Influence of Thermophoresis and Brownian Motion of Nanoparticles on Radiative Chemically-Reacting MHD Hiemenz Flow over a Nonlinear Stretching Sheet with Heat Generation

In this study, a radiative MHD stagnation point flow over a nonlinear stretching sheet incorporating thermophoresis and Brownian motion is considered. Using a similarity method to reshape the underlying Partial differential equations into a set of ordinary differential equations (ODEs), the implications of heat generation, and chemical reaction on the flow field are described in detail. Moreover a Homotopy analysis method (HAM) is used to interpret the related mechanisms. It is found that an increase in the magnetic and velocity exponent parameters can damp the fluid velocity, while thermophoresis and Brownian motion promote specific thermal effects. The results also demonstrate that as the Brownian motion parameter is increased, the concentration values become smaller.


Introduction
Nanofluids are nanometer-sized particles less than 100 nanometres in size that are introduced into base fluids such as oil, water, bio fluids, ethylene, and lubricants. Despite their essential worth in industry, medicine, and a variety of other efficacious domains of science and technology, countless researchers have gained an interest in nanofluids as opposed to other fluids. However, nanofluids still occupy an indispensable key position in medical sectors, such as the use of gold nanoparticles in the screening of cancerous tumours and the processing of minuscule bombs that are exploited to eradicate cancerous tumours. Choi [1] was the one who came up with the idea of nano materials. He inferred from his observations that infusing these particles strengthens the thermal conductivity of the fluid. Hayat et al. [2] produced analytical solutions for MHD nanofluid squeezing flow between two parallel plates. Hussain et al. [3] studied the dynamics of Jeffery nano-fluid across an exponentially stretched sheet, but also radiation consequences. In a magnetised nanofluid flow, Abbas et al. [4] investigated the role of thermal radiation and chemical reaction response. Hayat et al. [5] studied the characteristics of heat and mass transfer by imposing the convective conditions. Ganesh Kumar et al. [6] have investigated the boundary layer flows and melting heat transfer of a Prandtl fluid over a stretching surface in the presence of fluid particle suspensions. Jalali et al. [7] conducted a numerical investigation of the competition between viscosity and thermal conductivity about their effects on heat transfer by Al 2 O 3 -water nanofluid. Much insight on this theme can be found in [8][9][10][11][12].
Modern metallurgical and metal-working technologies rely heavily on understanding MHD flow of an electrically-conducting fluid. Mabood et al. [13] acknowledged the Laplace transform outcomes for the unsteady natural convective motion of revolving magnetohydrodynamics motion in a permeable medium over an oscillating sheet. Mahantesh et al. [14] inspected the effect of the chemical reaction of a magnetohydrodynamic free convective motion of a moving liquid over a vertical sheet. The problem of Marangoni mixed convection in the presence of an inclined magnetic field with uniform strength in a nanofluid is addressed numerically by Sastry et al. [15].
The Hiemenz flow pattern, including its applications 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's also essential in various of fields, including hydrodynamic processes, electronic fan cooling, and nuclear device freezing, to name a few. The optimum values to the aforementioned phenomenon were provided by Ariel [16]. Motsa et al. [17] ascertained numerically the Maxwell fluid outcomes for two-dimensional Hiemenz flow on the way to a diminishing sheet. Parand et al. [18] assessed Hiemenz flow with heat transfer through a porous medium of an incompressible non-Newtonian Rivlin-Ericksen fluid.
In many industries, the significance of thermal radiation on MHD flow and heat transfer is rapidly getting crucial. Heat transfer by thermal radiation has substantial applications in space technology and projects that grasps high temperatures. Aamir Hamid et al. [19] looked at the implications of variable thermal conductivity on MHD Williamson nanofluid flow. A porous mechanism is a material that has a solid matrix and has an interconnected void that allows fluid to flow through it. Pandey et al. [20] reported the collective impact of thermal radiation and porous medium nanofluid flow. Bandari [21] examined the steady state of the two-dimensional incompressible magnetohydrodynamics (MHD) flow of a micropolar nanofluid over a stretching sheet in the presence of chemical reactions, radiation and viscous dissipation. Inayat et al. [22] examined the two-dimensional nanomaterials based mixed flow. The consequences of thermal radiation on convective phenomena in MHD nanofluid over a non-linear stretching surface under heat generation and chemical reaction were studied by considering thermophoresis and Brownian motion using HAM [23][24][25][26][27].
In evident references to modern works, our investigation "The effects of thermal radiation and chemical reaction on MHD Hiemenz flow over a non-linear stretching sheet in presence of thermophoresis and Brownian motion" will certainly appear more viable now. The investigation is approved out for the 2D steady MHD Hiemenz flow. Using a similarity method to reshape the underlying Partial differential equations into a set of ordinary differential equations (ODEs), the implications of heat generation, and chemical reaction on the flow field are described in detail. Moreover a Homotopy analysis method (HAM) is used to interpret the related mechanisms. The influence of numerous parameters is graphically studied and numerically investigated.

Mathematical Formulation
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 ðmÀ1Þ=2 , where x is the distance around the plate from the plate's leading edge.
Under above assumptions, the governing equations are where ū, v are respectively the velocity constituents on the way to the x and y directions, r is the electrical conductivity of the fluid, q is the fluid density, s is the ratio of heat capacity of nanoparticles to the base fluid, 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 1 and C 1 are the ambient temperature and concentration of the fluid, k 0 is the dimensional chemical reaction, Q 0 is the heat source coefficient.
The appropriate boundary conditions are where u e x ð Þ is the potential velocity, u w x ð Þ is the velocity of the plate, T w and C w are the plate temperature and plate concentration.
The radiative heat flux is determined by using Rosseland approximation where r Ã is the Stefan-Boltzman constant and k Ã is the mean absorption coefficient. We recognise that the disparity in temperature within the flow makes sure that in a Taylor's sequence, T4 can be extended. Hence, established T 4 in Taylor series about T 1 , we obtain T 4 ¼ 4T 3 1 T À 3T 4 1 . Now, we introduce the following similarity transformations: Substituting Eq. (6) in Eqs. (2)- (5), we obtain 1 is the Lewis number, c ¼ k 0 u 1 x mÀ1 is the chemical reaction parameter.
The boundary conditions are where prime denotes differentiation with respect to f, V ¼ u 0 u 1 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 ¼ 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 the mobility is greater. V ¼ 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 ð Þ 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 get where Re x ¼ u e x m is the local Reynolds number.

HAM
We now adopt aforementioned initial guesses and linear operators to encapsulate the homotopic solutions of Eqs. (7)- (10): where C i ði ¼ 1 to 7Þ are the arbitrary constants.
We construct the zeroth-order deformation equations where where p 2 ½0; 1 is the embedding parameter, h 1 , h 2 and h 3 are non-zero auxiliary parameters and N 1 , N 2 and N 3 are nonlinear operators.
The nth-order deformation equations are follows: (20) with the following boundary conditions: where

Convergence of HAM Solution
The auxiliary parameters h 1 , h 2 and h 3 have a straightforward influence on the convergence and approximation rate of the findings drawn. h-curves are interpreted in Fig. 2 in accordance to establish the requisite quantities for both parameters. The principal scenario of the parameters is just about [−2.0, 0.0] as a consequence of such a comprehensive breakdown. For h 1 ¼ À0:85, h 2 ¼ h 3 ¼ À0:65, in the entire region of f, the series solutions are convergent. Table 1 implies the method's convergence.

Results and Discussions
For a wide range of physical characteristics, tables and charts are often performed to ascertain and describe the nature of flow, temperature, concentration, skin friction coefficient, and local Nusselt and Sherwood numbers. We check out the following values all across the exploration, apart from renovated quantities as revealed in the tables and charts.
Figs. 3-5 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,   Fluid velocity amplifies as the velocity ratio parameter V accelerates, whereas the fluid temperature and concentration drop. This is observed from Figs. 9-11. Figs. 12 and 13 reveal the implications of the Brownian motion parameter Nb on temperature and concentration fields. Brownian motion, in particular, aids in the heating of the fluid in the boundary layer and the restriction of particle evacuation from the fluid on the surface. As a result, the temperature goes up while the concentration lowers.
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 high temperature. We also identified that relatively high Nt values result in greater nanoparticle concentrations. This is shown in Figs. 14 and 15. Fig. 16 highlights temperature recuperation for diverse levels of the radiation parameter R. With altered measurements of R, temperature sketches accelerate as well. This is owing to the belief that heightened radiative heat transmission makes the establishment of thermal boundary layers simpler. The deviation of Prandtl number Pr on temperature is interpreted in Fig. 17. It is clear from the figure that uplifting values of Pr, temperature accelerates. Heat energy is accomplished in the flow region when the heat source parameter Q is risen, allowing the temperature to rise rapidly. Fig. 18 reflects it.  Figure 10: Effect of V on h f ð Þ Figure 11: Effect of V on f f ð Þ Figure 12: Effect of Nb on h f ð Þ Fig. 19 illustrates that as the Lewis number Le grows, the concentration distribution of nanoparticles drops. Larger values of Le relate to a poorer Brownian diffusion coefficient, leading in a diminution in the concentration distribution of nanoparticles. The consequence of a chemical reaction parameter on concentration profiles is visualized in Fig. 20. It has been recognised that as the chemical reaction parameter grows, the concentration lowers.
When the plate and fluid advance at the similar tempo V ¼ 1 ð Þ, the skin friction coefficient is constant (zero). In the scenario of a stationary plate V ¼ 0 ð Þ, it strengthens with M. When the plate gets faster than the free stream V ¼ 2 ð Þ, it slows down with M. This is depicted in Fig. 21. Nusselt number amplifies with R and Pr. This is noticed in Fig. 22. From Fig. 23, it is noticed that Sherwood number accelerates with Le and c.  Tables 2 and 3, and the stats indicate a strong and positive correlation.

Conclusive Remarks
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 HAM. The foregoing are the crucial insights reached from the graphical and numerical solutions to the problem: The temperature profile is significantly amplified by the heat source parameter. Thermal radiation and thermophoresis parameters lead enhance 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 c.