Radiation and Chemical Reaction Effects on Nanofluid Flow Over a Stretching Sheet

: The present research aims to examine the steady state of the two-dimensional incompressible magnetohydrodynamics (MHD) flow of a micropolar nanofluid over a stretching sheet in the presence of chemical reactions, radiation and viscous dissipation. The effect of particle rotation is taken into account. A conducting fluid passes over a semi-infinite plate with variable temperature while a magnetic field is directed in the transverse direction. Results for velocity, angular momentum, temperature and concentration profiles are obtained for various values of Eckert number, Schmidt number, Prandtl number, thermophosis parameter and Brownian motion parameters. A similarity approach is used to transform the original set of two-dimensional partial differential equations into a set of highly nonlinear-coupled differential equations in dimensionless form. A numerical solution is obtained with the help of the COMSOL multiphysics software in the framework of a finite element method. Our findings indicate that on increasing Brownian motion and the chemical reaction rate, the fluid temperature becomes higher. An increase in the values of other physical parameters has the opposite effect. A variation in the boundary layer thickness typically results in changes in the concentration distribution in the flow. The angular velocity is deeply affected by the Eckert number, material parameter and magnetic number.


Introduction
Incomprissible magnetohydrodynamics (MHD) flow plays an important role in various industrial applications. MHD flow may be appicabale in the extursion of polymer sheet from a die or in drawing of plastic films. The heat and mass tranfer for micropolar MHD flow past though a stretching sheet has the varities of applications such as polymer blends, porous rocks, aerogels, alloys and microemulisions. Due to a large number of appliactions of magnetic fluid, the researchers of many disciplines have been attaracted in the field of MHD during the last few decades. In the early stage of MHD research, the problems of MHD flow were solved through analytical method. However, time to time, new methods of solutions came into existence. Now a days, the numerical methods have become the strong tool to solve a set of nonlinear coupled differntial equations of MHD flow. Sometimes the numerical approach of the solution is a big challenge, if the differential equations involved in the problems are highly nonlinear. Therefore, the researhers are finding numercial solution through mathematical modeling without sacrificing the relevant phenomena. In fact, resluts obatiend through mathematical modeling of the system of nonlinear coupled differntial equations are very relaible than numerical methods only. A problem of MHD flow with magntic and viscous dissipation effects towards the streching sheech has been studied [Hsaio (2017)]. MHD flow of micropolar nanofluid flow in the presence of some physical parameters have been been investigated ; Pal and Mandal (2017) ;Tabassum, Mehmood and Akbar (2017)]. The unsteady electrically conducting magnetic fluid flow past over an oscillating vertical plate is investigated [Sheikholeslami, Kataria and Mittal (2018)]. The boundary layer and stagnation -point flow of nanofluid over a stretching sheet and Soret effects on viscoelastic nanofluid flow over vertical stretching surface have been studied in the presence of viscous dissipation and radiation effects [Imtiaz, Hayat and Alsaedi (2016); Anwar, Shafie, Hayat et al. (2017); Ramzan, Yousaf, Farook et al. (2016)]. Various types of nanofluid with heat and mass transfer have been investigated through some physical properties of nanofluid [Sheikholeslami (2018); Sheikholeslami and Rokni (2017); Sheikholeslami, Haq, Shafee et al. (2019); Sheikholeslami (2019); Lu, Ramzan, Huda et al. (2018)]. Lattice Boltzman method and Brownian motion were the key parts for the study of nanofluid flow through numerical simulation [Sheikholeslami (2018); Sheikholeslami, Shehzad and Li (2018)]. Two-dimensional MHD stagnation point flow is presented to study the heat transfer of a viscous incompressible non-Newtonian nanofluid [Gupta, Kumar and Singh (2018)]. A three-dimensional nanofluid flow over a nonuniform sheet and an exponentially stretching surface in a porous medium with variable thermal conductivity have been investigated [Subhani and Nadeem (2017);Kumar, Raju, Sekhar et al. (2017); ]. The effect of acoustic streaming on nanoparticle motion and morphological evolution inside an acoustically levitated droplet using an analytical approach coupled with experiments is investigated [Saha, Basu and Kumar (2012)]. A mixed stagnation point flow of nanofluids over a stretching surface with thermal radiation and viscous dissipation effects have been analyzed [Pal, Vajravelu and Mandal (2014)]. Heat and mass transfer of nanofluid with motile gyrotactic effects have been investigated ] Jeffery nanofluid flow over a linearly stretched surface and inclined stretched cylinder with chemical reaction and slip conditions have been studied [Ramzan, Bilal, Chung et al. (2018, 2017. Effects of Hall current on MHD unsteady flow through a vertical plate in the presence of radiation and chemical reaction is investigated [Biswas and Ahmmed (2018)]. The entropy generation of micropolar fluid flow in a rectangular duct subjected to slip and convective boundary conditions is reported [Srinivasacharya and Himabindu (2017)]. Heat and mass transfer analysis on magnetohydrodynamic flow of viscous fluid by curved surface is presented with joule heating [Hayat Qayyum, Imtiaz et al. (2018)]. Laminar flow of nanofluid has been investigated numerically through Homotopy Analysis Method [Lu, Farooq, Hayat et al. (2018)].
Effects of viscous dissipation and Joule heating in MHD flow by rotating disk of variable thickness are investigated [Hayat, Qayyum, Khan et al. (2018)]. MHD mixed convection flow and heat transfer in a porous enclosure filled with a Cuwater nanofluid in the presence of partial slip effect are investigated in the presence of heat sink and source parameters [Chamkha, Rashad, Mansour et al. (2017)]. MHD viscous two-phase dusty fluid flow and heat transfer over permeable stretching sheet is studied taking linear deformation in the wall boundary [Turkyilmazoglu (2017)]. The heat and mass transfer of a MHD nanofluid on a stretching sheet and stretching cylinder have been analyzed with Brownian motion and thermophoresis effects [Atif, Hussain and Sagheer (2018) ;Hayat, Nassem, Khan et al. (2018)]. A mathematical model has been developed to examine the MHD micropolar nanofluid flow with buoyancy effects in the presence of nonlinear thermal radiation and dual stratification ]. The problems on Casson nanofluid over a stretching surface with various physical parameters have been investigated by the researchers [Rana, Mehmood, Narayana et al. (2016) (2017)]. Kellor box algorithm has been developed to study the behavior of nanofluid in the presence of various physical parameters [Mehmood, Rana and Maraj (2018)]. The velocity and temperature distribution are studied on twodimensional oblique Oldroyd-B flow on a stretching heated sheet [Mehmood and Rana (2018)]. The stagnation point flow using Jeffery nanofluid is studied in the presence of various physical parameters [Mehmood, Nadeem, Saleem et al. (2017)]. Homotopy perturbation transform method is also useful to solve Navier-Stokes equation for MHD flow [Kumar, Singh and Kumar (2015)]. Homotropy perturbation Sumudu mehod has been used to study two-dimensional viscous flow over a shrinking sheet [Rathore, Shisodia and Singh (2013)]. In the present problem, two dimesional MHD flow of micropolar nanofluid over a semiinfinite stretching plate is considered subjected to a transverse magnetic filed with variable temperature. The fluid is electrically conducting and flow is an incomprissible. Thermal and concentration buyoyancy effects and heat bsorption and radiation effects are also presnt in the nanofluid. The chemical concentration effect is also considered in the fluid. The nonlinear coupled differential equations involed in the problem are solved numerically using finite element method through mathematical modeling in COMSOL. The micropolar nanofluids are special kind of fluid which exhibts the microscopic and nano effects due to micromotion and Brownian motion of the fluid particles. The model for micropolar nanofluid are useful to study the behavior of collodial suspensions, liquid crystels, polymeric fluid, liquid crystals and animal bloods.

Mathematical formulation
Two dimensional incomprssible flow of micropolar nanofluid due to moving surface is considered here. The magnetic filed is directed perpendicular to the streching sheet and the x axis and y axis are along the surafce and perpendicular to the surface, respectively [Hiao (2017) and Singh and Kumar (2016)] .The governing equations are as follows: The equation of continuity The equation of motion The equation of angular momuntum The energy equation The equation of concentration In Eq. (2) is taken as reference length. The expression  represent the force due rotational motion. The boundary conditions are as follows: Eqs.
(1)-(6) can be written in the dimensionsless form as: The dimensionless quantities are presented as follwos: The boundary conditions are as follows: 3 Numerical solution Eqs. (7)-(10) are solved numerically through mathematical modeling in COMSOL. Under PDE interface, COMSOL solves only second order partial differntial equations. Therfore, the following transformation has been used to reduce the above system of nonlinear equations into second order differential equations: Hence, Eqs. (8)- (12) can be written as dp q dξ = dq r dξ = Eqs. (14)-(19) with the help of Eq. (20) has been modled in COMSOL and this solution is based on finite element method. These equations have been solved in COMSOL under Dirichlet boundary conditions. Extremely fine mesh has been used during the solution. Maximum size of elemet is 0.045 and the minimum element size is 0.00009. The shear stress at the surface can be calculated as: Using Eq. (11), The relation w u bx = represents the characteristic velocity, the skin friction velocity f C can be expressed as: The local Nusselt number can be calculated as:

Results and discussion
The main aim of this work is to study the heat and mass transfer on MHD flow of a micropolar nanfluid over a stretching sheet with various physical parameters. In this study, the effects of material parameter ( ) Q′ on the temperature, concentration and velocity distributions are investigated. Rotation of the particle has also been taken into consideration. Due to the difference between the rotation of the particle and the rotation of the fluid, a viscous torque is generated which is being equilibrated by magnetic torque. The rotation of the particles in the presence of magnetic field generate additional resistance in the flow. Due to which, additional viscosity known as rotational viscosity is generated in the flow. Figs. 1-7 represent the temperature distribution with the variation of different parameters. Fig. 1 is obtained for different values of radiation absorption coefficient. Increasing the value of this parameter, the heat will be absorbed. Therefore, it decreases the temperature; however, other parameters are taken same. Increasing the values of Thermal Grashof number makes the bond between the fluid to become weaker. This parameter decreases the internal friction resulting the gravity becomes stronger enough.  Q′ the concentration increases. The concentration distribution with the variation of material parameter is shown in Fig. 11. There is no significant impact of this parameter in concentration motion. However, at large values of ξ , the material parameter increases the concentration effects. Fig. 12 depicts the effect of Brownian motion parameter on the concentration profile. It is clearly observable from the results that increasing the Brownian motion parameter increases the concentration and temperature effects in the flow.   Fig. 20. The angular velocity decreases for increasing values of the material parameter. However, for increasing values of the magnetic parameter, the angular velocity increases as shown in Fig. 21. Magnetic field tries to rotate the particle in the fluid resulting an additional resistance is created in the flow, but it increases angular velocity. clear from the result that there is no effect of Sc in the velocity distribution. The velocity distribution remains same for the other physical parameter presented in the model. Therefore, these results are not presented in the graphical form. If the effects of radiation and chemical reaction parameters is excluded, the results reduce to the previous published model [ Hsaio (2017)].

Conclusions
In the present problem, a novel study has been carried out for an incompressible flow of a micropolar nanofluid past a stretching surface with the chemical reaction and radiation considering the effect of the particles motion. The following points can be concluded from the results: 1.
For increasing the values of the radiation absorption parameter, Prandtl number, Solutal Grashof number, thermophoresis parameter, the temperature in the flow decreases. However, temperature increases only for increasing the values of the Brwonian motion parameter and chemical reaction parameter.

2.
It is obserable that for incresing the values of Schmidt number, Chemical reaction parameter, Prandtl number, magnetic parameter and thermophoresis parameter decreases the concentration effects in the flow. However, variation in material parameter and Eckert number have mixed effects in the concentration distribution. For small boundary layer thickness, the concentration effects decrease but for large boundary layer, concentration effects increase. It is also noticeable that for increasing the values radiation absorption parameter, Brwonian motion parameter, thermal Grashof number, Solutal Grashof number, the concentration motion also increases. 3. Angular velocity changes significantly for different values of the Eckert number, material parameter and magnetic number, however other parameters do not have much effect on the angular velocity. For increasing values of the Eckert number and material parameter decrease the angular velocity distribution and the magnetic number increases in the angular velocity.

4.
It is also interesting to notice that only the Eckert number, material number and magnetic number disturbs the velocity distribution. It remains approximately same for other parameters.