A VARIATIONAL RAYLEIGH–RITZ SOLUTION FOR NONLINEAR MULTIPHASE MAGNETOHYDRODYNAMIC NANOFLUID FLOW AND HEAT TRANSFER OVER A PERMEABLE STRETCHING/SHRINKING SURFACE EMBEDDED IN A POROUS MEDIUM

Abstract

This study presents a variational Rayleigh–Ritz solution for nonlinear magnetohydrodynamic (MHD) multiphase nanofluid flow and heat transfer over a permeable stretching/shrinking surface embedded in a porous medium. The mathematical model incorporates the combined effects of porous medium permeability, thermal radiation, wall transpiration, and higher-order Navier slip conditions. Blood-based nanofluids containing Silver (Ag) and Copper (Cu) nanoparticles are considered to investigate the influence of nanoparticle-enhanced thermophysical properties on momentum and thermal transport. The governing partial differential equations are transformed into coupled nonlinear ordinary differential equations using similarity transformations. A modified constrained Rayleigh–Ritz variational formulation is developed by constructing suitable exponential trial functions for the momentum and temperature fields. The higher-order slip and thermal boundary conditions are incorporated through constraint elimination, and exact analytical integration of the variational functionals yields nonlinear algebraic systems for the unknown Ritz coefficients. The resulting semi-analytical solutions are used to examine the effects of the magnetic parameter, porous medium permeability, nanoparticle volume fraction, wall transpiration, slip parameters, and thermal radiation on the velocity and temperature distributions. The results demonstrate that the magnetic field suppresses the fluid velocity while enhancing the thermal boundary layer, whereas higher-order slip significantly modifies the flow and heat transfer characteristics. The proposed constrained Rayleigh–Ritz method provides an accurate and computationally efficient semi-analytical framework for solving coupled nonlinear multiphase fluid flow problems in porous media and offers an effective alternative to conventional numerical methods.