Modeling and optimization of unsteady MHD Casson-dusty nanofluid in a Darcy-Forchheimer variable porous medium via response surface methodology

Abstract

This study presents an optimized analysis of unsteady magnetohydrodynamic (MHD) flow of a Casson-dusty nanofluid through a Darcy-Forchheimer porous medium with variable porosity and permeability. However, most existing studies focus uniform porous medium under steady flow and neglect the complex behavior of variable porosity and permeability in Darcy-Forchhimer space for unsteady flow with sensitivity analysis of it, especially under magnetic influences. The novelty lies in integrating variable porosity and permeability with nonlinear drag forces in a single model, addressing a gap in the existing literature. To bridge this gap, the present study investigates the unsteady magnetohydrodynamic (MHD) flow of a Casson nanofluid through a Darcy-Forchheimer space incorporative with dust particle. The governing boundary-layer flow PDEs are reduced to coupled nonlinear ODEs via similarity transformations and solved numerically using the Galerkin finite element method with appropriate boundary conditions. To optimize and analyze the effects of key parameters, response surface methodology (RSM) is employed with the local Nusselt number as a thermal performance metric. ANOVA tables and residual plots validate the model accuracy, showing a high coefficient of determination R 2 = 99.2 % , an adjusted R 2 = 98.4 % . Sensitivity analysis identifies the ϕ as a dominant factor influencing on N u x . The Pareto chart shows that the quadratic term ϕ 2 is the most influential factor on N u x (27.20%), followed by the linear terms ϕ (23.88%) and β t (19.90%). The key findings indicate that an increase in the parameters α , and ϕ enhances the fluid velocity profile F ′ ( ξ ) , whereas M and d 2 reduce it. The dust phase velocity H ( ξ ) increases with α but decreases with M . The temperature profile Θ ( ξ ) rises with increasing R m and γ , but decreases with higher values of Pr , β m , and β t . Furthermore, the dust phase temperature Θ p ( ξ ) increases with κ r , while it declines as γ , β m , and β t increase. These findings highlight the potential of tuning porosity, permeability, and magnetic field effects to optimize nanofluid-based thermal systems for advanced cooling and energy technologies. • Unsteady MHD Casson-dusty nanofluid flow over paraboloid surfaces is modeled. • Variable porosity, permeability, and nonlinear Darcy–Forchheimer drag included. • Galerkin FEM applied with RSM optimization; model accuracy R 2 = 99.2 % . • Nanoparticle volume fraction ( ϕ ) is the most influential heat transfer factor. • Findings guide design of nanofluid-based cooling and energy technologies.