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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.

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 [

Here, we considered steady magnetohydrodynamic boundary layer nanofluid flow with a uniform velocity

The flow map diagram.

Here,

The appropriate extreme values are

Utilizing the Rosseland diffusion approximation [

Substituting (

Introduce the similarity transformations:

The stream function

Equation (

The governing variables appearing in (

The local Reynolds number is given by the equation

Using similarity variables in

The nonlinear flow expressions (ODEs) in (^{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

BVPh2 routine algorithm in Mathematica software.

Let us introduce the transformation variables as ^{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

Graphical comparison for two solutions in case of the velocity profile.

Graphical comparison for two solutions in case of the temperature profile.

Graphical comparison for two solutions in case of the concentration profile.

Numerical solution via the analytical solution for the velocity

Numerical solution | HAM solution | Absolute error | |
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Numerical solution via the analytical solution for the temperature

Numerical solution | HAM solution | Absolute error | |
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Numerical solution via the analytical solution for the nanoparticle concentration

Numerical solution | HAM solution | Absolute error | |
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Total residual error via the order of approximation.

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 (

Influence of the velocity

Influence of the velocity

Influence of the temperature

Influence of the temperature

Influence of the temperature

Influence of the

Influence of the

Influence of the temperature

Influence of the

Influence of the

Skin friction/drag force via

Nusselt number via

Sherwood number via

Figure

Effects of pertinent variables against physical quantities

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:

Velocity field

Velocity profile augmented with larger velocity parameter

Thermal field

A similar feature is viewed qualitatively for higher thermophoretic parameter

Solutal field

Larger values of radiation parameter

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

The authors declare that they have no conflicts of interest.

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).