The present investigation aims to examine the heat flux mechanism in the hagnetohydrodynamic (MHD) mixed convective flow of Williamson-type fluid across an exponential stretching porous curved surface. The significant role of thermal conductivity (variable), non-linear thermal radiation, unequal source-sink, and Joules heating is considered. The governing problems are obtained using the Navier–Stokes theory, and the appropriate similarity transformation is applied to write the partial differential equations in the form of single-variable differential equations. The solutions are obtained by using a MATLAB-based built-in bvp4c package. The vital aspect of this analysis is to observe the effects of the curvature parameter, magnetic number, suction/injection parameter, permeability parameter, Prandtl factor, Eckert factor, non-linear radiation parameter, buoyancy parameter, temperature ratio parameter, Williamson fluid parameter, and thermal conductivity (variable) parameter on the velocity field, thermal distribution, and pressure profile which are discussed in detail using a graphical approach. The correlation with the literature reveals a satisfactory improvement in the existing results on permeability factors in Williamson fluids.

This research work describes and investigates Williamson nanofluid flow over an exponentially stretching permeable vertical plate with temperature-dependent thermal conductivity and viscosity. The governing non-linear partial differential equations (PDEs) are metamorphosed into coupled non-linear ordinary differential equations (ODEs) by using similarity transformation. The succeeding equations were numerically solved using MATLAB function bvp4c for various values of parameters. For velocity, temperature, concentration, the skin friction coefficient, and the local Nusselt number, data are presented in the form of graphs and tables. It is noted that for increasing values of magnetic parameter $M$, Williamson parameter $\lambda ,$ and viscosity parameter $\alpha$, the boundary layer thickness of the velocity profile decreases, while it increases for the temperature profile. The findings of the present work are validated through the published results.

The present study concerns investigating the two-dimensional Magnetohydrodynamics (MHD) boundary layer flow of Williamson nanofluid over a non-linear stretching sheet. The focus of this study is based on the global influence of the non-Newtonian Williamson fluid parameter $\left(\lambda \right)$ rather than the local one that exists in the literature for linear and non-linear stretching cases. The mathematical model of the problem is based on the law of conservation of mass, momentum, and energy. The derived partial differential equations are transformed into ordinary differential equations by applying an appropriate similarity transformation. The subsequent equations are solved numerically by using the Shooting method. The physical quantities Skin friction coefficient, as well as the Sherwood and Nusselt numbers are computed locally. To validate the implemented shooting method, a comparison is made with the results obtained by Matlab function bvp4c, and good agreement is found. The Prandtl number, $Pr$, has an increasing impact of 25.14% on the wall temperature gradient. The impact of various physical parameters are presented through graphs and tables.