Chemical MHD Hiemenz Flow over a Nonlinear Stretching Sheet and Brownian Motion Effects of Nanoparticles through a Porous Medium with Radiation Effect

<jats:p>In this paper, the numerical solutions for magneto-hydrodynamic Hiemenz fluid over a nonlinear stretching sheet and the Brownian motion effects of nanoparticles through a porous medium with chemical reaction and radiation are studied. The repercussions of thermophoresis and mass transfer at the stagnation point flow are discussed. The plate progresses in the contrary direction or in the free stream orientation. The underlying PDEs are reshaped into a set of ordinary differential equations employing precise transformation. They are addressed numerically using the successive linearization method, which is an efficient systematic process. The main goal of this study is to compare the solutions obtained using the successive linearization method to solve the velocity and temperature equations in the presence of m changes, thereby demonstrating its accuracy and suitability for solving nonlinear differential equations. For comparison, tables containing the results are presented. This contrast is significant because it demonstrates the accuracy with which a set of nonlinear differential equations can be solved using the successive linearization method. The resulting solution is examined and discussed with respect to a number of engineering parameters. Graphs exemplify the simulation of distinct parameters that govern the motion factors.</jats:p>

section_content: In a wide range of engineering applications, non-Newtonian fluids are frequently encountered. Industries such as paper, food, personal care, textile coating, and suspending solutions all use some of these applications, which are noteworthy. Three categories, differential, rate, and integral fluids, make up the majority of this classification. Recent advances in technology and engineering have led to the creation of a wide variety of non-Newtonian fluids with many significant variations from viscous fluids. In addition, modern breakthroughs in nanotechnology have made it much easier to make nanoparticles or nanostructures using nuclear or molecular methods that have better thermo-physical properties than their bulk equivalents. Nanofluids are created by distributing nanoparticles in a base fluid such as water, oil, or ethylene glycol, for example. In a two-dimensional container equipped with a Cu-water based nanofluid, the role of inclination angular position on natural convection heat transfer flow was explored by Abu-Nada and Oztop [1]. They concluded that heat transfer accumulation slows down as the Rayleigh number expands. Zargartalebi et al. [2] analyzed how a nanoparticle's stagnation point and volume fraction evolved over an isothermal stretching sheet. In the existence of thermophoresis and Brownian motion, Makinde and Aziz [3] explored the heat and mass transfer flow of an electrically conducting nanofluid over a radially stretching surface. 

Knowledge regarding the MHD flow of an electrically conducting fluid is pivotal in modern metallurgical and metalworking technologies. There has been a surge of attention Math. Comput. Appl. 2023, 28, 21 2 of 20 towards inspecting MHD flow and heat transfer in any medium because of the strong influence of a magnetic field on boundary-layer flow control and the efficiency of many systems using electrically-conducting fluids. Ahmad et al. [4] acknowledged the Laplace transform outcomes for the unsteady natural convective motion of revolving magnetohydrodynamics motion in a permeable medium over an oscillating sheet. Mbeledogu and Ogulu [5] inspected the effect of the chemical reaction of a magnetohydrodynamic free convective motion of a moving liquid over a vertical sheet. The exact solutions employing the Fourier sine and Laplace transform for magnetohydrodynamic transient rotating flow in a permeable medium in the existence of a magnetic field have been analysed by Salah et al. [6]. Additional interesting studies can be seen in [7] [8] [9] [10] and cross references. 

A chemical reaction is the process by which one or more substances, referred to as reactants, transform into one or more other substances, referred to as products. Substances consist of chemical components, or compounds. Chemical reactions produce a variety of products via rearrangement of the atoms in the reactants; see references [11] [12] [13] [14]. 

The Hiemenz flow pattern, including its employment in the monitoring of flows over submarine tips, ship tips, and aeroplanes, occupies a crucial role in the exploration of many industrial and natural phenomena. It is also essential in various of fields, including hydrodynamic processes, electronic fan cooling, and nuclear device freezing, to name a few. The optimum values of the aforementioned phenomena were provided by Ariel [15]. Motsa et al. [16] numerically ascertained the Maxwell fluid outcomes for two-dimensional Hiemenz flow on the way to a diminishing sheet. Parand et al. [17] assessed the Hiemenz flow with heat transfer through a porous medium of an incompressible non-Newtonian Rivlin-Ericksen fluid. 

Thermal radiation, the impact of thermophoretic diffusion on the Darcy-Forchheimer flow of nanofluid and melting heat transport and nanofluid in a nozzle of liquid rocket engine have all been studied [18] [19] [20]. 

Thermal radiation's ramifications on MHD flow and heat transfer have become crucially influential in many sectors. Thermal radiation heat transfer has a variety of potentials in space technology and projects that require a significant amount of heat. Hashim et al. [21] studied the implications of variable thermal conductivity on MHD Williamson nanofluid flow. A porous mechanism is a material that has a solid matrix and an interconnected void that allows fluid to flow through it. Zhang et al. [22] and Pandey and Kumar [23] reported the collective impact of thermal radiation and porous medium nanofluid flow. Additional fascinating and detailed work is available in [24] [25] [26] [27]. 

According to the existing literature, no attempt has been made to investigate the electrically conducting Hiemenz fluid over a nonlinear stretching sheet and Brownian motion. Thus, the purpose of this paper is to apply the findings of reference [28] to a broader problem, such as the effects of nanoparticles through a porous medium with chemical reaction. The motivation of this work is to compare the results obtained using the SLM technique to solve velocity, temperature and concentration equations in the presence of the power-law velocity exponent parameter changes, thereby demonstrating its accuracy and suitability for solving nonlinear differential equations. This work tabulates the effects of numerous flow parameters found in the governing equations and visually illustrates their effects. Employing SLM [29, 30], this study strives to pinpoint the consequences of thermal radiation on convective phenomena in MHD nanofluid over a non-linear stretching surface under heat generation and chemical reaction through a porous medium. This paper is structured as follows: Section 1 contains the literature survey; Section 2 contains the mathematical formulation; Section 3 contains the methodology; Section 4 presents the results; and Section 5 contains the conclusion.
section_name: Introduction

section_content: We review a non-linear continuously stretched horizontal plate impinging on a steady, two-dimensional, incompressible stagnation-point flow. The plate and free stream velocities are analogous to x m, while the magnetic field and mass transfer velocity are analogous to x (m-1)/2, where x is the distance around the plate from the plate's leading edge. Figure 1 describes the flow model. The relevant interpretations developed in this study are:
section_name: Mathematical Formulation

section_content: x and y axes are taken as the way of sheet motion and normal to the motion.
section_name: •

section_content: The nonlinear stretching velocity of the flat plate is assumed as u w (x) = u 0 x l m, where u 0 is a constant indicating the direction of the plate along the positive or negative side of the x axis, depending on whether u 0 > 0 or u 0 < 0, and a stationary plate when u 0 = 0, m is the power-law velocity exponent, and l is the characteristic length.
section_name: •

section_content: The ambient fluid's moving velocity has the form

where B 0 is a constant is assumed along the plate. 

• The nonlinear stretching velocity of the flat plate is assumed as

where 0 u is a constant indicating the direction of the plate along the positive ative side of the x axis, depending on whether 0 0 u > or 0 0 u <, and a st plate when 0 0 u =, m is the power-law velocity exponent, and l is the ch istic length. 

• The ambient fluid's moving velocity has the form

, wher a constant. 

• A variable magnetic field ( )

where 0

B is a constant is a along the plate. Under the above assumptions, the governing equations so obtained are giv for example, [28, 31]. The formulation of the present problem is modelled with respect to following presumptions:

1. 

Stagnation point flow.
section_name: •

section_content: Micropolar liquid model.
section_name: 2.

section_content: Joule heat, radiation, source/sink, porous medium and chemical reaction effects are deemed. 4. 

Thermophoresis and Brownian motion effects are taken into account. 

Under the above assumptions, the governing equations so obtained are given; see, for example, [28, 31]. 

∂u ∂x

where u, v are, respectively, the velocity constituents on the way to the x and y directions, σ is the electrical conductivity of the fluid, ρ is the fluid density, τ is the ratio of heat capacity of nanoparticles to the base fluid, D T is the thermophoretic diffusion coefficient, D B is the Brownian diffusion coefficient, α = k ρC p is the thermal diffusivity, k is the thermal conductivity, T is the fluid temperature, C is the fluid concentration, C p is the specific heat at constant pressure, T ∞ and C ∞ are the ambient temperature and concentration of the fluid, respectively, k 0 is the dimensional chemical reaction, and Q 0 is the heat source coefficient. 

The appropriate boundary conditions are

where u e (x) is the potential velocity, u w (x) is the velocity of the plate, and T w and C w are the plate temperature and plate concentration, respectively. The radiative heat flux is determined using Rosseland approximation

where σ * is the Stefan-Boltzmann constant and k * is the mean absorption coefficient. We recognise that the disparity in temperature within the flow ensures that in a Taylor's sequence, T 4 can be extended. Hence, establishing T 4 in a Taylor series about T ∞, we obtain

∞. Now, we introduce the following similarity transformations:

Substituting Equation ( 6 ) in Equations ( 2 )-( 5 ), we obtain

where Pr = ν α is the Prandtl number, M =

is the thermophoresis parameter,

is the heat generation/absorption parameter, Le = ν D B is the Lewis number, and γ = k 0 u ∞ x m-1 is the chemical reaction parameter. The boundary conditions are

where prime denotes differentiation with respect to η, V = u 0 u ∞ is the velocity ratio parameter, V > 0 reflects that the plate is progressing in the identical manner as the free stream velocity, V < 0 implies that the plate is heading in the contrary side of the free stream, and V = 0 stands for static plate. The case 0 < V < 1 designates that the plate's mobility is slower than that of the free-flowing fluid and V > 1 designates that the mobility is greater. V = 1 is the case when the plate and the fluid proceed at the same velocity. 

Non-dimensional skin friction coefficient C f and Nusselt number Nu x are

where τ w = µ(∇u) y=0, Nu x = xq w k(T w -T ∞ ) and the Sherwood number

where q w and q m are the heat flux and mass flux at the surface, respectively, given by

Substituting q w and q m in the preceding equation, we obtain

where Re x = u e x ν is the local Reynolds number.
section_name: 3.

section_content: We now adopt aforementioned initial guesses and linear operators to encapsulate the numeric solutions of Equations ( 7 )- (10). For the SLM solution we select the initial guesses functions f (η), θ(η), and θ(η) in the form

and the boundary conditions are

We construct the linearization equations

subject to the boundary conditions

where

and
section_name: Methodology (SLM)

section_content: The convergence of SLM solutions for different orders of approximations is presented in Table 1. To justify the accuracy of our method, we compare our numerical results with Uddin et al. [28] in Table 2, and Uddin et al. [28] and Yih [31] in Table 3. The comparisons are satisfactory. Table 4 shows the numerical values of the skin friction coefficient, local Nusselt number and local Sherwood number for different emerging parameters; the Nusselt number and Sherwood number decrease when there is an increase in the velocity ratio parameter V. For a wide range of physical characteristics, tables and charts are often assembled to ascertain and describe the nature of flow, temperature, concentration, skin friction coefficient, and local Nusselt and Sherwood numbers. We check the following values across the exploration, apart from renovated quantities, as presented in the tables and charts.
section_name: Results and Discussion

section_content: -f " (0) -θ ' (0) -φ ' (0) Figures 2-4 illustrate the impression of magnetic parameter M on distributions. It is insinuated that as M strengthens, the velocity distribution of the fluid degrades. Whenever a magnetic field is imparted to an electrically conducting fluid, the Lorentz force generates, and this energy contradicts the flow pattern, forcing velocity drawings to deteriorate. This is because an increase in the Lorentz force creates resistance to fluid flow, resulting in a drop in the velocity profile, while the thermal and solutal boundary layer thickness improves. As the space variable turns away from the boundary surface, modifications in the velocity exponent parameter m restrain the growth of the momentum boundary layer, which reaches zero. The plots of temperature and concentration find similar results. This is shown in Figures 5 6 7. As the space variable turns away from the boundary surface, modifications in the velocity exponent parameter m restrain the growth of the momentum boundary layer, which reaches zero. The plots of temperature and concentration find similar results. This is shown in Figures 5 6 7. As the space variable turns away from the boundary surface, modifications in the velocity exponent parameter m restrain the growth of the momentum boundary layer, which reaches zero. The plots of temperature and concentration find similar results. This is shown in Figures 5 6 7. Fluid velocity amplifies as the velocity ratio parameter V grows, whereas fluid temperature and concentration drop as the velocity ratio parameter boosts. This is observed in Figures 8 9 10. Fluid velocity amplifies as the velocity ratio parameter V grows, whereas fluid temperature and concentration drop as the velocity ratio parameter boosts. This is observed in Figures 8 9 10. Fluid velocity amplifies as the velocity ratio parameter V grows, whereas fluid temperature and concentration drop as the velocity ratio parameter boosts. This is observed in Figures 8 9 10.
section_name: Order

section_content: Nt to appear. The inclusion of nanoparticles is correlated to the thermal conductivity liquids. When the amplitude of Nt is improved, the thermal conductivity of the flu boosts, and this greater thermal conductivity leads to a high temperature. We also iden fied that relatively high Nt values result in greater nanoparticle concentrations. This shown in Figures 13 and 14. The inclusion of nanoparticles externally allowed the thermophoresis parameters Nt to appear. The inclusion of nanoparticles is correlated to the thermal conductivity of liquids. When the amplitude of Nt is improved, the thermal conductivity of the fluid boosts, and this greater thermal conductivity leads to a high temperature. We also identified that relatively high Nt values result in greater nanoparticle concentrations. This is shown in Figures 13 and 14.
section_name: The inclusion of nanoparticles externally allowed the thermophoresis paramet

section_content: Nt to appear. The inclusion of nanoparticles is correlated to the thermal conducti liquids. When the amplitude of Nt is improved, the thermal conductivity of th boosts, and this greater thermal conductivity leads to a high temperature. We also fied that relatively high Nt values result in greater nanoparticle concentrations. 

shown in Figures 13 and 14.
section_name: The inclusion of nanoparticles externally allowed the thermophoresis param

section_content: This paper presents the Brownian motion effects of nanoparticles through porous media with chemical reaction and numerical solutions of MHD Hiemenz flow over a nonlinear stretching sheet. The analytical exploration of two-dimensional steady forced convective flow of a Newtonian fluid past a convectively heated vertically moving plate towards the face of a variable magnetic field and the radiation factor is reviewed in this report using SLM. The impact of the Prandtl number Pr, magnetic parameter M, per- meability parameter Ω, radiation parameter R, Brownian motion parameter Nb, thermophoresis parameter Nt, heat generation/absorption parameter Q, Lewis number Le, and chemical reaction parameter γ are investigated and presented in tables. The validity of the current results was tested and compared with those that had been previously published [28, 31]. Tables 2 and 3 show limited examples where there is strong agreement. 

The crucial insights reached from the graphical and numerical solutions to the problem are:

• The temperature profile is significantly driven by the heat source parameter. 

• Thermal radiation and thermophoresis parameters lead enhanced temperature. 

• The concentration profile lowers as both the Lewis number and the chemical reaction parameters expand. 

• The rate of heat transfer elevates with R and Pr. 

• The rate of mass transfer elevates with Le and γ. Assessments of f (0) using the recent literature are undertaken in Table 2, and the statistics indicate a strong and positive correlation. Tables 2 and 3 compare skin friction andf (0), respectively, for various combinations of M, V and m.
section_name: Conclusions

section_content: This paper presents the Brownian motion effects of nanoparticles through porous media with chemical reaction and numerical solutions of MHD Hiemenz flow over a nonlinear stretching sheet. The analytical exploration of two-dimensional steady forced convective flow of a Newtonian fluid past a convectively heated vertically moving plate towards the face of a variable magnetic field and the radiation factor is reviewed in this report using SLM. The impact of the Prandtl number Pr, magnetic parameter M, permeability parameter Ω, radiation parameter R, Brownian motion parameter Nb, thermophoresis parameter Nt, heat generation/absorption parameter Q, Lewis number Le, and chemical reaction parameter γ are investigated and presented in tables. The validity of the current results was tested and compared with those that had been previously published [28, 31]. Tables 2 and 3 show limited examples where there is strong agreement. 

The crucial insights reached from the graphical and numerical solutions to the problem are:

• The temperature profile is significantly driven by the heat source parameter.
section_name: Conclusions

section_content: Thermal radiation and thermophoresis parameters lead enhanced temperature.
section_name: •

section_content: The concentration profile lowers as both the Lewis number and the chemical reaction parameters expand.
section_name: •

section_content: The rate of heat transfer elevates with R and Pr.
section_name: •

section_content: The rate of mass transfer elevates with Le and γ.
section_name: •

section_content: Funding: This research received no external funding. 

Math. Comput. Appl. 2023, 28,21

section_name: Data Availability Statement:

section_content: The data used to support the findings of this study are included within the article. The data generated using MATLAB code are presented in tables and figures included in the results and discussion section of the manuscript.
section_name: Data Availability Statement:

section_name: Conflicts of Interest:

section_content: Author Contributions: Conceptualization, F. S. and K. K. V. ; methodology, F. S. ; software, F. S. ; validation, F. S., A. O. M. S. and K. K. V. ; formal analysis, F. S. ; investigation, A. O. M. S. ; data curation, F. S. ; writing-original draft preparation, F. S. ; writing-review and editing, K. K. V. ; visualization, A. O. M. S. ; supervision, K. K. V. ; project administration, A. O. M. S. ; funding acquisition, K. K. V. All authors have read and agreed to the published version of the manuscript.

section_content: The authors declare that there are no conflicts of interest regarding the publication of this paper.
section_name: Conflicts of Interest: