Haar Wavelet Operational Matrix Method for the Solution of Magnetohydrodynamic Lubrication Problem of Porous Rough Plates for Squeeze Films

Objectives: In this study, a rough upper plate and a porous lower plate, which are lubricated with an electrically conducting couple-stress fluid in a magnetic field, are examined under magnetohydrodynamic squeeze-film lubrication. The goal is to create a precise and effective Haar wavelet-based computational framework. This framework will capture the combined effects of non-Newtonian properties, porosity, surface roughness, and magnetic influences on lubrication performance. Methods: The Haar Wavelet Operational Matrix of Integration (HWOMM) helps derive the modified MHD Reynolds equation, considering longitudinal roughness. This method offers strong numerical stability and fast convergence. It does this by turning the governing nonlinear partial differential equation into a compact nonlinear algebraic system. Findings: According to the numerical results, surface imperfections improve load support. Magnetic intensity increases hydrodynamic pressure, load-carrying capacity, and squeeze-film durability. Fluid infiltration into the substrate leads to increased permeability of the porous layer, which reduces pressure development. Couple tensions extend the lubrication period by delaying film thinning. The Haar wavelet formulation provides smoother solution profiles, better accuracy, and requires less processing effort than finite-difference methods. Novelty: The Haar Wavelet Operational Matrix of Integration for MHD squeeze-film lubrication with stochastic roughness, porous media, magnetic fields, and couple-stress fluid dynamics is used for the first time in this work. This new approach addresses tightly linked electromagnetic, rheological, and surface effects within a single wavelet framework. This is different from earlier wavelet-based lubrication studies, which focused on smooth geometries, Newtonian fluids, or non-magnetic conditions. The method's strong numerical stability and efficiency allow for the identification of lubrication features that traditional solution methods cannot capture. Keywords: Magnetohydrodynamic lubrication (MHD), Porous rough plates, Squeeze film, Hartmann number, Haar wavelet Operational matrix method (HWOMM)