Transfer of energy via third-grade fluid flow over an inclined stretched sheet under the influence of Lorentz force and thermal radiation

The third-grade fluid (TGF) flow across an inclined elongating sheet is investigated for heat and mass transfer, and the effects of a magnetic field and chemical reaction are reported. The TGF flow is examined in relation to activation energy, heat source/sink, and thermal radiation. TGF refers to fluids that, despite the boundary’s inflexibility, exhibit non-Newtonian (NN) features such shear thickening, shear thinning, and normal stresses. It also possesses elastic viscous fluid qualities. The TGF model in the suggested approach is created as a set of coupled, nonlinear partial differential equations (PDEs). Prior to using the bvp4c numerical software, the coupled equation system is reduced to a non-dimensional form. The Lobatto three-stage IIIa formula is specifically implemented by the finite-difference code bvp4c.
Tables and Figures illustrate how flow restrictions affect the velocity field, energy profile, Nusselt number, and skin friction. A numerical comparison using Table is made between the results and the published study to ensure the correctness of the findings. It is evident from the graphical results that the fluid velocity enriches as the TGF factor and Richardson number vary. The heat source parameter raises the fluid temperature by acting as a heating mediator for the flow system.
Because of its numerous applications in a variety of technical and industrial areas, the fluid flow through a stretching sheet is of great significance within the field of fluid dynamics. The behaviour of fluid flow was analysed by Abolbashari et al. 1. The flow scenario they studied involved a stretching sheet and a velocity slip condition. The impact of heat transfer on the mobility of a ferromagnetic fluid across an extended surface was discussed by Zeeshan et al. 2. This ferromagnetic fluid was made up of a mixture of magnetic solid particles that were well-blended. An electromagnetic dipole state was present for the whole process.
Sandeep and Sulochana created a new mathematical model to study energy and heat transfer in non-Newtonian fluids on a stretched surface; the Jeffrey nanofluid performed better in terms of heat transfer than the Maxwell and Oldroyd-B nanofluids; Besthapu et al. investigated velocity slip on an extending sheet with convectively non-uniform characteristics. Shit et al. studied the dynamics of unsteady boundary layer magnetohydrodynamic flow and convective heat source.
A 3D simulation of the MHD behaviour of hybrid fluid flow across double stretching surfaces was carried out by Alqahtani et al. 6. Chu and colleagues (2017) examined the two-dimensional continuous laminar flow of a TGF across a decreasing surface that was home to gyrotactic bacteria. An applied electric field made the flow electrically conductive, and mathematical modelling was done using the Buongiorno nanoliquid model. Chemical processes having activation energy effects were also included in the study. Li et al.9 and Kumar et al.
examined the rate of energy transmission through an absorbent stretched sheet in a hydromagnetic Williamson nano liquid flow. A conversation about the hybrid nano liquid flow containing copper and nanoparticles in water was held by Khan et al. 10. This flow originated from a surface that was centrifugally porous and capable of stretching or contracting. Elattar et al. (2011) examined the continuous flow of hybrid nanoliquid across a stretchy, impermeable sheet.
In order to increase the rates of energy transference and the efficacy and efficiency of thermal energy propagation, a mathematical model was created. The entropy and temperature studies of the nanoliquid flow inside a porous cylinder were reported by Dogonchi et al. 12. Some amazing findings that Ref. recently provided.

A mixture of solid and liquid particles flows in a naturally complex way that is influenced by a wide range of factors. One popular method to better comprehend and investigate these complex flows is to treat the mixture as a NN fluid. A significant amount of research has been devoted to the study of different transport processes that take place in non-Newtonian fluids, such as coal slurries.
Heat transfer is one of these processes that is especially important to handle and process when working with these fluids. It is essential to the effective handling and treatment of these complicated mixtures.19. The laminar flow and steady state of a TGF over a permeable flat tube were studied by Ariel20. Ellahi and Riaz21 conducted a study to look at the TGF in a conduit with varying viscosity. In the framework of the investigation, this study also took the fluid’s characteristics of heat diffusion into account.

The MHD motion of a Carreau Yasuda liquid started by an exponentially increasing surface was investigated by Bilal et al. in reference 22. Adesanya and colleagues (2019) investigated the inherent irreversibility associated with the flow of a third-grade fluid along a conduit subjected to convective heat. This study acknowledges that the heat produced causes the channel’s entropy to continuously generate. An research was carried out by Reddy et al.24 to determine how the Prandtl number affected TGF surrounding a uniformly heated vertically oriented cylinder.
In the presence of nanoparticles, Mahanthesh and Joseph25 investigated the steady-state behaviour of a third-grade liquid running over a pressure-type die. Because the fluid is dissipative, its characteristics are taken into account consistently throughout the analysis. Refs contains up-to-date and creative material about non-Newtonian (third-grade fluid).
The study of magnetohydrodynamics (MHD) examines how electrically conductive materials, such as liquid metals, ionised gases, and plasmas, react to magnetic fields. This field of study explores the relationship between electromagnetic forces and fluid motion, with broad implications in astronomy, engineering, geophysics, and plasma physics. Al-Habahbeh et al.31 examined the effects of varying viscous flow within a narrower channel. A thorough description of the application of MHD to biological systems was given by Rashidi et al. 32.
Given its significance and applicability in the medical sciences, the study of MHD fluid motion in various orientations related to human anatomical structures is an important area of science. The simultaneous effects of MHD, heat transmission, and slip over a flat plate in motion were investigated by Ellahi et al. 33. Additionally, the impact of entropy formation in this particular situation was evaluated in this study. Lv et al.34 investigated the effects of different physical processes in the setting of MHD free convective rotating flow of nanoliquids, such as radiation-absorption and diffusion-thermo.
The MHD Radiative Unsteady Fluid Flow with the Outcome of Heat Source over a Channel Placed in Absorbent Medium was explored by Kumam et al. 35. Tian et al.36 investigated the combined effects of forced and natural convection while studying the energy transfer through fluid flow in a rectangular enclosure with a heat sink filled with hybrid nanofluids. Bhatti et al.37 investigated the erratic flow inside parallel spinning spherical discs submerged in a permeable material.
Due to their significant roles in a variety of industrial applications, the impact of magnetism on lubrication has drawn attention. Their growing application in liquid metal lubricants for high-temperature bearings is one noteworthy example. A computer analysis of the effects of various geometric parameters on an extending cylinder was conducted by Alharbi et al. 38. Hamid and Khan39 looked on how NN Williamson fluid flow was affected by magnetic flux. An elongating cylinder in the presence of nanocomposites caused the flow.

The effects of chemical reactions on Couette-Poiseuille nanoliquid flow over a gyrating disc were investigated by Shamshuddin et al. 40. The MHD unsteady radiative flow was investigated by Kumam et al.41. Optimising entropy and comprehending heat transport in the flow of a magneto-nanomaterial were the main goals of Khan and Alzahrani42. The impact of MHD in the fluid was considered in this experiment. Li et al. (44) and Adnan and Ashraf (43) assessed the nanoliquid flow across a permeable surface.
Examining the heat and mass transfer through the TGF flow over an inclined elongating sheet is what makes the suggested model unique. The TGF flow is examined in relation to the effects of thermal radiation, activation energy, and magnetic field. TGF fluids exhibit normal stresses, shear thickening, and shear thinning, among other NN features, even in the presence of an inflexible boundary.
The TGF model is expressed as nonlinear coupled PDEs in the suggested model. Prior to using the bvp4c numerical software, the coupled equation system is reduced to a non-dimensional form. Tables and Figures illustrate the importance of flow variables on the velocity field, energy profile, and Nusselt number.
Using Table, a numerical comparison is made between the results and the published, previous study to ensure the validity of the findings. The problem is structured as a set of PDEs and is solved numerically in the next section.
We have examined the mass and energy transfer across an inclined elongating sheet via the steady and incompressible flow TGF. Under the influence of a chemical reaction, magnetic field, activation energy, and thermal radiation, the two-dimensional TGF flow is examined. It is believed that the sheet’s surface is Darcy permeable. As seen in Fig. 1, the x- and y-axes represent the horizontal and normal axes of an inclined stretched sheet. The gravitational acceleration, surface temperature, and concentration are represented by the letters g, Tw, and Cw, respectively. The TGF flow equations are given as45, 46, taking into consideration the aforementioned assumptions.