Impact of chemical reaction on Eyring–Powell fluid flow over a thin needle with nonlinear thermal radiation

The thin needle is viewed as a revolutionary object since it has a thinner thickness than a boundary layer. As a consequence, scientific and engineering applications for instance electrical equipment, hot wire anemometers and geothermal power generation are significantly impacted by the flow deformed by a thin moving needle. MHD Eyring–Powell fluid flow over a thin needle perceiving heat source, chemical reaction and nonlinear thermal radiation is the subject of the current investigation. In addition, the present study utilizes the Buongiorno model to examine the special effects of the fluid's Brownian and thermophoretic forces. The solution of the dimensionless form of ODEs is produced by applying exact renovations to the given problem, which is determined by the structure of PDEs. The bvp4c algorithm, based on the finite difference approach is utilized to numerically solve such modified ODEs. For validation, the results obtained indicate good agreement when compared to the literature. Finally, a detailed graphical analysis of key parameters is shown and explained while keeping in mind the physical significance of flow parameters. The results show that as magnetic and fluid parameter values improve, the velocity gradient falls. Increasing heat source and radiation parameters optimises heat transfer rate. The augmentation of the Lewis number and chemical reaction accelerates the rate of mass transfer on the surface. Brownian motion and thermophoresis provide enhanced thermal performance for the fluid temperature. Growing the thermophoresis parameter from 0.1 to 0.3 upsurges the Nusselt number by 5.47% and the Sherwood number by 12.26%.

stream.Malik et al. 7 conducted an analytical study of heat transmission of Eyring-Powell liquid over a stretching cylinder with Vogel's and Reynolds forms of changing viscosity.The radiation's impact on Eyring-Powell fluid across an unstable oriented stretched sheet with a heat source/sink was evaluated by Hayat et al. 8 .The unsteady Eyring-Powell flow in a conduit with a porous surface was examined by Zaman et al. 9 .Rosca and Pop 10 looked at how an Eyring-Powell fluid flows and transfers heat across a shrinking surface.Relevant components of the Eyring-Powell fluid flow have been examined in several studies [11][12][13][14] .
Magnetohydrodynamics (MHD) is the learning of the magnetism and behaviour of conducting liquids.It is sometimes known as hydro-magnetism or magnetohydrodynamics. Magnetic fluids include liquid metal, plasma, electrolytes and salt water.The basic theory underlying MHD is that a magnetic field creates an electric current, which generates Lorentz force, which influences the flowing liquid.The optimal pre-use factor is required for both controlling the cooling rate and achieving acceptable product quality 15 .Afterwards, Eldabe et al. 16 provided the MHD-free convective heat and mass transfer of Eyring-Powell fluid via a porous medium.The significance of magnetic flux over a thin needle moving vertically was searched by Salleh et al. 17 .MHD Williamson nanofluid moving thin needle flow was hypothetically directed by Khan et al. 18 .View some other notable research on MHD in [19][20][21][22] .
Common liquids are often not used extensively in a range of scientific and technical sectors due to their poor heat conductivity.Due to their undeniable thermal impact and unique applications in industries, biological research, and engineering sciences such as nuclear power, paper production, insolation collectors, glass-fibre manufacture, geothermal energy pipe cooling systems and heat transmission of heating nozzles in aircraft equipment, nanoparticles are gaining a lot of attention in the twenty-first century.Nanoparticles are small metallic particles (1-100 nm) having enhanced thermo-physical properties.Nanoparticles are sensitive to many interrelationships that may aid in the formation of particular turbidity patterns or the delimitation of density as well as the generation of nanoparticles and buoyant forces.Recently it was determined that nanofluids have higher thermal conductivity ratings than regular fluids.Choi 23 established the existence of such a concept by providing experimental validation of nanofluids.Dhanai et al. 24 utilised the shooting methodology to present a solution of MHD power-law nanofluids caused by a shrinking/stretching surface.The numerical analysis of the MHD micropolar fluid flow on a contracting surface was done by Lund et al. 25 .The mixed convection of nanofluid contained in a thin needle was analyzed by Soid et al. 26 .A thin needle-induced variable fluid characteristic MHD flow of ceramic nanofluid was explored by Nayak et al. 27 .Alsenafi et al. 28 discussed the heat transfer analysis with Blood-Fe 3 O 4 over a thin needle.Scientists worldwide have been fascinated by the dominant qualities of nanofluid, and as a result, amazing investigations have been documented over the past two decades [29][30][31][32][33] .
Thin needle geometry elaborates the blurring surface generated by spinning a parabola about its axis.In such geometries, physical events take place near the disparaging cylindrical tube with quasi-stiffness.The practical value of this particular geometry in a variety of areas, like blood flow challenges, cancer treatment, metal spinning, and led to its acceptance.Many researchers have investigated heat transmission and flow through the use of a moving thin needle.The boundary layer (BL) flow across a narrow needle was invented by Lee 34 .Ishak et al. 35 accomplished the double solution on a narrow needle.Waini et al. 36 looked at the radiative flow rate of a fluid flowing on a thin needle.Afridi et al. 37 exploited heat dissipation to induce entropy for nanofluid flow upon this thin needle.Trimbitas et al. 38 examined the transmission of heat by convection on a vertical needle theoretically.0][41][42][43][44][45] .
There are many applications and importance of chemical processes in engineering, geophysics, and industry.Several chemical reactions are carried out in a reactor during industrial chemical operations to renovate less expensive crude resources into higher-standard products.The design of chemical processing machinery, blood pumping, food processing, blood pumping, glass production and other industrial processes all depend heavily on homogeneous/heterogeneous chemical reactions.Due to its benefit, distinct researchers have conducted evaluations for the thermal and mass flux that take into account how chemical reactions influence the flow system.Matin and Pop 46 used chemical reactive phenomena to describe the convective heat flow for nanofluid over the permeable surface.Mabood et al. 47 used chemically reactive effects over a needle.Ramzan et al. 48examined the stimulus of chemical process upon flow system on fluid flow across a needle.The MHD flowing for nanoparticles upon a horizontal surface has only been described by Makind and Animasaun 49 .Khan et al. 50have drawn a connection between the outcomes of a chemical reaction and a Casson fluid moving across a surface.2][53][54][55][56] .
To the extent that we are aware, no authors have taken into account this study.Until now, no research has been directed to demonstrate the 2-dimensional flow of an MHD Eyring-Powell fluid over a thin-needle in the dignity of a chemical reaction.Brownian motion, heat sink/source, thermophoresis and nonlinear thermal radiation are introduced in the report.The interaction of mass, momentum and energy culminates in the entire formulation of a nonlinear mathematical problem.The built-in finite difference method was used to perform nonlinear analysis on the velocity, Sherwood numbers, surface drag force, temperature, local Nusselt and nanoparticle concentration profiles.The authors must respond to the following research queries by the study's conclusion: • What is the Eyring-Powell fluid's behaviour in the occurrence of a magnetic field parameter, and which fluid is most significantly impacted by the magnetic parameter?• What is the behaviour of the Eyring-Powell fluids in the bearing of chemical reaction, and which fluid is most severely impacted by a chemical reaction?• How do the heat sink/source, Brownian motion, and thermophoresis and for which fluid have a dominant impact?• How does the nonlinear thermal radiation affect the Eyring-Powell fluids?

Formulation
Consider the laminar, 2D, steady, MHD, and chemically reactive flow of Eyring-Powell nanofluid over the moving thin-needle in the availability of heat sink/source, thermophoresis, Brownian motion, and nonlinear thermal radiation.As presented in Fig. 1, the needle radius is r = R = χxν f /U, where r is the radial coordinate, ν f is the kinematic viscosity, χ is the size or shape, U = u w + u ∞ is the composite velocity, u w denoted as the needle is moving horizontally and x is the axial coordinate.The heat and momentum BL are thicker than the thin needle, which is thinner yet.The magnetic strength B 0 is imposed in the radial direction, and there is no pressure gradi- ent on the surface.Furthermore, T w (K) and C w (mol/m 3 ) are the corresponding amounts at the needle's surface, whereas T ∞ and C ∞ are considered to be the temperature and concentration at the free stream.Note that the fluid at the surface is thought to contain a concentration and temperature that are higher than the ambient level.
The BL equations for continuity, thermal energy, momentum, and concentration can be stated in cylindrical coordinates under the aforementioned assumptions 7,37,38,41,42 .

Quantities of physical interest
The rate of heat transmission and surface drag force are acknowledged as and m w = ∂C ∂r r=χ are denoted correspondingly as the shear force, thermal, and mass flux.

Solution procedure
The bvp4c technique is used to show the numerical solution of the altered Eqs. ( 7) to (9) with BCs (10).As the resulting problem has a two-point boundary value and is highly nonlinear, we must first convert it to first order.Let's take f = y 1 , f ′ = y 2 , f ′′ = y 3 , θ = y 4 , θ ′ = y 5 , φ = y 6 , and φ ′ = y 7 , with boundary condition becomes, In the present scenario, we choose an appropriate finite η ∞ value in order to asymptotically satisfy the far-field boundary requirements.In the present study, it is considered that η ∞ should have a definite value of less than 5 (7) , and (Re) www.nature.com/scientificreports/ in order for the established conditions to be equally satisfied and for numerical solutions to not just alter.During the computation, the CPU time to calculate the values of profiles in the modelled problem is up to 2.15 s, and the convergence criterion is 10 − 6 with the step size �η = 0.001 41,42 .

Results and discussion
The exploration's focus on determining the manipulate of Brownian motion, nonlinear radiation and thermophoresis on the heat and mass transport properties of Eyring-Powell fluid flow via a thin-needle including chemical process.To get the desired results, the MATLAB-built bvp4c approach is used.For choose various values of the parameters such as M = 0.2,Pr = 15,θ r = 1.2,Nr = 0.5,H = 0.1,α = 0.3, = 0.4,a = 0.2,χ = 0.2,Le = 0.3, Nt = 0.2, and Kr = 0.5.The results of Nusselt number, flow rate, drag force, temperature and Sherwood number were established in the form of tables and figures.
The findings for the code validation are remarkably comparable to the earlier research by Hamid 51 , Ishak et al. 35 , Nadeem et al. 42 , and Song et al. 29 see Table 1.
Figure 2a,b depicts the consequence of α on the f ′ (η) and θ(η).The increase in α has resulted in a drop in momentum BL while an increase in heat flux.Physically, increasing the causes it to oppose the free moment of the needle in the flow because resistance forces acting in the opposite direction of the needle movement enhance the surface area of the object.The internal energy and therefore the heat transfer rate develop as a result of these resistive forces.Figure 3a,b depicts the influence of on the f ′ (η) and θ (η).As grow, the velocity profile upsurge whereas fluid temperature decline due to the thermal and momentum BL thickness expands.Eyring-Powell fluid displays shear-thinning properties, therefore their viscosity drops as the shear rate rises.An inverse relationship exists between this characteristic and a non-Newtonian fluid's dynamic viscosity.With an improvement in , the flow resistance reduces, and as it does so, the fluid velocity rises.Consequently, it permits the fluid particles to disperse from the surface and reduces the thickness of the thermal BL.The effect of M on f ′ (η) and θ(η) are depicted in Fig. 4a,b.We discovered that increasing the M reduces fluid velocity while an increase in fluid temperature.The drag force caused by the produced Lorentz force, which slows down the haphazard motion of fluid particles, is felt by the immersed nanoparticles as the magnetic flux strength steadily increases.Fluid velocity on a needled surface is slowed down by the resistance nanoparticles encounter.The heat released into the system and the thermal boundary profiles exhibit corresponding expansion when the slower fluid increases friction between the fluid layers.Figure 5a,b depicts the impact of the Nr and θ r on the θ(η).When θ r and Nr are increased, the fluid temperature increases.A larger θ r indicates that the needle wall and its surrounds are considerably different in temperature.With a temperature rise, the BLs thickness escalates.The radiative component promotes small particle mobility by creating collisions between randomly moving particles, which transforms frictional energy Table 1.Comparative inspection for −f ′′ (χ) when M = Nr = 0. χ Ishak et al. 35 Nadeem et al. 42 Song et al. 29      www.nature.com/scientificreports/into heat energy.Also, the heat transmission rate to the fluid by radiation increases as the radiation parameter is elevated because it lowers the mean absorption coefficient.The accomplish of the H and Pr on the θ(η) are exem- plified in Fig. 6a,b.It is well known that as the H and Pr heighten, so does the heat flow while the temperature and thermal BL thickness drop.The heat-generating mechanism boosts the fluid temperature in the BL zone of the needle by transmitting a substantial amount of heat energy from the needle to the liquid.The influence is more pronounced with lower Pr because its heat transmission rate is falling as the BL thickness, because the Pr is inversely related to thermal diffusivity.Figure 7a,b depicts the performance of the Nb and Nt on θ (η).Strengthening the Nb and Nt, raising the fluid temperature.The promptly moving molecules or atoms in the base fluid strongly influence the arbitrary moving of nanoparticles spread in it, known as Brownian motion.It occurs when molecules of liquids or gases and nanoparticles interact.Physically, a rise in the Nb is accompanied by a noticeable movement of nanoparticles, which raises their kinetic energy and causes them to produce more heat.In addition, this raises the liquid's temperature and the thickness of the thermal BL.This is because particles close to a hot surface produce a force called thermophoresis, which raises the liquid's temperature in the BL area.Thermo-phoresis is a process that pulls tiny particles from a hot surface to a cool one.This concept is supported by the fact that an elevation in Nt produces a stronger thermophoretic force, promoting further nanoparticle migrating from a heated surface to a cold ambient liquid, boosting the heat flux and escalating the thermal BL thickness.Figure 8a  Table 2 displays the impacts of the various constraints on the drag force, heat and mass transfer rate.When the values of alpha and lambda grow, the drag force on the surface increases, but for M, the reverse behaviour occurs.As a consequence, the rapidly rising forces that are inspired by the influences of the strong viscosity are decelerated.This causes the heated fluid motions that are addressing the wall to begin.The heat transfer rate on the surface can be realistically boosted by raising M, lambda, Nt, and Nb, which helps transport more heat into the liquid.The opposite phenomenon has been observed for higher values of alpha, kr, and Le.A faster response rate is indicative of greater values for M, lambda, Nt, kr, and Le.The Sherwood number subsequently rises, however, mass transfer rate exhibits the reverse behaviour for the alpha and Nb parameters.

Conclusion
The current work used a binary chemical reaction to investigate a steady 2D MHD Eyring-Powell nanofluid flow over a thin needle that is moving horizontally.The system also includes a heat source/sink, the Begnano model, and nonlinear radiation.The similarity transformation and the bvp4c technique were operated to explain this system of equations, which resulted in the translation of the controlling PDEs into a pair of ODEs.Graphs  • The Le, alpha and Kr increase the shoowerd number while demonstrating the opposite tendency.
• The effects of varied thermal conductivity and variable viscosity can be added to the aforementioned inves- tigation in the future.In addition, hybrid nanofluid may be used in place of the standard nanofluid in the current work. https://doi.org/10.1038/s41598-023-48400-1

Table 2 .
Result for Cf x Re −0.5x , Nu x Re 0.5x , and Sh x Re 0.5x .