Numerical analysis of MHD Casson boundary layer nanofluid flow over porous stretching surface with the effects of radiation and chemical reaction

In this article numerical analysis of incompressible, two dimensional and mixed convective MHD Casson fluid flows over a stretching surface with porosity under the influence of radiation and chemical reaction as well as viscous dissipation considered. By utilizing suitable similarity analysis, the governing PDEs (Partial differential equations) with its respective boundary conditions were transformed to dimensionless forms. The resulting ODEs (Ordinary differential equations) along with corresponding boundary conditions were solved via shooting technique combined with Runge-kutta-Fehlberg method. The outcomes of this study illuminates that velocity, temperature and concentration fields decreases due to the thickness of the boundary layer as we go away from the stretching sheet surface and falling of velocity observed in Casson fluid parameter. Under some restriction the resultant outcome were compared with previous published results and is found in admirable agreement.


INTRODUCTION
Depends on viscosity the ratio of shear stress with the rate of strain inside the fluid can be categorized as linear, nonlinear or plastic. The shear stress is proportional linear to the rate of strain, in Newtonian fluid, hear the proportionality constant is the dynamic viscosity coefficient. While in case of non-Newtonian the shear stress is nonlinearly proportional to the rate of strain, here the constant coefficient doesn't defined as even it is reliant on time. Further, plastic deformation retains if shear stress not depends on rate of strain. When undergoing deformation the material exhibit both combine characteristics of viscous and elastic property is known as viscoelastic. The viscoelastic property disappears with the rate of shear, time and temperature.
In fluid dynamics, dissipation is the energy conversion from one form to another. For example if fluid flow needs kinetic energy i.e. velocity of the fluid changes energy in to internal energy by taking it from the motion. The kinetic energy is dissipated. The rise of the fluid velocity outcomes the rise up the temperature of the fluid.
Depending upon the Eckert number (nondimensional quantity) viscous dissipation varies. Viscous dissipation is applied in geophysical flows and industries. Santoshiet al. [1] discussed the effects of chemical reaction on boundary layer flow upon a stretching sheet along with viscous dissipation. Dash et al. [2] explored the impact of heat and mass transfer on boundary layer fluid flow past through a moving vertical porous plate along with magnetic field in the transverse direction in the presence of heat source and chemical reaction. Tiegang et al. [3] researched an unsteady incompressible, boundary layer flow of stagnation-point with mass transfer. Zigta [4] analyzed the effect of unsteady, incompressible, magneto hydrodynamics filled with viscoelastic fluid in an infinite vertical porous channel inserted in a porous medium. Numerous investigators talked over different traits of stretching flow problem [5]. Reddy et al. [6] studied an unsteady MHD free flow of a Casson fluid past an oscillating vertical plate with constant wall temperature. Reddy et al. [7] analyzed the impact of radiation and heat generation on MHD boundary layer fluid flow over a stretching porous surface through porous medium with mass www.psychologyandeducation.net transfer. Raja Sekhar et al. [8] studied an unsteady oscillatory MHD slip fluid flow over a planer channel with the impact of chemical reaction and heat source. Chiam [9] discussed the boundary layer flow on stretching plate with distribution of velocity by power law in the presence of magnetic field in transverse direction. Elbashbeshy et al. [10] developed Maxwell nanofluid flow with heat and mass transfer over a stretching surface inserted in a porous medium in the existence of a heat source. Suresh et al. [11] analyses the impact of thermal radiation on free convective MHD fluid flow over a vertical porous moving plate in presence of chemical reaction. Liao et al. [12] explained comprehensive account of the boundary layers through a vertical plate inserted in a porous medium. Reddy et al. [13] discussed the impact of heat radiation on boundary layer MHD nanofluid slip velocity flow, over a stretching surface with boundary conditions. Shateyi et al. [14] explained the free convection of combined heat and mass transfer effects toward an unsteady stretching porous sheet along with radiation, viscous dissipation and chemical reaction. Chamka et al. [15] studied thermophoretic MHD heat and mass transfer stream above plane surface. Gebhart et al. [16] discussed some general characteristics of similarity circumstances for flow over in convection plumes and surfaces. Vajravelu et al. [17] considered the phenomenon of heat transfer and flow through a nonlinearity enlarging sheet.
Rafael [18] considered transfer of heat and fluid flow in a stretching surface through a porous medium. Khan et al. [19] examined boundary layer flow of viscoelastic and transfer of heat over continuous exponential stretching sheet. Akyildiz et al. [20] developed the viscoelastic fluid flow with mass transfer of species which are chemically reactive over an extending sheet. Pop et al. [21] developed the steady boundary layer nanofluid flow with heat transfer over a widening surface. Aziz et al. [22] investigated the effect of heat transfer on boundary layer nanofluid flow with convective boundary condition over an enlarging surface.
The main objective of the present study is to analyze the effect of MHD Casson boundary layer nanofluid flow upon a stretching porous sheet in the existence of radiation, chemical reaction along with viscous dissipation. The governing PDEs transformed to ODEs then the resulting ODEs are solved by shooting technique along with Rungekutta-Fehlberg method.

Mathematical formulation
Consider the numerical investigation on an incompressible, steady, two-dimensional, laminar, viscous boundary layer flow of an electrically conducting MHD Casson fluid towards a stretching surface with chemically reactive species experiencing chemical reaction.

Fig.1. Geometry of the flow under consideration
The plate has been taken in x-direction along with stretched velocity uw(x) = ax, (a>0) as shown in Fig.1 Here The boundary conditions for Equations (2), (3), (4), (5) and (6) are given as Using Rosseland approximation equation (4) can be expressed as here prime indicates differentiation w.r.to η.
with the boundary conditions (0) , The Nusselt number Nu x , the skin friction coefficient C f, and the Sherwood number Sh x, can be expressed as: Where, τ w is the surface shear stress, q m is the wall surface mass flux, and q w is the heat flux at the wall surface, given by: Using the dimensionless variables, we get where Re x indicates the Reynolds number and is stated as: Fig.2depicts the impact of the magnetic parameter (M), on velocity profiles. For both the cases in suction and injection, it is clear that for rising values of magnetic field parameter (M), the velocity profiles decreases. This is because of the fact that the influence of magnetic field on electrically conducting fluid generates a drag force which cultivates the body force known as Lorentz force. This force diminishes the fluid motion hence; it results to reduce the velocity. Fig.3 depicts the effect of various values of permeability parameter on velocity profiles. It is observed that for the enhancing values of permeability parameter, the velocity raises due to porous medium increases the resistance to the flow which cases the fluid to decrease. The influence of the Casson fluid parameter () on the velocity profiles are shown in Fig.4. It is noticed www.psychologyandeducation.net that for the raising values of Casson fluid parameter () the fluid velocity decreases, the reason is that the fluid become more viscous with the growth of Casson fluid parameter (), therefore more resistance is offered, which reduces the boundary layer thickness. Fig. 5 depicts the temperature profile for different values of radiation parameter (R). It is observed that the temperature enhances with an increasing values of radiation parameter. Because of the reason, for the raising surface heat flux leads to temperature grow up.

RESULT AND DISCUSSION
The variation of Prandtl number w.r.to the temperature profiles as shown in Fig.6. As Prandtl number Pr is in inversely related with thermal conductivity. So that for an enhancing value of Prandtl number, the capability of material to conduct heat falls so, the temperature of the fluid diminutions. Fig.7. represents the influence of slip parameter (A) on the velocity profiles. The slip parameter growths, the velocity decreases due fluid flow boundary layer thickness. Fig.8 shows the variations in velocity w.r.to suction parameter (S). It is noted that the suction parameter rises, velocity falls. As the fluid particles tense into the wall with the increase of suction parameter hence the boundary layer reduces. Figs. (9) and (10) reveal the influence of thermophoresis parameter (Nt) on the distribution of temperature and concentration profiles respectively. It is identified that the temperature upturns with the growing the thermophoresis parameter. The heightening of thermophoretic effects outcomes the nanoparticles migration from the hot to the cold surface ambient fluid as a result of this the temperature rises in the boundary layer. Also it is noticed that the concentration drops with an growing thermophoresis parameter. Figs. (11) and (12) prepared to show the Brownian motion parameter (Nb) impacts on the temperature and concentration profiles respectively. It is observed that both the temperature and concentration increases with an increasing the Brownian motion due to the thickness of thermal boundary layer. Fig. 13 represents the influence of Eckert number on the temperature distribution and it is visualized here to increase the Eckert number increases, temperature increases. Due to friction, the Eckert number values accelerate with the heat energy, which consequences in the enrichment of the temperature profile. Fig.14 depicts the concentration profiles with the impact of Lewis number. It is observed that the concentration decreases with an enhancing Lewis parameter. This is due to the ratio of diffusivity of momentum to Brownian diffusion. The influence of chemical reaction on concentration profiles discoursed with the assistance of Fig. 15. It can be identified that for an enhancing chemical reaction, the concentration profiles are decreases depletion in concentration. The impact of Biot number on the temperature profiles as shown in Fig.16. As Biot number evaluates the heat transfer resistance quotient from outside and on the surface, which causes the temperature profile increases.
For the validity of our results, the skin friction factor, the Nusselt and Sherwood number have been compared with those already published in literature as shown in Tables I and II. From tables it can be seen that the results achieved by the present code are found convincingly much closed to the published results [1].

Conclusions:
From the above discussion, we can make the following conclusions.