Radiative MHD Casson Nanofluid Flow with Activation energy and chemical reaction over past nonlinearly stretching surface through Entropy generation

In the present research analysis we have addressed comparative investigation of radiative electrically conducting Casson nanofluid. Nanofluid Flow is assumed over a nonlinearly stretching sheet. Heat transport analysis is carried via joule dissipation, thermal behavior and convective boundary condition. To employ the radiative effect radiation was involved to show the diverse states of nanoparticles. Furthermore entropy optimization with activation energy and chemical reaction are considered. Thermodynamics 2nd law is applied to explore entropy generation rate. Nonlinear expression is simplified through similarity variables. The reduced ordinary system is tackled through optimal approach. Flow pattern was reported for wide range of scrutinized parameters. Computational consequences of velocity drag force, heat flux and concentration gradient are analyzed numerically in tables. Results verify that conduction mode augments with enhance of magnetic parameter.Increasing radiation boosts the temperature and entropy. Activation energy corresponds to augmented concentration. Heat transmission rate augments with the consideration of radiation source term.


Problem Formulation
We assumed two dimensional electrically conducting thermally radiative steady Casson nanofluid flow through a past non-linear stretching surface. Two equal and opposite forces are applied to stretch the surface along x-direction (Fig. 1). The exponential velocity is defined as u ax w m 1 = . The magnetic field B o is applied normally to the stretching sheet where the electric filed for low magnetic Reynolds number Rm ( 1 )  is not considered. Energy presentation is demonstrated in existence of chemical reaction, activation energy, Joule heating and www.nature.com/scientificreports www.nature.com/scientificreports/ thermal flux. Furthermore, Ludwig-Soret effect of the nanoparticles is considered. Entropy analysis is taken and for it Thermodynamic second mechanism is utilized to explore. The governing equations can be written as 1,5,17,49 v y The applied boundary layer constraints are The velocity profile is u v ( , , 0) in the directions of the x y ( , ),where in µ ρ υ = , nf , υ µ represents kinematic and viscosities, ρ f show nanofluid density, σ nf is conductivity of nanofluid due to electric current and B represents the magnetic force. In Eq. (4), T is the temperature of the fluid, in relation c ( ) f α ρ κ = , α use for thermal diffusivity, κ for thermal conductivity, c represents heat capacity. In relation q r , q r is represents radiative heat flux, κ R is mean absorption coefficient and σ e is the Stefan-Boltzman. The terms D D , B T signifies Brownian and thermophoretic individually. In Eq. (5), C represents concentration, the relation x represents mutable heat transmission coefficient, K r is the rate of chemical reaction rate, Ε α is the activation energy, n o is fitted rate constant.

entropy Generation and Modeling
The entropy generation is mathematically expressed as Which after simplification give the form G

Physical Quantities
Surface drag force. The physical quantities Skin friction coefficients C Fx is defined as w y y y 0 The dimensionless form is x Fx 1/2 In which Re x 1/2 indicates the local Reynold number. The dimensionless form is (2020) 10:4402 | https://doi.org/10.1038/s41598-020-61125-9

Heat transfer rate. The Local Nusselt number or temperature gradient
www.nature.com/scientificreports www.nature.com/scientificreports/ Mass transfer rate. Sherwood number Sh x are stated as ( ) hx w

Solution via HAM
Here we have used optimal approach to get computational results. Due to couple nonlinear system of governing differential equations system homotopy analysis scheme is proposed to compute the solutions. Homotopy analysis scheme is independent of small or large parameters and needs no discretization. This scheme has no stability issues like seen in numerical approaches. This scheme needs the choice of linear operator and initial guess. Initial guess is selected in such a manner that it satisfies the given boundary conditions. Initial guesses are The corresponding to linear operators are= F these linear operators conform following features Here Ε ∑ = , n n 1 7 with n 1, 2, 3 = … are subjective constants.

Discussion
The steady radiative electrically conducting Casson nanofluid flow through a nonlinearly stretching sheet is investigated with joule dissipation, thermal behavior and convective boundary condition. Optimal approach is used due to couple nonlinear system of governing differential equations system to get computational results. Momentous features of various intriguing parameters on entropy, velocity, concentration and temperature are deliberated through graphs. Surface drag force, temperature gradient and mass transmission rate are numerically intended versus various engineering variables.
Velocity. Figure 1 presented the significant effect of M on f ( )  Figure 3 is present the impact of M on θ η ( ). The augmented values of M augmented the heat transfer rate and temperature profile ( ) θ η increases. Apparently for higher (M) the Lorentz forces increases which augments the opposing forces to the fluid particles and hence the temperature enhances. Figure 4 presented the impact of a very imperative parameter N r on temperature profile ( ) θ η . Physically N ( ) r is the relative involvement of heat transmission conduction to thermal radiation transfer. We observed enhancement in temperature field θ η ( ) with higher value of radiation parameter N ( ) r . Augmentation in N ( ) r generates more heat which turn increases the nanofluid temperature. The impacts parameter Ν ( ) b and parameter ( ) www.nature.com/scientificreports www.nature.com/scientificreports/ cold region and rise temperature there and collective temperature of the whole system rises. Effect of P ( ) r on ( ) θ η is presented in Fig. 7. Clearly temperature is a decreasing function of P ( ) r . Relation between E ( ) c Eckert number and temperature function ( ) θ η is shown in Fig. 8. Eckert number is relation amongst the variance boundary layer enthalpy and flows of kinetic energy, which described the transmission dissipation. Increasing E ( ) c augmented the internal energy of nanofluid which in turn, augmented the heat transfer rate.    ( ) φ η while increasing (N t ) augmented φ η ( ). Boosting (N t ) enhances the motion of nanoparticles from higher to lower temperature gradient which in turn, exploit the concentration of nanoparticles. Figure 11 is drawn to scrutinize the behavior of Sc ( ) on ( ) φ η . Increasing Sc ( ) the mass diffusivity decays and thus concentration is declined. Figure 12 is sketch to examine the behavior of reaction rate σ ( ) 1 on ( ) φ η . It is observed that concentra-  www.nature.com/scientificreports www.nature.com/scientificreports/ tion φ η ( ) is increasing function for σ ( ) 1 . The impact of activation energy parameter E ( ) on concentration φ η ( ) is presented in Fig. 13. The higher value of activation energy augmented the concentration φ η ( ) of nanofluid.
Entropy. The significant effects of various parameters modelled from concentration Eq.   www.nature.com/scientificreports www.nature.com/scientificreports/ increasing function. In Fig. 17 the influence of Brinkman number ( ) r Β is described. Actually Brinkman number is a heat generated source within the fluid moving region. The heat generated together with the heat transfer from the wall increases the entropy optimization.    Table 3. From Table 3 it is observed that the mass transfer rate varies with augmentation of Ε Ν S ( ), ( ), ( ) c t while reduces for Ν ( ) t .   www.nature.com/scientificreports www.nature.com/scientificreports/

Validation of Results by Comparison
In this section comparison amongst current and published results for validation are presented. Tables 4 and 5 32 and ref. 48 . Clearly the result is in good agreement.

Mean findings
Here we scrutinize the entropy optimization investigation in electrically conducting Casson nanofluid over nonlinear stretchable surface. Novel behavior of Brownian motion and thermophoresis are also studied. Furthermore physical features activation energy with convective conditions and entropy optimization are studied.The physics of heat and mass transmission are explained and associations were developed for them. The significant observations of present study are given below: • For greater value of (M) the Lorentz forces enhances which rises the resistive force to the nanofluid motion and in result the velocity η η f ( ) reduces.     = . .