A Mathematical and Analytical Solution for MHD Boundary Layer Flow and Heat Transfer of a Nanofluid Over a Stretching Sheet Using the Homotopy Analysis Method

In this work our purpose is to seek universal ,to investigate the flow characteristics of MHD nanofluid past a porous stretching/shrinking sheet with heat transfer phenomena globally by exploiting universal analytic solutions in presence of thermal dispersion effect, shear-thinning/Thickening behavior viscosity and convective boundary conditions. Water based with water as base fluid and aluminum, copper nanoparticles. That (PDE's) of flow dynamics and heat transfer are represented by a system of ordinary differential equation's(ODE's), which govern the behavior.which is expressed by the dimensionless similarity vector induced in this study. Hence, the transformed dimensionless stream function and temperature profiles can be solved analytically. Algorithms for solving the system of ODEs obtained, utilizing an analytical and a numerical approach are constructed via Identification of Parameters alongside Runge-Kutta-Shooting to have a versatile solution method applicable over small values to large values in magnetic parameter analysis. The present study is analyzed for the solution structure has complex, Two identical solutions or no (or single — unique) solution ranges of constant coefficients limit incompressible boundary layer flow past a viscous fluid over stretching sheet. For stretching/shrinking sheet the hybrid nanofluid shows a better option of improving cooling when some parameters are changed. Especially for the shrinking sheet, the first solution in presence of hybrid nanofluid is detected as stable and meaningfully reasonable while second one proved unphysical both with mixed (with-outer-flux-heater) or without replacing to usual fluids. Tabulated results provide an extension and a generalization of previous studies on nanofluids, while new data refers to the hybridnanofluid stability as well as heat transfer properties in different conditions. These results have wide implications for the industries like cooling systems and material processing where efficient thermal management is critical. The novelty of this analysis is the consideration of mutual effect on fluid dynamics, stability characteristics and heat transfer profile due to magnetic field temperature for hybrid nanofluids, which offer foundation unexplored yet a comprehensively correlated scenario.