ROLE OF MAXWELL VELOCITY AND SMOLUCHOWSKI TEMPERATURE JUMP SLIP BOUNDARY CONDITIONS TO NON-NEWTONIAN CARREAU FLUID

The forthright aim of this correspondence is to examine the conduct of MHD, viscous dissipation and Joule heating on three dimensional nonNewtonian Carreau fluid flow over a linear stretching surface. Impact of non-linear Rosseland thermal radiation and homogenous/heterogenous reaction process have been also considered to examine the heat and mass transfer process during fluid flow. The velocity and thermal slip effect at the surface have also been scrutinized in detail. By utilizing a suitable transformation, the modelled partial differential equations (PDEs) are renovated into ordinary differential equations (ODEs) and furthermore solved with the help of the numerical procedure namely the RK-4 method embedded with shooting procedure. The behaviour of the velocity, temperature and concentration profiles against various parameters are portrayed in the form of figures and tables. In the presence of the Maxwell velocity slip effect, the velocity profile is found to diminish. It is experienced that the temperature profile depreciates as a result of an augmentation in the Smoluchowski temperature slip effect and moreover concentration profile depreciates as a result of an improvement in the homogeneous and heterogeneous reaction parameters. To affirm the reliability of the proposed numerical technique, a comparison with already published work is also taken into account. A remarkable agreement amongst the accomplished and the existing outcomes has been obtained.


INTRODUCTION
During the past two decades, widespread interest is developed in researchers to analyze the fluid flow behaviour and energy transfer phenomenon in the non-Newtonian fluid models like Bingham, power-law, Ellis, Casson, Eyring-Powell and Carreau fluid models. Examples of non-Newtonian fluids are ketchup, genetic liquids, shampoo, blood , tooth paste etc. Nagalakshm and Vijaya (2020) scrutinized the impact of MHD on Carreau nanofluid past a nonlinear stretching sheet. Their main finding was that an augmentation in magnetic parameter force guides to an abatement in the velocity profile. Cortell (2011) studied the impact of the power-law fluid flow towards an elastic porous stretching sheet embedded with the inclusion of effects like suction, thermal radiation and viscous dissipation. He observed that the thermal boundary-layer thickness diminishes on the account of an increment in the thermal radiation parameter. Ghiasi and Saleh (2019) scrutinized the two dimensional Casson fluid past a stretching sheet by incorporating the effects of Joule heating and non-uniform heat source/sink. They concluded that the temperature field augments by rising the Eckert number. Ali and Sandeep (2017) studied the MHD Casson-ferro fluid with the aid of effect like nonlinear thermal radiation and found that the nonlinear thermal radiation is the major factor responsible for regulation of the temperature at the thermal boundary layer. Three dimensional fluid accompanied with variable thermal conditions for the case of stretching surface was scrutinized by † Corresponding author. Email: tanveer.sajid15@yahoo.com Liu and Andersson (2008) with the conclusion that the Nusselt number decreases by rising the internal heat parameter. Hayat et al. (2017) pondered three dimensional nanofluid flow over a stretching sheet accompanied with heat and mass flux boundary conditions. Khan et al. (2017a) studied the behaviour of 3D magneto Carreau nanofluid past a stretching sheet. They observed that by enhancing the magnetic parameter, a decrement in the velocity gradient is resulted. Megahed (2019) studied the impact of variable thermal conductivity and thermal radiation on Carreau fluid past a nonlinear stretchable surface. Raju and Sandeep (2016) investigated the impact of MHD, nonlinear thermal radiation and heat source/sink on unsteady Carreau fluid moving over a stretching sheet. To learn a bit more about the discussed issues, some more relevant articles are cited in Turkyilmazoglu (2014); Mukhopadhyay et al. (2013); Misra and Adhikary (2017); Narayana et al. (2014); Azam et al. (2017); Olajuwon (2011); Machireddy and Naramgari (2018).
Thermal radiation is actually an electromagnetic radiation produced by the thermal motion of particles in matter. These radiations actually move in the form of electromagnetic waves like those from electric burner and room heater. Thermal radiation has immense applications in agriculture, space exploration, law enforcement, polymer preparation, furnace design, electricity generation etc. The linear thermal radiation can be achieved by linearizing the Rosseland approximation and furthermore linear thermal radiation has unique Prandtl number Magyari and Pantokratoras (2011). The problem with linear thermal radiation is that it looses its effectiveness at high temperature difference. To overcome this obstacle, researchers recently introduced the nonlinear thermal radiation. The temperature becomes highly nonlinear in the occurrence of nonlin-ear thermal radiation. In the case of nonlinear thermal radiation three key factors Prandtl number, temperature ratio parameter and thermal radiation are responsible for heat transfer analysis. Nonlinear thermal radiation has been applicable where high temperature difference is required and has distinguished applications in high temperature energy sector like polymer production, thermal furnaces, nuclear reactor, space craft etc. Impact of nonlinear thermal radiation and activation energy on Maxwell nanofluid past a stretchable surface have been debated in detail by Sajid et al. (2018). Magyari and Pantokratoras (2011) studied the impact of linear thermal radiation on various boundary layer flows. The only parameter which is helpful to linearize the Rosseland approximation is the radiation parameter. Krishnamurthy et al. (2016) studied the impact of nonlinear thermal radiation on water based nanofluid accompanied with slip effects. They perceived that the velocity profile diminishes by rising the velocity slip parameter. Astuti et al. (2019) contemplated the impact of thermal radiation on nanofluid past a variable stretchable surface and observed that an amplification in radiation parameter guides to a augmentation in the temperature field. Mohamed and Wahed (2017) studied the impact of nanoparticles, MHD and nonlinear Rosseland thermal radiation on a continuously moving surface. They observed that the Nusselt number rises by enhancing the radiation parameter. Cortell (2014) pondered the impact of thermal radiation on the fluid flow over a stretching surface. It was established that the temperature profile embellishes for the larger values of the temperature ratio parameter but behaviour is quite opposite in the case of thermal radiation parameter. Mahanthesha et al. (2016) delineated the water based nanofluid flow over a stretching surface with the combination of effect like nonlinear thermal radiation and canvassed that the nonlinear thermal radiation has higher impact on the flow field as compared to the linear thermal radiation. The impact of variable thermal conductivity and viscous dissipation on radiative fluid past a stretching sheet was investigated by Shateyi and Muzara (2020) with the conclusion that the temperature profile upsurges because of an escalation in the radiation parameter. The impact of nonlinear thermal radiation on 3D Carreau fluid past a stretchable surface was investigated by Khan et al. (2017b). The influence of nonlinear thermal radiation and bio-convection on Carreau fluid was deeply investigated by . Jyothi and Reddy (2019) studied the effect of magnetite nanoparticles and zero mass flux boundary condition on radiative Carreau fluid past a stretching sheet.
A catalyst is a substance which speeds up the chemical reaction without itself being consumed. Catalysts are categorized into two types namely the homogeneous and the heterogeneous catalysts. In case of the homogenous reaction, the catalyst and the reactant are in same phases both are solids or liquids, but reverse in the case of heterogenous reaction where both occur in the different phase (one is solid other is liquid). Different chemical reaction structures involve homogenous/heterogenous reactions like polymer production, biochemical systems, combustion and catalysis. Ziabakhsh et al. (2010) investigated a steady incompressible fluid along with chemically reactive species past a porous stretching sheet. Kameswaran et al. (2013) studied the dynamics of a fluid past a porous stretching sheet with the inclusion of nanoparticles and chemical reaction and claimed that the temperature field decreases with an enrichment in the Weissenberg number. Merkin (1996) considered the model over a boundary flow with homogeneous/heterogeneous reaction. Nandkeolyar et al. (2014) contemplated a viscous incompressible fluid flow past a stretching surface under the effects of homogeneous/heterogeneous reaction and MHD. They deliberated that an augmentation in the magnetic field parameter guides to a decrement in the velocity profile. Bilal et al. (2017) observed the behaviour of non-Newtonian Williamson fluid past a stretching cylinder under the effect of stagnation point, homogeneous/heterogeneous reaction and MHD. Mansur et al. (2016) pondered a fluid past a stretching surface with the inclusion of nanoparticles and homogeneous/heterogeneous reaction phenomenon and found a reduction in the mass fraction due to an increment in the homogeneous/heterogeneous reaction parameter. The behaviour of Casson-Carreau fluid in addition to homogeneous/heterogeneous reaction is inspected by Gireesha et al. (2017) with the main finding that the temperature profile boosts up by mounting the temperature ratio parameter. Bachok et al. (2011) explored the effect of a stagnation point flow on the fluid flow over a stretching sheet combined with homogeneous/heterogeneous reaction and came up with the conclusion that a variation in the heterogeneous reaction parameter brings about a diminishment in the concentration profile. Chaudhary and Merkin (1995) considered a simple isothermal model with the inclusion of homogeneous/heterogeneous and stagnation point. Irfan et al. (2018) scrutinized Carreau fluid past a stretching sheet embedded with homogeneous/heterogenous reaction, variable thermal conductivity and heat source sink. The impact of homogeneous/heterogenous reaction on 3D Carreau fluid past a stretchable sheet was discussed by Khan et al. (2018).
In recent years, various researchers investigated the behaviour of the slip boundary layer phenomenon over different types of surfaces. The velocity of the fluid tends to zero and no surface friction takes place with regard to no slip condition, but in the case of slip condition, the fluid velocity normal to the wall is zero whereas its tangential velocity is nonzero. The slip conditions are considered important to study the microelastic mechanical systems, artificial heart cell etc. It is experienced that by including the velocity slip condition at the boundary, the heat transfer rate is increased. Recently, Aljoufi and Ebaid (2016) scrutinized a fluid flow towards a stretchable surface along with slip effects. They determined that the temperature profile diminishes by augmenting the slip parameter and Biot number. Zheng et al. (2012) scrutinized the viscous fluid flow towards a stretchable surface along with the velocity slip condition. They profound that the mass transfer rate upsurges by mounting Prandtl number. The effects of slip conditions, binary chemical reaction on stagnation point flow in a porous medium with convective boundary condition are considered by Sivasankaran et al. (2017). They concluded that an enrichment in the chemical reaction parameter reduces the heat transfer rate. Devakar et al. (2014) scrutinized the couple stress fluid between parallel plates with inclusion of slip effects. They perceived that the velocity in the couette flow is enhanced when the lower and the upper plates move in the same direction. The effects of nonlinear velocity slip over a stretching permeable surface is deliberated by Xinhui et al. (2017). They determined that the boundary layer thickness of velocity becomes thinner by rising the magnetic field parameter. Fang et al. (2009) deliberated the MHD viscous fluid flow over a stretching sheet in the presence of the slip effect. They analyzed that the wall slip velocity boosts in the case of an improvement in the magnetic parameter. The behaviour of velocity slip, thermal slip and solutal slip toward a stretching surface is pondered by Ibrahim and Shankar (2013) and established that the thermal boundary layer thickness reduces by augmenting the slip parameter. The difference between the fluid velocity of the wall and the velocity of the wall itself is directly proportional to shear stress and this proportional factor named as slip length. The slip boundary condition is given by |u| wall = ls| ∂u ∂y |, where ls represents the slip length Hak (2001). The behaviour of velocity slip |u| wall = 2−σv σv λ0 ∂u ∂y and temperature jump T wall = 2−σ T σ T 2r r+1 λ 0 P r ∂T ∂y , σv and σT denote momentum accommodation and temperature accommodation coefficient are studied by Maxwell (1879) and Smoluchowski (1898). Khan et al. (2013) studied the impact of velocity and temperature jump slip conditions on double diffusive nanofluid past a vertical plate and found that the velocity field increases on the behalf of an improvement in the velocity slip parameter. Rahman and Eltayeb (2011) pondered the conduct of rarefied fluids flow over a wedge along with Smoluchowski temperature slip condition and came up with the conclusion that the temperature field abates owing to an augmentation in temperature slip parameter. The impact of Maxwell velocity and Smoluchowski temperature slip boundary condition on rarefied gas fluid moving over a wedge is investigated by Das et al. (2017). Hayat et al. (2015) have scrutinized the impact of MHD on three dimensional nanofluid flow towards a stretching sheet embedded with Maxwell velocity slip boundary condition. This paper is extension of work done by Hayat et al. (2015) under the inclusion of various effects.
In the light of above mentioned study Das et al. (2017) and Hayat et al. (2015) specially , the purpose of current research is to explore the conduct of nonlinear thermal radiation, viscous and Ohmic dissipation, homogeneos/heterogenous reactions, Maxwell velocity and Smoluchowski temperature slip boundary conditions on three dimensional rarefied Carreau fluid past a stretching sheet. According to best of author's knowledge no work have been reported yet to study the impact of slip conditions along with nonlinear thermal radiation and homogeneos/heterogenous reactions on rarefied fluid flow over a three dimensional stretchable surface.

MATHEMATICAL FORMULATION
The physical model used to represent the time independent three dimensional incompressible Carreau fluid flow behaviour is shown in Fig.  1. A non-uniform magnetic field B0 is executed vertical to the sheet and furthermore the induced magnetic field effectiveness is amitated by assuming the small Reynolds number. The surface coincides with the plane z = 0 and the flow is restrained in the region z > 0. Flow considered in the current problem is optically thick and temperature difference of the fluid flow is large, therefore Rosseland radiative heat flux is not linearized with the help of Taylor's series. The surface is also assumed to exhibit the velocity slip conditions. The stretching surface velocities acting along x -axis and yaxis are Uw(x) = lx and Vw(x) = my. The  of Carreau fluid is defined by The shear rate at zero and infinity is represented by µ0 and µ∞. Another important parameter n in the above equation represents the power-law index. The fluid exhibits the shear thickening behaviour for n > 1 , shear thinning as n < 1 and furthermore Newtonian in the case n = 1. In the current formulation of the problem µ∞ is considered as zero. Then the Carreau fluid equation reduces to The apparent viscosity for Carreau fluid model can be expressed as The term β * = µ0 µ∞ represents the infinite shear-rate viscosity to the zero shear-rate viscosity and taken less than one in the current study. The homogeneous as well as the heterogeneous reaction between species A and B is premeditated byMansur et al. (2016). Cubic homogeneous chemical reaction is considered within the flow field whereas heterogeneous chemical reaction is taken at the catalyst surface. Homogeneous reaction is heterogenous chemical reaction on the catalyst surface is premeditated by The The Rosseland approximation Mohamed and Wahed (2017) is defined as The Stefan-Boltzmann and the mean absorption coefficients are expressed as σ * and κ * respectively. By adopting the following similarity transformation Khan et al. (2017b), Equations (6)-(10) get the following non-dimensional form: The dimensionless form of the boundary conditions (11) is given below: Various dimensionless parameters appearing in Equations (14)- (19) are characterized as By taking DA=DB =1 and δ = 1 Mansur et al. (2016) h(η) + q(η) = 1.
By using Equation (21) in Equation (17), we get subject to the boundary conditions The heat transfer coefficient is given by The dimensionless form of the heat transfer coefficient is bestowed by The skin friction coefficients are where the terms τxz and τyz are given by The skin friction coefficients in the dimensionless form are formulated as:

SOLUTION METHODOLOGY
The nonlinear, non-dimensional Equations (14)- (16) and Equation (22) along with the boundary conditions (19) and (23) can be comprehended with the assistance of the shooting strategy Na (1979) utilizing the RK4 method. For numerical solution, the unbounded domain [0, ∞) has been replaced by [0, ηmax] where ηmax is a real number chosen in such a way that the solution doesn't show significant variations for η > ηmax. It is observed that ηmax = 7 guarantees an asymptotic convergence for all the results presented in this article. For convenience in the choice of missing conditions and computational efficiency, first the momentum Equations (14)-(15) will be solved numerically by the shooting method. Later on, using f and g as known functions, the energy Equation (16) will be treated numerically. Finally the concentration equation will be solved by the same technique by using the available solution of Equations (14)-(15). The momentum equations have been converted into a system of first order ODEs signifying f by y1, f by y2, f by y3, g by y4, g by y5, g by y6, y2(0) by γ1 and y5(0) by γ2. The resulting system of equations is: (1 + nW e 2 1 y 2 3 ) , y 4 = y5, (1 + nW e 2 2 y 2 6 ) , with initial conditions are y1(0) = 0, y2(0) = 1 + γγ1, y3(0) = γ1, y4(0) = 0, To reach close enough to the missing initial conditions, Newton's method is iteratively applied until the following criteria is met.
where depicts a small positive number.
In the case of temperature equation, the transformed ODE (16) is converted into first order ODEs by denoting θ by w1, θ by w2. As a result of Frontiers in Heat and Mass Transfer (FHMT), 14, 28 (2020) DOI: 10.5098/hmt.14.28 Global Digital Central ISSN: 2151-8629 the introduction of theses new variables, the following system of ODEs is achieved.
with initial conditions For the refinement of the initial guess γ3, Newton's method is applied unless the condition underneath is fulfilled.
The concentration equation (16) is converted into the first order ODEs by denoting h by z1, h by z2. The first order ODEs are then given by: having initial conditions For the improvement of the initial guess γ4, the iterative scheme called Newton's method is used until the criteria given below is achieved.
In Table 1, a comparison of the presently computed values of different physical quantities with those already published in literature.

RESULTS AND DISCUSSIONS
In the present section, the numerical solution of the dimensionless mathematical model has been presented and analyzed. The skin friction coefficient and h (0) against various parameters is delineated in Table 3 by keeping Rd = 1, θw = 1.5, Ecx = 0.5, Ecy = 0.5, P r = 1, δ = 1 as fixed. According to Table 2, an enrichment in magnetic parameter M and the stretching rate parameter β prompts a decrease in the skin friction coefficient but an inverse behaviour is observed on account of the power-law index n (shear thickening) and the slip parameter γ. An improved behavior is seen in the skin friction coefficient along y-axis for the magnetic parameter M and the stretching rate β. However for n and γ a reverse behavior is experienced for skin friction coefficient. Table 3 demonstrates the behavior of the Nusselt number against different parameters, for fixed values of W e1 = 0.1, W e2 = 0.1, Sc = 0.2, K = 2,K1 = 1. It is quite clear that boosting the values of n (shear thickening), the radiation parameters Rd, the temperature ratio parameter θw and Prandtl number P r bring about an enlargement in the Nusselt number but situation is reverse on account of the remaining parameters like the magnetic parameter M , Eckert number Ec, stretching rate β, slip parameters γ and δ.
Figures 2-17 exhibit the behaviour of velocity, temperature and concentration profiles against various parameters arising in the present problem. Fig. 2 exhibits the dynamics of the velocity profiles f and g under the effect of the magnetic parameter M . It is noted that the magnetic field in the presence of electric current develops a force called Lorentz force which lessens the fluid velocity inside the boundary layer. Fig. 3 delineates the impact of the power-law index n on the velocity profiles f and g . The values of n have been taken for all the three cases shear thinning n < 1, shear thickening n > 1 and Newtonian n = 1. From this figure, it can be clearly seen that a boost in f and g happens due to an embellishment in the power-law index. An increment in the power law also guides to an improvement in the momentum boundary layer thickness. The impact of Weissenberg number W e1 on the velocity profile is manifested in Fig. 4. Weissenberg number is described as the ratio of the shear rate time to the relaxation time. Liquids turn out to be more thicker because of an augmentation in the Weissenberg number. Therefore velocity profile is reduced with an enrichment in the Weissenberg number. The execution of the velocity profile against the slip parameter γ is displayed in Fig. 5. Rising the values of the velocity slip over the stretching sheet lessens the fluid velocity. The impact of the Wessenberg number W e2 on the velocity profile is considered in Fig. 6. It is observed that the liquid turns out to be more viscous on the behalf of an amplification in the Weissenberg number which reduces the momentum boundary layer thickness and the velocity profile. Fig. 7 demonstrates the dynamics of the velocity field versus the slip parameter γ. The physical reason behind the velocity decrement is actually an increment in the frictional resistance existing between the surface and the fluid particles which reduces the fluid flow and the velocity profile. Fig. 8 displays the impact of the stretching parameter β on the velocity profile. An enlargement in the value of the stretching parameter brings about a diminishment in the velocity profile. Fig. 9 and Fig. 10 depicts the performance of viscosity ratio parameter β * on the velocity profiles f (η) and g (η). It is quite cleat the fluid behaves like shear thickening on the behalf of an amplification in the β * which reduces the fluid velocity and guides to a reduction in the velocity profiles. Fig.  11 displays the significant role of the frequently used parameter Prandtl number P r, which is actually the ratio of the momentum diffusivity to the thermal diffusivity. It is well established that Prandtl number reduces the fluid temperature. Thermal diffusivity reduces for the larger values of P r which guides to a thickness in the thermal boundary layer. Fig.  12 portrays the behavior of the velocity profile versus the Eckert number Ecx. By boosting the Eckert number, more heat in the liquid is generated due to the frictional heating which improves the temperature behaviour. From Fig. 13, it is probed that by mounting the Eckert number Ecy, an additional heat in the liquid is generated on account of the frictional force which is responsible for an augmentation in the temperature field. Fig. 14 demonstrates the effect of the magnetic parameter M on the temperature field. Lorentz force which is basically a drag force, causes a decrement in the fluid velocity which amplifies the temperature field. The behavior of radiation parameter Rd on the temperature profile is deliberated in Fig.  15 It is quite abvious that heightening the value of Rd escorts the temperature rise inside the boundary layer, and hence the temperature profile rises. Fig. 16 determines the impact of the temperature ratio parameter θw on the temperature profile. By ascending θw, an enlargement in the wall temperature is observed. As a result the temperature ratio parameter escalates and a rise in the temperature profile take place. Fig. 17 sketches the impact of the Schmidt number Sc on the concentration profile. When Schmidt number enhances, the viscous diffusion also increases which the concentration profile. The behavior of the concentration field against the homogeneous reaction parameter K is depicted in Fig. 18. The flow diffusivity fluctuates by mounting the value of the homogeneous reaction, which is the prominent factor for diminishment of the concentration profile. Fig. 19 examines the association between the heterogeneous reaction K1 and the concentration field. An increment in the homogeneous reaction brings about a reduction in the concentration field.                Frontiers in Heat and Mass Transfer (FHMT), 14, 28 (2020) DOI: 10.5098/hmt.14.28 Global Digital Central ISSN: 2151-8629

FINAL REMARKS
A three dimensional Carreau fluid flow past a stretching sheet along with different effects like viscous dissipation, homogeneous/heterogeneous reaction, Ohmic dissipation, velocity slips, temperature slip, nonlinear thermal radiation is analyzed. The shooting method is utilized to solve the system of nonlinear ordinary differential equations. Some decisive comments from the concluding work are as below: • The temperature profile increases with an increment in the Eckert number Ecx along x-axis and Ecy along y-axis.
• By enhancing the Weissenberg numbers W e1 and W e2, a decrement in the velocity profiles is prompted.
• The temperature profile depicts an enhancing behavior for the magnetic parameter M and the Schmidt number Sc.
• The velocity profiles f (η) and g (η) are decreased for the higher values of the velocity slip parameter..
• A decay in the concentration profile is observed for the larger values of the homogeneous reaction parameter K.
• The concentration field depreciates in the case of an enrichment in the heterogeneous reaction parameter K1 • The concentration profile depreciates as a result of an augmentation in the homogeneous and heterogeneous reactions.

DATA AVAILABILITY
The data supporting the findings of this study are available within the article.

CONFLICTS OF INTEREST
The author(s) declare(s) that there is no conflict of interest regarding the publication of this paper.

ARTICLE FUNDING
The corresponding author pay article processing charges.

W e1
Weissenberg number along x − axis W e2 Weissenberg