The present study is carried out to examine the behavior of magnetohydrodynamic Williamson nanofluid flow and heat transfer over a non-linear stretching sheet embedded in a porous medium. In the current work, the influence of heat generation and viscous dissipation has been taken into account. The considered phenomenon in the form of partial differential equations is transformed into ordinary differential equations by utilizing an appropriate similarity transformation. The reduced form is solved by using rigorous MATLAB built-in solver bvp4c. The numerical solutions for the velocity field, temperature field, and mass concentration along with the skin friction coefficient, Nusselt number, and Sherwood number are computed. The obtained solutions are shown in graphs and are discussed with physical reasoning. It is noted that by increasing Williamson fluid parameter W, the velocity decreases and concentration profile increases. It is deduced that increasing Eckert number Ec leads to a rise in temperature and mass concentration. It has been viewed that with the increment in heat generation parameter Q, the temperature field increases and concentration decreases. The results show that an increasing magnetic field parameter M leaves a decreasing trend in the velocity field and an increasing trend in the temperature field and concentration profile. The present results are compared with the existing solution which shows good agreement and endorses the validation of current solutions.

The current investigations are carried out to study the influence of the Darcy–Forchheimer relation on third-grade fluid flow and heat transfer over an angled exponentially stretching sheet embedded in a porous medium. In the current study, the applied magnetic field, Joule heating, thermaldiffusion, viscous dissipation, and diffusion-thermo effects are incorporated. The proposed model in terms of partial differential equations is transformed into ordinary differential equations using suitable similarity transformation. The reduced model is then solved numerically with the help of MATLAB built-in function bvp4c.The numerical solutions for velocity profile, temperature profile, and mass concentration under the effects of pertinent parameters involved in the model are determined and portrayed in graphical form. The graphical effects of the skin friction coefficient, the Nusselt number, and the Sherwood number are also shown. From the displayed results, we conclude that when the Joule heating parameter is enlarged, the velocity and the temperature of the fluid are increased. We observed that while enhancing the viscous dissipation parameter (Eckert number) the fluid’s velocity and temperature increase but decreases the mass concentration. By increasing the values of the thermal-diffusion parameter, the velocity distribution, the temperature field, and the mass concentration increase. When the diffusion–thermo parameter rises, the velocity field and the temperature distribution increase, and the reverse scenario is seen in the mass concentration. The results of the current study are compared with already published results, and a good agreement is noted to validate the current study.

The current study aims to investigate the thermal-diffusion and diffusion-thermo effects on heat and mass transfer in third-grade fluid with Darcy–Forchheimer relation impact over an exponentially inclined stretching sheet embedded in a porous medium. The proposed mechanism in terms non-linear and coupled partial differential equations is reduced to set of ordinary differential equations by employing an appropriate similarity variable formulation. The reduced form of equations is solved by using the MATLAB built-in numerical solver bvp4c. The numerical results for unknown physical properties such as velocity profile, temperature field, and mass concentration along with their gradients such as the skin friction, the rate of heat transfer, and the rate of mass transfer at angle of inclination $\alpha =\pi /6$ are obtained under the impact of material parameters that appear in the flow model. The solutions are displayed in forms of graphs as well as tables and are discussed with physical reasoning. From the demonstration of the graphical results, it is inferred that thermal-diffusion parameter $Sr$ velocity, temperature, and concentration profiles are augmented. For the increasing magnitude of the diffusion-thermo parameter $D_f$ the fluid velocity and fluid temperature rise but the opposite trend in mass concentration is noted. The current results are compared with the available results in the existing literature, and there is good agreement between them that shows the validation of the present study.

Nanotechnology is well-known for its versatile and general thermal transport disciplines, which are used in semiconductors, spacecraft, bioengineering, functional electronics, and biosensors. As a result, process optimization has attracted the interest of scientists and technologists. The main aim of the current analysis is to explore the enhancement of energy/heat transfer via the dispersion of cylindrical-shaped nanoparticles of alumina and copper in ethylene glycol as a base fluid using a non-Newtonian Maxwell fluid model. In the current study, the effects of solar radiation, plate suction, and magnetohydrodynamics on a Maxwell hybrid nanofluid are encountered. The flow is induced by linearly stretching a sheet angled at $\xi =\pi /6$, embedded in a porous space. The proposed problem is converted into a mathematical structure in terms of partial differential equations and then reduced to ordinary differential equations by using appropriate similarity variables. In the similarity solution, all the curves for the velocity field and temperature distribution remain similar, which means that the symmetry between the graphs for the velocity and temperature remains the same. Therefore, there is a strong correlation between similarity variables and symmetry. The obtained model, in terms of ordinary differential equations, is solved using the built-in numerical solver bvp4c. It is concluded that more nanoparticles in a fluid can make it heat up faster, as they are typically better at conducting heat than the fluid itself. This means that heat is transferred more quickly, raising the temperature of the fluid. However, more nanoparticles can also slow the flow speed of the fluid to control the boundary layer thickness. The temperature field is enhanced by increasing the solar radiation parameter, the magnetic field parameter, and the porous medium parameter at an angle of$\xi =\pi /6$, which serves the purpose of including radiation and the Lorentz force. The velocity field is decreased by increasing the values of the buoyancy parameter and the suction parameter effects at an angle of$\xi =\pi /6$. The current study can be used in the improvement of the thermal efficiency of nanotechnological devices and in renewable energy sources to save energy in the energy sector. The present results are compared with the published ones, and it is concluded that there is excellent agreement between them, which endorses the validity and accuracy of the current study.

In this work, a novel enhanced model of the thermophysical characteristics of hybrid nanofluid is introduced. An innovative kind of fluid called hybrid nanofluid has been engineered to increase the heat transfer rate of heat and performance of thermal system. A growing trend in scientific and industrial applications pushed researchers to establish mathematical models for non-Newtonian fluids. A parametric study on theheat transfer and fluid flow of a Williamson hybrid nanofluid based on AA7075-AA7072/Methanol overincessantly moving thin needle under the porosity, Lorentz force, and non-uniform heat rise/fallis performed. Due to similarity variables, the partial differential equations governing the studied configuration undergo appropriate transformation to be converted into ordinary differential equations. The rigorous built-in numerical solver in bvp4c MATLAB has been employed to determine the numerical solutions of the established non-linear ordinary differential equations. It is worthy to note that velocity declines for both AA7075/Methanol nanofluid and AA7075- AA7072/Methanol hybrid nanofluid, but highervelocitymagnitudes occur for theAA7075/Methanol whilethe Williamson fluid parameters increased. It is alsoconcluded that as the porosity parameter isincreased, the flow intensity decreases gradually. It is worthy to note that for both non-uniform heat-rise and fall parameters, the temperature of the fluid gets stronger. Mounting valuesof needle thickness parameter leads to reduction in fluid speed and temperature. It is noticedthat as volume fractions of both types of nanoparticles are augmented then fluidvelocity and temperature amplify rapidly. A Comparison of current and published results is performed to ensure the validity of the established numerical model.