Numerical Investigation of MHD Carreau Nanofluid Flow with Nonlinear Thermal Radiation and Joule Heating by Employing Darcy–Forchheimer Effect over a Stretching Porous Medium

,


Introduction
Despite their complex stress-strain relationships, in recent years, non-Newtonian fuids have attracted researchers' attention on account of their multidisciplinary applications (see Ref. [1][2][3][4]).Non-Newtonian fuids are abundant in several engineering, industrial, and biological activities, such as in food processing, in arterial blood fow, in mud-drilling machines, in polymer suspensions, and in heavy oil lubrications.Te notion of improving the thermal conductivity of nanofuids by dispersing nanoparticles in conventional fuids was frst initiated by Choi and Eastmann [5].Nanofuids stand for a mixture of nanosized particles (e.g., Cu, Ti, Al 2 O 3 , and TiO 2 normally with size < 100 nm) scattered stably in host fuids (e.g., water, blood, ethylene glycol, and vegetable oil).As compared to host fuids, nanofuids possess advanced thermophysical properties.Such fuids being enhanced heat transfer agents, their applications extend to cancer and hyperthermia treatment, solar energy collectors, safety management in nuclear reactors, and cooling processes of electronics, computer chips, and transformers.However, the study of nanofuid is still not at its end, and it seems there is still a need to have an outweighing theory on how to prepare and use nanoparticles to culminate the efciency of thermal conductivity.A nonuniform heat source/sink in the form of surface temperature and wall heat fux was considered by Ramesh et al. [6] to investigate the stagnation point fow of MHD non-Newtonian dusty fuid fow.Growth in fuid-particle interaction parameter improves the velocity feld of dusty fuid and diminishes this distribution for clean fuid as discussed by the authors.Bouslimi et al. [7] used the Runge-Kuttabased shooting method to investigate heat and mass transfer properties of mixed convection nanofuid under Soret diffusion and nonlinear heating caused by an extending surface.Te thermal behavior of a hybrid nanofuid in a doublepipe heat exchanger was tested by Jalili et al. [8].Te result indicated that water/Al 2 O 3 nanofuid has superior convective heat transfer than water/TiO 2 nanofuid and pure water.Tey also found that raising nanofuid concentration from 0.4% to 6% increased the heat transfer rate by 12%.Te use of nanoparticles is also an attractive and promising feld in the development of diagnostics and treatments for cardiovascular disease.Te authors in [9] examined the stenosed artery heat transfer behavior of hybrid nanofuid in a slanted orientation.Stenosis is a condition that results from the narrowing of blood vessels due to the buildup of cholesterol or any other plaque that blocks the smooth fow of blood in the artery.Enhancing the Al 2 O 3 nanoparticle volume fraction decreases the value of the temperature profle, as elaborated in their work.Additional studies involving the importance of nanofuids can be taken care of through the investigations in [10][11][12][13].
Magnetohydrodynamic (MHD) explains the study of dynamic interactions between magnetic felds and conducting fuids.An opposing electric current is induced owing to liquid motion, which alters mechanical properties of the fuid.MHD analysis regulates many natural phenomena and engineering and industrial problems on almost every scale, such as the formation of stars, proper mixing of alloying parts, electrolysis process, the MHD generator, and magnetic fltration and separation.Ishak et al. [14] studied hydromagnetic fow and the transfer of heat energy above a vertical and expanding surface with a varying magnetic feld.Tey noticed that strengthening magnetic intensity brings a reduction of heat energy loss and local skin friction.Te accounts of heat source/sink, chemical reaction, and nonlinear thermal radiation on the magnetohydrodynamic fow of Williamson nanofuid was explored by Bouslimi et al. [15].Teir result verifed that the heat transfer rate of Williamson nanofuid is adversely afected by rising values of temperature ratios, thermal radiation, and heat generation/ absorption parameters.
As a result of the diversity of fow in nature, investigators have recommended various non-Newtonian models based on their constitutive rheological variables, such as Casson fuid, power-law fuid, Williamson fuid, Oldroyd-B fuid, Maxwell fuid, Carreau fuid, and Eyring-Powell fuid.Among these models are Carreau fuids, which are realized as generalized Newtonian fuids having Newtonian (n � 1), shear-thickening (or dilatant with n > 1), and shear-thinning (or pseudoplastic with 0 < n < 1) characteristics, where n stands for the power-law index.Examples of these fuids are detergents, fuid crystals, pulps, blood, and synovial fuids.Carreau fuids have gained awareness on account of their importance in the extrusion of polymers, tumor therapy, cosmetics, capillary electrophoresis, and petrochemical industries.Te impact of melting and heat source/sink has been analyzed by Khan et al. [16], taking the Carreau nanofuid over a wedge.Tey recognized that the rise in the above parameters deteriorates the nanoparticle concentration curve in both shear-thinning and thickening regions.Te Haar wavelet quasilinearization approach was utilized by Che Ghani and Siri [17], in contemplating the MHD Carreau nanofuid model with the impact of velocity slip boundary and suction/injection above an expanding surface.Babu et al. [18] depicted the nonlinear MHD convective Carreau nanofuid fow with the efect of viscous dissipation and Arrhenius activation energy above an exponentially expanding surface.Many other studies of Carreau nanofuids are detailed in [19][20][21][22].
According to Buongiorno's [23] inspection, there exist various slip mechanisms in nanoparticle-based fuid interactions.He identifed in the laminar fow region that the prominent dissipative energy sources associated with a nanoparticle-based fuid slip are Brownian motion and thermophoretic difusion.Contemplating the Rayleigh-Benard problem, the impact of thermophoresis and Brownian movement on CuO-water nanofuid heat transfer nature was studied by Haddad et al. [24].Teir study revealed the infuence of thermophoresis and Brownian motion is more amplifed at low-volume fractions of nanoparticles.Te infuence of thermophoretic difusion and Brownian movement on the mixed convection fow of Carreau nanofuid was studied by Irfan [25].Te importance of thermophoretic and Brownian difusion on MHD nanofuid fow over a stretching circular cylinder with the insertion of a variable magnetic feld, free stream velocity, and multiple slips was examined by Majeed et al. [26].Termal analysis on the efect of thermophoresis, Brownian motion, and Hall currents in micropolar nanofuids was presented by Jalili et al. [27].Lately, the authors in [28] considered the fow behavior of micropolar nanofuid subjected to electromagnetism, thermophoresis, and Brownian motion in a rotating realm.Further investigations on Brownian difusion and thermophoresis in light of various non-Newtonian fuids can be addressed in [29][30][31].
Many scientists are motivated to understand heat and mass transfer phenomena associated with chemical reactions and chemical transportation processes by considering their industrial applications in combustion, catalysis, and biochemical systems.Tese chemically reacting systems involve homogeneous (or bulk) and heterogeneous (or surface) reactions.Representing homogeneous reactions by cubic autocatalysis and heterogeneous reactions by a frst-order process, Merkin [32] studied the boundary-layer fow of homogeneous-heterogeneous reactions.Te author has indicated that the surface reaction is the most infuential mechanism near the front edge of a uniform stream fow over a fat surface.In [33], Khan et al. picked the MHD fow of the Powell-Eyring fuid model to elaborate on the Newtonian heating impact on homogeneous and heterogeneous reactions.Tey observed that the skin friction coefcient advances for large values of magnetic and fuid parameters.In contrast, the mass transfer rate dampens for a homogeneous reaction parameter.Additional furtherance on chemical reactions for various non-Newtonian fuids is reported in [34][35][36][37].

2
International Journal of Diferential Equations High-temperature gradient MHD convective fow problems are remarkably infuenced by thermal radiation.Its signifcance can be seen in nuclear power plants, electrical power generators, solar power collectors, glassblowing, and manufacturing of plastic and rubber sheets.To analyze the energy production feature of fow between two circular plates, Jalili et al. [38] studied the thermal radiation infuence of unsteady compressing fow of magnetohydrodynamic Casson fuid.Tey identifed that non-Newtonian fuid temperature improves by 20%, responding to a surge in the squeezing factor value.In light of the 2D cavity and 3D cavity, the impact of magnetic feld and thermal radiation on the transport process of magnetohydrodynamic convection fow was exploited by Zhang et al. [39].Te study unfolded when strong thermal radiation and weak magnetic feld are set forth, and the diference in fow and thermal radiation between 2D and 3D cavities is notable as compared to weak thermal radiation and strong magnetic feld combination efects.Shaw et al. [40] investigated MHD hybrid nanofuid fow exposed to quadratic and nonlinear thermal radiations.By considering radiation properties and convective phenomena, the infuence of Joule heating on magnetic Carreau nanofuid has been explored in [41].Recently, employing hybrid analytical and numerical techniques, the authors in [42] presented the heat and mass transport features of axisymmetric micropolar fuid constricted to the magnetic feld in a cylindrical polar system.Teir work showed that increasing the radiation parameter from 0 to 8% changes the shape of the temperature profle while keeping the maximum and minimum temperatures unaltered.
Te transport process of non-Newtonian fuid via porous media is an attractive area of research owing to its vast applications, such as packed bed reactors, geothermal industries, enhanced oil recovery, drying of paper pulp, gel chromatography, and soil structures.A pioneering work regarding the fow of fuid in a porous medium was initiated by Darcy in 1856.A mixed convection power-law fuid fow drenching the porous space was discussed by authors in [43].Te nanofuid fow in a Darcy-Forchheimer porous medium was studied by Rasool and Zhang [44], taking into account the Cattaneo-Christov heat and mass fux model.Tey disclosed that resistive force resulting from the porosity factor increases the temperature feld.Conversely, the concentration feld of nanoparticles decays in response to an increase in the inertial force.Te efect of using diferent hybrid nanofuids as solar energy absorbers in a Darcy-Forchheimer porous medium was studied by Alzahrani et al. [45].Te account of the non-Darcy-Forchheimer law on MHD Carreau fuid fow subjected to a heat source/sink and thermal radiation above a stretching sheet was anticipated by Siddiq et al. [46].
We have surveyed the above literature and many more not listed here and found that there does not exist a study that demonstrates the combined impacts of nonlinear thermal radiation, Ohmic heating, heat source/sink, chemical reaction, and Darcy-Forchheimer inertial efect on MHD fow of Carreau nanofuid embedding linearly stretching porous surface with Brownian motion and thermophoretic events.

Problem Formulation
Te mathematical model of the present study is developed by assuming a steady, 2D, viscous, and incompressible nonlinear radiative fow of magneto Carreau nanofuid above a stretching sheet.Te sheet is expanding with the velocity u w � a 0 x, where a 0 is a stretching constant.A constant magnetic intensity B 0 is administered orthogonal to the surface.We suppose a small magnetic Reynolds number in order to detach the infuence of the induced magnetic feld.Furthermore, the efects of Hall current and ion slip are ignored.Te porous sheet is kept at a temperature T w , and the fuid's free stream temperature is T ∞ with T w > T ∞ .Te x-axis is aligned in the direction of extension of the sheet, while the y-axis is normal to it.Te physical sketch of the fow is represented in Figure 1.
In two-dimensional steady fow, the velocity, temperature, and concentration felds take the following form: where u and v represent the respective x-and y-direction velocities.Depending on the above assumptions, we execute the boundary-layer analysis that leads us to the set of equations governing the conservation of continuity, momentum, energy, and nanoparticle concentration expressed, respectively, as follows: International Journal of Diferential Equations Here, τ � ((ρc p ) p /(ρc p ) f ) denotes the efective heat capacity ratio of nanoparticles to base fuid.All the other notations are indicated in nomenclature.As in [17], the boundary conditions suitable for the current problem are as follows: where v w , h t , T w are the mass transfer velocity, heat transfer coefcient, and wall temperature of the fuid, respectively.
For an optically thick medium, the Rosseland approximation for the radiative heat fux provides where k e is the coefcient of the mean absorption and σ ⋆ is the Stefan-Boltzmann constant.For nonlinear radiation, We adopt the following similarity transformations: where η is the similarity variable and ψ(x, y) denotes the stream function satisfying the equation of continuity u � zψ/zy, v � −zψ/zx.
Using equations ( 1)-( 3), (11), and ( 12), the transformed nondimensional form of governing equations ( 5)-( 8) and corresponding boundary conditions (9) are expressed as follows: With the boundary conditions, where a 0 ρc p is the heat source/sink parameter, Sc � ]/D B is the Schmidt number, c � k 0 /a 0 is the chemical reaction parameter, and is the Biot number.

Numerical Approaches
Bvp4c is a residual error-based mesh adaptive fnite difference program that executes the three-stage Lobatto IIIa formula.Te conventional method in Bvp4c provides a guess to the missed initial conditions.However, we found our problem very sensitive to the initial conditions and boundary values.Terefore, we adopted the continuation technique to resolve this problem.It is a tactic developed by Robert and Shipman [50].Tey used the method to solve boundary value problems that cannot be addressed by conventional shooting methods.Introducing variable χ i for i � 1, 2, . . .7, we reduce the order of equations ( 13)-( 15) into a system of 7 ODEs: For numerical computation, we have used the tolerance error, RelTol � 1e − 8. Next, to implement the continuation method, the following steps are taken in the Bvp4c MATLAB program: (1) Using a dummy parameter δ, write the system as a sum of its linear and nonlinear parts (2) Using normal guess and transformed boundary conditions (19), approximate the solution for (18) by the linear part (3) Employing computed values of step 2 and taking a small positive fraction of δ as a coefcient of nonlinear part, approximate the solution (4) Repeat step 3 with small increment in δ until we come up with approximate solution to the problem with in the prescribed error tolerance International Journal of Diferential Equations To examine the surface drag force, heat, and mass transfer rate, we have determined the skin friction, Nusselt number, and Sherwood number for Carreau fuid as follows: In a dimensionless form, the local Reynolds number is Re � u w x/]; the local skin friction, Nusselt, and Sherwood number, respectively, take the following form: .

Results and Discussion
Te transformed frst-order ODE (18), along with the corresponding boundary conditions (19), has been solved using the continuation technique implemented on MATLAB package bvp4c.Te infuence of leading parameters on 6 International Journal of Diferential Equations dimensionless fuid velocity, temperature, and nanoparticle concentration is discussed for both shearing cases in graphical and tabular forms.For the accuracy and verifcation of the method, a comparison in local shear stress has been made for the present and published results in [46] under some restricted conditions.Te Lorentz force efect resulting from the presence of applied magnetic feld is noticed in Figures 2-4.Tis force regulates the way of fuid fow. Figure 2 indicates for both dilatant and pseudoplastic fuids, an increase in the magnetic parameter value deteriorates the velocity feld.Here, the velocity curve downturn in the latter fuid type is higher than in the former one.Across the boundary layer, a sharp surge in the temperature feld is achieved in response to increasing values of M. A similar behavior is noticed in the nanoparticle concentration at the free stream, and it follows the reverse direction near the wall (see Figures 3 and 4).Tese results coincide with the physical meaning of increasing magnetic feld that intensifes the Lorentz force, which in turn has the capability of increasing the nanoparticle volume fraction in the motion of nanofuid and fuid temperature as well.Figures 5-7 illustrate the Weissenberg number (W e ) efect.Practically, We measures the time taken by the fuid to relax before regaining its original shape.An increase in W e amplifes the gap between shear-thinning and thickening regions of velocity, temperature, and concentration felds.
Enhancing W e stimulates velocity in shear-thickening cases and declines it in shear-thinning regions.Te opposite scenario is demonstrated in temperature and concentration felds.
Te local inertia coefcient and porosity parameter infuence on the momentum boundary layer are realized in Figures 8 and 9, respectively.For both regions, strengthening these parameters discouraged momentum thickness.
Temperature distribution responses to porosity, thermophoresis, and temperature ratio parameter values are demonstrated in Figures 10-12.Te increment in these parameters augmented the temperature feld and its corresponding thermal thickness.Te result in Figure 11 agrees with the fact that an increase in the thermophoresis parameter results in extra particles pushing away from the stretchable wall.As a result, the temperature distribution is facilitated.In Figure 12, the improvement of θ w initiates the conductivity of the fuid fow.Consequently, θ uplifts.However, as noticed in the fgures, shear-thinning fuids are more infuenced than shear-thickening fuids for the mentioned parameters.Te behavior of the temperature feld corresponding to variations in the Eckert number is captured in Figure 13.Te rise in E c imparts excess viscous heating to the fuid.Following this, kinetic energy is transformed into internal energy, which in turn pushes up the temperature curve.Figures 14-16 display the impacts of other pertinent parameters on temperature boundary layer thickness.Figure 14 describes that θ decreases with a rise in the value of the Prandtl number in the two fuid regions.Figure 15 shows us that a signifcant rise in temperature distribution can be achieved by a small increment in the Biot number, ζ 1 .Te efect of thermal radiation is analyzed in Figure 16.Here, the temperature distribution and its corresponding thermal thickness are escalated by the rise of R d .Substantially, the growth in radiation ray transports more heat energy to the running fuid, and in turn, a rise in thermal boundary layer thickness ensues.Here, we bear in mind that thermal boundary layer thickness is higher in the case of shear thinning than in shear-thickening behavior.Near the wall and in the free stream, their variation brings reversed efects.Figure 18 displays a growing behavior of the concentration feld for the rise in N t .Amplifying the thermophoretic parameter causes the microscopic transfer of nanoparticles from warmer to cooler regions, which in turn grows the nanoparticle concentration.An increase in the Brownian motion parameter declines the Figure 20 shows the expected response of nanoparticle concentration for an increase in the Schmidt number value.Te coefcient of mass difusion is inversely related to the Schmidt number.As a result, a decline in the distribution of nanoparticle concentration and its corresponding solutal thickness is realized.Figure 21 presents the nanoparticle concentration distribution to deliberate the infuence of variations of the chemical reaction parameter.It reveals a decaying nature in the nanoparticle concentration feld for the rise in c.International Journal of Diferential Equations Table 1 compares the shear stress values of the present result with the work of Siddiq et al. [46], under certain restricted conditions.It can be concluded that an excellent agreement is achieved.From Table 1, we notice that increases in parameters M, p, and α infated wall shear stress.On the contrary, an increase in n and W e defated it.
Table 2 executes the local heat transfer rate against various values of listed parameters.Except for the Eckert number, the surge in the heat transfer rate occurred with the rise in parameters P r , R d , θ w , and ζ 1 for both shear-thinning and thickening behaviors.Here, we perceive that the heat transfer rate in the shear-thickening region is better than in the shear-thinning region.International Journal of Diferential Equations Table 3 demonstrates the mass transfer rate response for various values of parameters N b , N t , c, Sc, and p r .For both behaviors, the rise in N t and p r encouraged the mass transfer rate, as opposed to the parameters N b , c, and Sc.

Conclusion and Future Directions
In the present work, we have numerically analyzed the boundary layer incompressible MHD Carreau nanofuid fow above an extending sheet under the efect of nonlinear thermal radiation, Joule heating, heat absorption/generation, chemical reaction, and Darcy-Forchheimer law with the incorporation of thermophoresis and Brownian motion, issuing shear-thinning and thickening behavior.Te continuation technique in the MATLAB bvp4c algorithm has been utilized to obtain numerical results.Te fndings of this study are summarized as follows: (1) Increasing M strengthens thermal and concentration boundary layers, while in contrast, the momentum boundary layer is decayed.(2) As the Weissenberg number scales up, the velocity feld for shear-thickening, the temperature and concentration felds for the shear-thinning case advance.On the other hand, the rise in this parameter results in a reverse efect on the velocity curve for the shear-thinning case and on the temperature and concentration profles for the shearthickening case.In our current work, we have employed the Fourier heat fux model and ignored the induced magnetic feld.Nonetheless, in numerous empirical endeavors, such as hightemperature plasma, power generation, and purifcation of crude oil, the induced magnetic feld efect plays a vital role.In future investigation, this problem can be extended for hybrid and trihybrid Carreau nanofuid convection and bioconvection by considering the induced magnetic feld as well as the Cattaneo-Christov heat fux model.Example.

Figure 1 :
Figure 1: Geometry of the fow problem.

Figure 11 :
Figure 11: Temperature variations contrasted with N t .

( 3 )( 7 )
Te rise in porosity and local inertia coefcient parameter penalizes the velocity feld.(4) An increment in R d from 0.1 to 0.4 lifted wall temperature by 20.62% at n � 1.5 and by 46.76% at n � 0.5.(5) Increasing the values of p, θ w , ζ 1 , and ω uplifts temperature distribution.On the contrary, an increase in P r diminishes this profle.(6) Te concentration feld is decayed by the rising value on N b , Sc, and c, whereas a rise in N t encouraged this profle.It is witnessed that increasing E c from 0 to 1.5, the average rise in temperature is 70.1% for shearthickening fuid and 21.76% for shear-thinning fuid.(8) By increasing thermophoresis from 0.1 to 0.4, the average rise in boundary layer temperature is 1.46% at n � 1.5 and 21.84% at n � 0.5.(9) When the values of M, p, and α are uplifted, wall friction aggravates.Te opposite result is achieved by surging the values of n and W e .

Table 1 :
[46]arison of shear stress −f ″ (0) of the present result with the work in[46]for diferent values of M, p r , α, p, and W e fxing θ w � 1, N b � N t � 0.1.

Table 2 :
Variations of local heat transfer rate Re − 1/2 Nu x for various values of p r , R d , E c , θ w , and ζ 1 at n � 0.5 and n � 1.5.

Table 3 :
Variations of local mass transfer rate Re − 1/2 Sh x for various values of N b , N t , c, Sc, and p r at n � 0.5 and n � 1.5