The main objective of this investigation to examine the momentum and thermal transportation of rotating dusty micropolar fluid flux with suspension of conducting dust particles across the stretched sheet. The novelty of the flow model is the exploration of the significance of boosting the volume concentration of dust particles in fluid dynamics. The governing PDEs of the problem for both phase models are transmuted into nonlinear coupled non-dimensional ODEs by utilizing suitable similarity modifications. The bvp4c technique was utilized in MATLAB script to acquire a graphical representation of the experimental results. This study illustrates the analysis of repercussions of pertinent parameters on non-Newtonian fluid and the dusty phase of fluid. By improving the volume concentration of dust particles and rotating parameters, the axial velocity for both phases depreciates, whereas temperature and transverse velocity for both phases have the opposite behavior. The micro-rotation distribution rises with higher contributions of rotating and material parameters, whereas it decreases against larger inputs of volume concentration of dust particles. The growing strength of the dust volume fraction (${\varphi }_d$) caused the coefficient of skin friction to decrease along the x direction, and the skin friction coefficient is raised along the y direction.

In this article, we probe the multiple-slip effects on magnetohydrodynamic unsteady Casson nano-fluid flow over a penetrable stretching sheet, sheet entrenched in a porous medium with thermo-diffusion effect, and injection/suction in the presence of heat source. The flow is engendered due to the unsteady time-dependent stretching sheet retained inside the porous medium. The leading non-linear partial differential equations are transmuted in the system of coupled nonlinear ordinary differential equations by using appropriate transformations, then the transformed equations are solved by using the variational finite element method numerically. The velocity, temperature, solutal concentration, and nano-particles concentration, as well as the rate of heat transfer, the skin friction coefficient, and Sherwood number for solutal concentration, are presented for several physical parameters. Next, the effects of these various physical parameters are conferred with graphs and tables. The exact values of flow velocity, skin friction, and Nusselt number are compared with a numerical solution acquired with the finite element method (FEM), and also with numerical results accessible in literature. In the end, we rationalize the convergence of the finite element numerical solution, and the calculations are carried out by reducing the mesh size.

This study investigates the Darcy-Forchheimer stagnation point flow and heat transfer characteristics of Williamson hybrid nanofluid (Cu-TiO₂ -Kerosene) over a stretching cylinder under the influence of convective boundary conditions and Cattaneo-Christov heat flux. The mathematical model is formulated by considering nonlinear heat source/sink, activation energy, Brownian motion, thermophoresis and porous medium effects. Appropriate similarity transformations are utilized to convert the governing partial differential equations into a system of nonlinear ordinary differential equations. The transformed equations are solved numerically using the MATLAB BVP4C technique. The novelty of the present work lies in the simultaneous investigation of Williamson hybrid nanofluid flow with Darcy-Forchheimer resistance, Cattaneo-Christov heat flux, nonlinear heat generation/absorption and stagnation point effects over a stretching cylinder. The obtained numerical results are validated through comparison with previously published studies, showing excellent agreement. The impacts of important physical parameters on velocity, temperature and concentration profiles are examined graphically and numerically for both nanofluid and hybrid nanofluid cases. The results reveal that increasing the Darcy-Forchheimer parameter and porosity parameter reduces the velocity profile due to enhanced resistance within the porous medium. The temperature profile increases significantly with higher Brownian motion, thermophoresis and Biot number parameters, while the Prandtl number reduces thermal boundary layer thickness. Concentration distribution decreases with increasing Schmidt number, whereas activation energy enhances concentration behavior. It is further observed that the hybrid nanofluid exhibits superior thermal performance compared to conventional nanofluid, making it more effective for advanced thermal management and industrial heat transfer applications.