Mixed Convection and Thermally Radiative Flow of MHD Williamson Nanofluid with Arrhenius Activation Energy and Cattaneo–Christov Heat-Mass Flux

Department of Mathematics, Dr. N. G. P. Arts and Science College, Coimbatore, Tamil Nadu, India Department of Mathematical Sciences, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia Department of Computer Engineering, Bolu Abant Izzet Baysal University, Bolu, Turkey Department of Humanities and Management, Jigme Namgyel Engineering College, Royal University of Bhutan, Dewathang, Bhutan


Introduction
Cooling and heating procedures are essential in many industries, and fluids make this process. e effectual cooling techniques are essential for cooling a higher thermal system in a short time. However, ordinary fluids such as ethylene glycol, engine oil, and water have poor thermal conductivity and do not fulfill the demand for powerful heat transfer cooling agents. Considering the needs of modern industry, including microelectronics, chemical production, and power generation plants, we need to establish a new type of fluids that will be efficient in cooling thermal systems. Nanofluid is a fluid consisting of nanoparticles (nanosized particles) such as oxides, nitrides, carbides, and metals stably and uniformly suspended in a base fluid. ese fluids overcome the difficulty of the base fluids and act as an agent of efficient cooling. e nanofluid flow on a stretchy sheet was reported by Khan and Pop [1]. ey noticed that the fluid temperature grows when the quantity of thermophoresis parameter is greater. Kuznetsov and Nield [2] addressed the natural convective flow of nanofluid on a plate. ey noted that the heat transfer rate becomes less in the presence of the Brownian motion parameter. Goyal and Bhargava [3] derived the numerical solution of viscoelastic nanofluid on a sheet under velocity slip condition. eir outcomes clearly show that the thermophoresis parameter leads to deceleration in the fluid temperature.
e Titania nanofluid flow in a cylindrical annulus was illustrated by Mebarek-Oudina [4]. e problem of bioconvective flow of MHD tangent hyperbolic nanofluid subject to Newtonian heating was solved by Shafiq et al. [5]. ey detected that the nanoparticle concentration suppresses when rising the thermophoresis parameter. Mabood et al. [6] illustrate the consequence of MHD flow of hybrid nanofluid on a wedge with thermal radiation. ey proved that the fluid velocity uplifts when enriching the magnetic field parameter.
In nature, heat transference occurs due to the temperature difference between one body to another body or within the same body. In the past, the heat transfer phenomenon was mostly addressed by using "Fourier's law of heat conduction." However, this law is not sufficient to express the fundamental characteristic of heat transfer.
at is, each part of the entire object having an initial disturbance. In general, there is no material satisfying this property. To overcome this complication, Cattaneo [7] incorporated the thermal relaxation in Fourier's theory which implements the heat transport is identical to the propagation of thermal waves with normal speed. Christov [8] upgraded the Cattaneo model by recommending the thermal relaxation time with upper convected Oldroyd's derivatives for the frame-invariant formation. e timedependent flow of nanofluid with Cattaneo-Christov double diffusion was examined by Ahmad et al. [9]. ey noticed that the thermal relaxation parameter declines the fluid temperature. Reddy and Kumar [10] delivered the impact of Cattaneo-Christov heat flux of micropolar fluid on carbon nanotubes. e 2D incompressible flow of heatgenerating/-absorbing Oldroyd-B fluid with Cattaneo-Christov heat flux on an uneven stretching sheet was portrayed by Ibrahim and Gadisa [11]. ey proved that the heat flux relaxation time parameter leads to thinning the thermal boundary layer thickness. Kumar et al. [12] analyzed the significance of Cattaneo-Christov flow on a cone. ey detected that the smaller heat transfer gradient occurred in the wedge than the cone for varying the thermal relaxation time parameter. Some recent developments on this concept are collected in [13][14][15][16][17][18][19]. e convective fluid flows on a porous medium play a vital role in many science and engineering systems. Some examples are crude oil production, heat exchanger layouts, groundwater systems, grain amassing, nuclear waste disposal, warm insurance outlining, fossil fuels beds, and many others. Darcy developed a semiempirical equation that uses in low porosity and low-velocity conditions. ese empirical equations were not sufficient for a larger Reynolds number. In this situation, Forchheimer [20] was developed a new model named as the Darcy-Forchheimer model, which includes the square velocity term in the Darcian model. Pal and Mondal [21] derived the numerical solution of MHD fluid flow on Darcy-Forchheimer porous medium. ey discovered that the mass transfer gradient accelerates for more availability of local inertia parameters. e dual solution of forced convective stagnation-point flow on a Darcy-Forchheimer porous medium over a shrinking sheet was derived by Bakar et al. [22]. ey achieved that the fluid temperature declines in the first solution and enhances the second solution when raising the porosity parameter. Meraj et al. [23] inspected the Darcy-Forchheimer flow of Maxwell fluid with Cattaneo-Christov heat flux theory. ey acknowledged that the thermal boundary layer thickness becomes high for a larger quantity of the porosity parameter.
In recent decades, many researchers are willing to study the chemical reactions and activation energy because they have more industrial applications. Few applications are fog formation, fibrous insulation, thermal oil recovery, cooling of nuclear reactors, etc. e mixed convective flow of Carreau nanofluid flow with activation energy was illustrated by Javed et al. [36]. Zaib et al. [37] explored the consequences of a binary chemical reaction and activation energy of a nonlinear radiative flow of Casson nanofluid on a Darcy-Brinkman porous medium. ey detected that the thickening of the solutal boundary layer thickness when raising the activation energy parameter. e impact of activation energy of an electrically conducting Carreau nanofluid flow in a stagnation point was discussed by Hsiao [38]. Time-dependent MHD natural convective flow with Arrhenius activation energy was analyzed by Maleque [39]. He noticed that the activation energy is enhancing the nanofluid concentration. Mabood et al. [40] portray the outcomes of Arrhenius activation energy effect on micropolar fluid on a thin needle. ey concluded that the Sherwood number decelerates for upsurging values of the activation energy parameter. e results of Arrhenius activation energy of a tangent hyperbolic fluid was revealed by Kumar et al. [41]. A variety of studies on this direction was found in [42][43][44][45].
e primary objective of this paper is to portray the 2D Darcy-Forchheimer radiative flow of Williamson nanofluid with subject to activation energy and heat absorption. e thermophoresis and Brownian motion effects are taking into account. e energy and mass equation models are constructed via Cattaneo-Christov heat-mass flux theory. e Darcy-Forhemmier flow of radiative Williamson nanofluid with activation energy and Cattaneo-Christov dual flux was not examined yet. So, we fill this gap and will give a significant contribution to the existing investigations. Generally, Williamson nanofluid has a wide range of usages in biological engineering; especially, it is used for computing 2 Journal of Mathematics the heat and mass transmission through the vessels in blood and hemodialysis, see [46]. e impact of pertinent parameters of the governing model of velocity, temperature and nanofluid concentration, local skin friction, local Nusselt number, and local Sherwood number are examined in terms of tables, charts, and figures.

Mathematical Formulation
We exhibit the steady mixed convective flow of 2D Williamson nanofluid on a Darcy-Forchheimer porous medium over a stretchy sheet. Let x− axis is considered in the flow direction and y− is perpendicular to the flow. e uniform magnetic effect B 0 is applied in the y− direction and the induced magnetic effect excluded becomes a small quantity of Reynolds number. e fluid temperature and nanofluid concentration nearby the boundary is T w and C w which is larger than the ambient fluid temperature T ∞ and concentration C ∞ , respectively. e Cattaneo-Christov model replaced Fourier's heat conduction law. e consequences of activation energy and binary chemical reaction are considered for our study. In addition, the fluid is heat consumption/generating. Under the above considerations, the governing flow problems are (see [47]) where All symbols are defined in the nomenclature part.
With the boundary conditions, Define Journal of Mathematics e corresponding ODE's are 1 All parameters are defined in the nomenclature part. e corresponding boundary conditions are e dimensionless form of wall shear stress, heat, and mass flux are expressed as

Numerical Solutions
e ODE models (5)-(7) with associative conditions (8) are numerically solved by implementing MATLAB bvp4c procedure. In this regard, first, we change the 2 rd and 3 rd order ODE into a system of first-order ODE. Let Journal of Mathematics e system of first-order ODEs is as follows: under the boundary conditions, e numerical procedure needs initial calculation with tolerance 10 − 6 .

Results and Discussion
is section scrutinises the consequences of pertinent parameters on velocity, temperature, nanofluid concentration, skin friction coefficient, local Nusselt number, and local Sherwood number with a fixed quantity of Prandtl and Schmidt numbers. Table 1 provides the comparison of our numerical results and Mustafa et al. [47] results. We achieved that our results are exactly matched with Mustafa's results. e estimation of We, λ, Fr, M, Ri, and Nr on skin friction coefficient, local Nusselt number, and local Sherwood number was presented in Table 2. We noticed that the surface shear stress accelerates when enhancing the We and Ri values, and it decelerates for heightening the quantity of λ, Fr, M, and Nr values. e heat transfer gradient grows when growing the values of Ri, and it diminishes when upgrading the We, λ, Fr, M, and Nr values. Quite the opposite results are attained in the local Sherwood number. Table 3 Table 4. We found that the mass transfer gradients decelerate for the small quantity of σ * * , and after that, it enriches for higher magnitudes. e LSN develops when developing the values of n, Nb, and δ, and the opposite trend was obtained for the more presence of E, Γ C , and Nt. Figures 1(a) and 1(b) explain the impact of We and M on DFF and NDFF in velocity profile. We found that the fluid velocity decelerates for rising the values of We and M. Physically, a higher Weissenberg number leads to enriching the fluid relaxation time, and this causes to slow down the motion of the fluid particles. e higher magnitude of the magnetic field parameter develops the fluid resistance and this causes to suppress the motion of the fluid particles. Also, we have seen that higher momentum boundary layer thickness occurs in NDFF compared to the DFF.      We exposed that the fluid temperature develops for increasing the Nt values and the opposite trend was obtained for Nb values. Also, we noticed that the reaction rate leads to suppressing of the thermal boundary layer thickness. e effects of Hg and R on temperature distribution were plotted in Figures 6(a) and 6(b). We noted that the fluid temperature raises for rising the Hg and R values. In addition, we found the larger thermal boundary layer thickness attains in the FHF model compared to the CCHF model. Figures 7(a) and 7(b) portray the consequences of fw and Nt for nanofluid concentration profile. We concluded that the nanofluid concentration enhances near the plate and falling-off away from the plate. e Nt values lead to enriching the nanofluid concentration boundary layer thickness. e nanofluid concentration distribution for different values of Nb and Γ C were shown in Figures 8(a) and 8(b). ese figures clearly show that the nanofluid concentration is an increasing behavior for Γ C and quite the opposite occurs for Nb values. Also, we noticed that the reaction rate leads to suppressing of the nanofluid concentration boundary layer thickness.

Conclusions
is analysis clearly shows the consequences of thermal radiation of a Darcy-Forchheimer flow of Williamson nanofluid on a stretchy plate with a magnetic field. e energy and nanoparticle concentration equations were framed with Catteneo-Christov heat-mass flux theory. Additionally, the mass transfer analysis is made by activation energy and binary chemical reaction. e governing PDE problems were converted into ODE problems by applying suitable variables, and these equations were solved using MATLAB bvp4c algorithm. e salient outcomes of the current analysis are outlined as below: (i) e fluid velocity decelerates when enhancing the Williamson fluid, magnetic field, and porosity parameters, and it accelerates by increasing the Richardson number. (ii) e fluid temperature accelerates when strengthening the heat generation/absorption radiation and thermophoresis parameters, and it declines when increasing the Brownian motion parameter.
(iii) e fluid concentration suppresses when increasing the Brownian motion parameter, and it enhances when escalating the thermophoresis and mass relaxation time parameters.
(iv) e smaller SFC occurs in the non-Darcy-Forchheimer flow of Williamson nanofluid. (v) e larger heat transfer gradient exists in viscous nanofluid without radiation.