#### Abstract

The increasing global demand for energy necessitates devoted attention to the formulation and exploration of mechanisms of thermal heat exchangers to explore and save heat energy. Thus, innovative thermal transport fluids require to boost thermal conductivity and heat flow features to upsurge convection heat rate, and nanofluids have been effectively employed as standard heat transfer fluids. With such intention, herein, we formulated and developed the constitutive flow laws by utilizing the Rossland diffusion approximation and Stephen’s law along with the MHD effect. The mathematical formulation is based on boundary layer theory pioneered by Prandtl. Governing nonlinear partial differential flow equations are changed to ODEs via the implementation of the similarity variables. A well-known computational algorithm BVPh2 has been utilized for the solution of the nonlinear system of ODEs. The consequence of innumerable physical parameters on flow field, thermal distribution, and solutal field, such as magnetic field, Lewis number, velocity parameter, Prandtl number, drag force, Nusselt number, and Sherwood number, is plotted via graphs. Finally, numerical consequences are compared with the homotopic solution as a limiting case, and an exceptional agreement is found.

#### 1. Introduction

In the recent development, nanofluid has gained considerable attention from researchers, engineers, scientists, and mathematicians due to its significant implementations in diverse fields of sciences. These applications cover the following areas: chemical engineering, space science, nuclear science, solar energy collection, and several other areas. The nanofluid applications can also be employed in other real-world problems which include engine oils, heat exchangers, and thermal conductivity [1]. The word nanofluid is considered to incorporate small nanoparticles whose dimension is up to 1–100 nm in the base liquid; biofluid, lubricants, oil, and ethylene are the common examples of nanofluids [2]. Eastman et al. [3] studied to develop the thermal behavior of nanofluids by incorporating various nanosized material particles to base fluids. Chamkha et al. [4] examined radiation effects on mixed convection in view of the vertical cone embedded in the porous medium with the nanoliquid. The influence of hydromagnetic free convective and heat transfer was analyzed by Sheikholeslami et al. [5]. The consequences of MHD flow and viscous dissipation on the momentum boundary layer of the nanoliquid were evaluated by Abbas and Sayed [6]. The hydromagnetic flow of nanofluids over a revolving disk was reported by Mahanthesh et al. [7]. Later on, various potential investigations have been carried out by many researchers and engineers into the development and implications of these fluids [8–11]. Heat transfer rheology in convective flow nanofluids with thermal conductivity and electrical behavior received exceptional importance for their fruitful applications in life sciences and engineering development. Such applications include solar energy, nuclear reactor, and cooling and heating mechanisms. Using Buongiorno’s model, Shehzad et al. [12] examined the effect of convective heat flow of the nanoliquid. Shen et al. [13] investigated the heat flow of a nanoliquid by a stretching surface with thermal radiation effect and velocity slip. Jahan et al. [14] evaluated the numerical solution to understand heat transfer aspects in nanomaterials over a convectively permeable stretching surface with radiation effect. Hamad and Pop [15] described thermal radiation effects on unsteady nanoliquid over an oscillatory moving plate with the heat reservoir. Sheikholeslami and Ganji [16] conducted a numerical consequence to analyze the three-dimensional nanoliquid with thermal radiation effects in a revolving system. Hussain et al. [17] illustrated analytical results by employing Laplace transform to explore unsteady hydromagnetic flow over a rotating system subject to chemical reaction and Hall current. Several mathematicians and investigators reported the thermal conductivity and electrical conductivity nature of nanofluids under different conditions with various geometry convective heat transfer effects in [18–23]. Mahanthesh et al. [24] evaluated the impact of suspended nanoparticles on the convective flow of nanomaterial in view of the vertical surface with radiation effects. Kumar et al. [25] conducted a numerical study by employing the RK-4 method to analyze radiative Jeffrey nanofluid flow with convective boundary conditions. Raza et al. [26] numerically studied the influence of viscous dissipation and magnetic field of molybdenum disulfide nanoliquid with the shape effect. Al-Odat et al. [27] addressed the interaction of magnetic effects and boundary layer flow of the fluid by an exponentially stretchable sheet. Chamkha and Aly [28] presented the numerical solution to magnetohydrodynamic free convection flow of a nanoliquid by a vertical plane in view of the porous medium and radiation effects. Aliakbar et al. [29] studied the analytical solution to hydromagnetic flow of upper convective Maxwell fluid past a stretchable extended surface. Khan et al. [30] evaluated the numerical solution by utilizing the explicit finite difference method (FDM) with stability analysis to unsteady nanofluid flow by a stretching surface. Ibrahim and Shankar [31] employed the shooting technique to inspect magnetohydrodynamic boundary layer flow of nanomaterial over the stretching sheet with the slip condition. Sparrow and Cess [32], Ozisik [33], Siegel and Howell [34], Howell [35], Takhar et al. [36], and Hossain and Takhar [37] all contributed well to radiative heat transfer analysis.

#### 2. Basic Flow Equations

Here, we considered steady magnetohydrodynamic boundary layer nanofluid flow with a uniform velocity moving towards an infinite plate. The velocity of the infinite plate is defined by the relation ; here, denotes the velocity parameter. The nanofluid flow is confined at . The coordinate system is chosen in the form such that the is normal to the direction of flow, and the magnetic interaction is employed normal to the plate. Let be the fluid temperature and be the concentration at the wall, and free-stream numbers are and . The proposed model is described by the following set of differential equations. The flow map and coordinates axes are presented in Figure 1.

Here, are the velocity components in coordinates axes, is the kinematic viscosity, is the thermal conductivity parameter, denotes the heat flux, represents Brownian diffusion coefficient, represents the thermophoresis diffusion coefficient, denotes the field strength, is the electrical conductivity parameter, represents the ratio of the nanoparticle heat capacity to the base fluid heat capacity, is the shear stress, represents thermal diffusivity, is the velocity parameter, corresponds to the downstream motion of the plate from the origin, and corresponds to the upstream motion.

The appropriate extreme values are

Utilizing the Rosseland diffusion approximation [35], thermal flux is defined aswhere is the Stefan–Boltzman constant and is the Rosseland mean absorption coefficient. The difference in the nanofluid temperature within the fluid is sufficiently small such that can be written as a linear function of temperature:

Substituting (6) and (7) in (3), we attained

Introduce the similarity transformations:

The stream function may be expressed as

Equation (1) is justified automatically, while equations (2)–(5) have the forms

The governing variables appearing in (11)–(13) are defined as follows.

, , , , , , , , and label the radiation constraint, Prandtl number, Lewis number, Hartmann number, Brownian parameter, thermophoretic force, drag force, and Nusselt and Sherwood numbers.

The local Reynolds number is given by the equation .

Using similarity variables in , we get the dimensionless form as

#### 3. Numerical Solution and Convergence Analysis

The nonlinear flow expressions (ODEs) in (11)–(13) subject to boundary conditions in (14) are first transformed into 1^{st}-order ODEs and then tackled numerically by employing a built-in computational algorithm BVPh2 in Mathematica software. The routine flow numerical code is demonstrated in Figure 2. Step size , and relative tolerance error is set; in addition, the choice of confirms that all numerical approximations approach correctly to asymptotic values.

Let us introduce the transformation variables as ; hence, the following system of 1^{st}-order seven differential equations are generated:

The transfer conditions are

For authentication purpose, the computational results are further tested by the use of an analytical scheme (HAM), and a reasonable agreement has been obtained in two solutions. The attributes of two solutions via graphs are shown in Figures 3–5, and the tabularized data for velocity and thermal and solutal fields are presented in Tables 1–3. Finally, the residual error analysis has been evaluated and shown in Figure 6. A decrease in error is perceived for higher-order deformations.

#### 4. Discussion

The current computational results accomplished by a numerical algorithm BVP2 unveil the influence of pertinent governing constraints on velocity, thermal field, and concentration profile. The impact of various emerging parameters in flow equations (11)–(13) is plotted through Figures 7–19. The numerical values of these flow factors are regarded as .

Figure 7 describes the Hartmann number effect on the nanofluid velocity profile . As anticipated, dwindles when subject to upsurge in . In reality, this figure revealed that augmentation in boosts Lorentz force. In consequence, velocity diminishes. The contribution of velocity parameter on nanofluid velocity profile is evaluated through Figure 8. As perceived, in this figure, fluid velocity enhances when upsurges. Hence, upsurges. The attributes of thermal field curves for magnetic field are disclosed in Figure 9. One can perceive that is a growing function of . In reality, the heat transfer rate of nanofluid particles boosts up through larger . Consequently, escalates. Such a scenario is perceived because higher implies larger Lorentz force provides more resistance which makes increases the fluid flow. In consequence, augments. Figure 10 explains variations in thermal field subjected to radiation parameter . This figure unveils enhancement for higher values of . In fact, working nanofluid acquires extra heat subject to the radiation factor. In consequence, upsurges. ;Figure 11 reveals variations in subject to thermophoresis parameter . Here, thermal field increases with increasing Nt. Physically, the thermophoretic force rises as parameter is escalated. Such force is responsible to move small size particles by hotter towards colder region. Consequently, escalates. Figure 12 explains the Brownian motion parameter effect on . As anticipated, thermal field enhances through larger parameter. In nanofluids, Brownian motion ascends due to small-size nanoparticles, and at this point, nanoparticle motion rate and its effect against the fluid have a vital vibrant role regarding heat transport. In consequence, upsurges in produce active nanoparticle motion within the base fluid. The result of disordered nanoparticle motion develops kinetic energy of the nanoparticles, and eventually, thermal behavior of the fluid augments. The contribution of Prandtl number on is evaluated through Figure 13. Here, thermal field diminishes against larger Prandtl number estimations. Attributes of on solutal field are interpreted in Figure 14. We perceived an increase in subjected to higher thermophoresis parameter estimations. In reality, an upsurge in thermophoresis force is viewed through greater parameter which is responsible for moving the fluid particles from higher temperature to lower temperature. In consequence, profile boosts. Solutal field curves for Hartmann number are unveiled in Figure 15. Clearly, the solutal field is the augmenting function of the Hartmann number. Mass transfer augments when is enhanced. Accordingly, increases. The attributes of Lewis number parameter on solutal field are interpreted in Figure 16. Clearly, diminishes when is increased. Physically, Lewis number signifies the influence of thermal diffusion on mass diffusion in the boundary layer region. Such a scenario is noticed because higher implies lessening in the solutal field and boundary layer.

Effects of pertinent variables against physical quantities are described in Figures 17–19. These figures highlight decay in for larger estimations.

#### 5. Closing Remarks

The aim of this research is to analyze two-dimensional incompressible viscoelastic magnetonanofluid flow with the Buongiorno model. This investigation further includes results of heat generation/absorption with convective conditions. Current investigation enables us to explain the following key outcomes:(i)Velocity field lessens when subject to increment in the Hartmann number , and thermal field develops with magnetic strength(ii)Velocity profile augmented with larger velocity parameter (iii)Thermal field upsurges when radiation parameter and Hartmann number are improved(iv)A similar feature is viewed qualitatively for higher thermophoretic parameter and Brownian motion variable (v)Solutal field boosts through larger Hartmann number , and field dwindles while Lewis number augments(vi)Larger values of radiation parameter and Hartmann number diminish skin friction (drag force), Nusselt number , and Sherwood number

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under the project number (RGP-2019-6).