process is used to regulate the sheet temperature f T . The nanoparticles concentration is w C which is assumed to be constant. For y goes to infinity, the concentration and temperature of nanofluid is represented by C  and T  respectively.

Energy equation
Physical principle: Energy is conserved

Mass transfer equation
The corresponding boundary conditions are:

Similarity Transformation
The following transformation [24] has been used to get ODEs from PDEs The following assumptions are made for the calculation of velocity components along x and y direction as , uv yx The non-dimensional form of the Eqs. (2), (3) and (4) are: The corresponding boundary condition becomes .
In the above Eqs. (8)-(11) R represents the radiation parameter, Pr stands for Prandtl number, Bi the Biot number, Ec for Eckert number, Nb represents Brownian motion parameter, Nt the thermophoresis parameter, Q represents the heat generation, Le stands for the Lewis number, A denotes velocity ratio number and S is the suction parameter, these parameter are formulated as:

Physical Quantities of Interest
Mathematical form of skin coefficient friction is And the Sherwood number is In the above equations We obtained the following dimensionless form for Nusselt number Sherwood number and skin friction coefficient: The Reynolds number can be defined as

Solution Methodology
The system of nonlinear ODEs (8)-(10) along with boundary condition (10) are converted into first order ODEs. The first order system of ODEs with appropriate boundary condition are solved by using shooting method. We adopt the following procedure: " ' " .
The above system of ODEs and the corresponding initial condition can be written as For the solution of above initial value problem, we use Adams Moulton Method of order four. For finding the initial condition we take  is chosen such that no considerable changes are obtained going beyond. The execution of the Newton's method can be presented in the following algorithmic form: Step-1: Choose an initial guess 0 rr  in the equation (8) and solve it by the Adams-Moulton method. Step The following steps are involved for the accomplishment of the shooting method.  (22). (iv) Repeat (i) and (ii). In a similar manner, the ODEs (9-10) along with the associated BCs can be solved by considering f as a known function. The stopping criteria for the shooting method is set as: where  is a small positive number. From now onward  has been taken as 8 10  whereas   is set as 7.

Results and Discussion
The numerical results of the equations in the previous sections are discussed in this section by using the graphs and tables.      Figure 2 analyzes the impact of  on dimensional velocity profile. The velocity of the fluid decreases by increasing the numerical value of  . Physically, this means that fluid viscosity increases due to accelerating values of  which in turn decelerate the nanofluid velocity profile. Furthermore, the present phenomena convert to Newtonian fluid as  approaches to infinity. Figure 3 illustrates the relationship between energy profile and  . It is seen that the temperature distribution of the fluid increases by gradually increasing the value of  . Actually, by increasing value of  the thermal boundary thickness increase due to which the surface temperature increases. Figure 4 demonstrates the behavior of  on the concentration field. The nanoparticle volume fraction is observed to be increased for the higher estimation of  . Figure 5 shows  Figure 8 illustrates the impact of Ec on temperature profile of the fluid. The graph clearly shows that the temperature distribution is enhanced by mounting values of Ec . Actually, Ec can be written as a ratio of kinetic energy of the fluid particle and thermal energy. The increasing value of Ec means, we have increased the kinetic energy of the fluid particle, as a result the thermal boundary layer thickness is enhanced. Figure 9 investigates the dependence of temperature distribution on Nt . The plot shows that the temperature profile of the fluid is escalating with boosting values of Nt . Actually, reason of this behavior is that the nanoparticle at the hot boundary side have been moved towards the cold boundary side and the thermal boundary layer become thicker in the existence of Nt . Figure 10 depicts the visualization of Nt on the concentration distribution. It is noticed that by gradually increasing Nt the concentration distribution also increases. Generally, in the presence of Nt exert forces on each other, as a result particle move from hotter to colder region of the fluid and has been noticed an increment in the concentration distribution. Figures 11 and 12 are drawn to analyze the impact of Bi on both energy and concentration distribution of the fluid respectively. Physically, Bi can be written as a ratio of convection to conduction. The convection is taking place on the surface while the conduction is taking place inside the surface. Thus, the boosting value of Bi accelerates both temperature and concentration profile. Figure 13 shows the relationship between slip parameter and dimensionless velocity distribution. The velocity is observed to be a reducing function of  . It can be generalized as the fractional resistance between fluid particles and the flow surface increases as a result the velocity profile of the fluid decreases. Figure 14 shows that the energy profile is accelerated by gradually uprising the value of  . Figure 15 analyzes the impact of R on energy distribution. The gradually rising value of R enhances the energy distribution of the fluid. Actually, the heat energy exhausted from the fluid due to large value of R and as a result the energy distribution increased. Figure 16 explores the impact of Pr on energy distribution. Since, the Pr can be written as a ratio of kinematic diffusivity to heat diffusivity. The gradually increasing value of Pr increase the fluid density and decreasing thermal diffusivity and as a result the energy distribution is enhanced. Figure 17 illustrates the relationship between Q and temperature profile of the fluid. The plot clearly shows a reduction in temperature distribution of the fluid for negative value of Q . In the same way heat generation occurs for the positive value of Q . Due to these behaviors the temperature of the fluid gradually increases.

Effect of Brownian Motion Parameter
Nb Figure 18 is drawn to illustrate the relationship between Nb and temperature distribution. The temperature distribution is enhanced with rising value of . Nb Physically, Nb is associated with movement of the fluid nanoparticles. The kinetic energy of the fluid particles increases with boosting values of Nb , due to which the temperature distribution of the fluid increases. Figure 19 is drawn to analyze the effect of Nb on concentration profile, which shows that the increasing value of Nb produce a reduction in concentration distribution.

Effect of Suction Parameter
S Figures 20-22 show the relationship between S and velocity, S and temperature and S and concentration profile respectively. From the graphs it is observed that increasing the numerical value of S a decrement in the velocity, temperature and concentration profile of the fluid occur. Figure 23 analyzed the relation between the Lewis number Le and concentration distribution. Concentration profile decreased for high value of Le and thus we have got a small molecular diffusivity. Generally, concentration profile is a decreasing function of Lewis number.

Conclusion
The overall conclusion drawn from the present work is summarized below.
 Decreasing behavior due to increasing the Casson fluid parameter in the velocity distribution and same behavior is noticed for enhancing the numerical value of suction parameter.  For the large value of magnetic parameter, the velocity field reduce but opposite trend is observed in the graph of temperature and concentration distribution.  The temperature distribution decelerate, and the concentration distribution accelerate due to boosting value of Prandtl number.  Temperature profile rises by increasing radiation parameter and same behavior is observed in the temperature field because of thermophoresis parameter.  Uprising of temperature field is observed for the increasing value of Eckert number.