Thermal analysis and magnetohydrodynamic flow with viscous dissipation and Joule heating effects over a nonlinear permeable stretching/shrinking sheet

Purpose This study aims to investigate magnetohydrodynamic (MHD) flow and heat transfer over a nonlinear permeable stretching/shrinking sheet, incorporating the effects of viscous dissipation and Joule heating. This work addresses the gaps in understanding dual solutions and stability in such systems, with implications for industrial and thermal management applications. Design/methodology/approach The governing partial differential equations are reduced to a system of ordinary differential equations through the application of similarity transformations. Analytical solutions for reduced skin friction and heat transfer coefficients are derived under specific parametric conditions. Physical insights are extracted through graphical and tabular representations of key parameters: suction strength, Prandtl number, magnetic field intensity, viscous dissipation, mass flux and sheet shrinking rate. Findings Dual solutions (upper and lower branches) emerge for the shrinking sheet case, with the upper branch extending as suction increases. Stability analysis confirms the lower branch’s instability. Parametric studies reveal that suction, viscous dissipation and Joule heating significantly influence temperature profiles and boundary layer thickness, whereas magnetic effects dominantly alter flow dynamics. Practical implications The findings are critical for industrial processes involving stretching/shrinking sheets, such as polymer extrusion, glass production and metal rolling. Unlike stretching flows, shrinking sheet flows exhibit backward-flow behavior, necessitating external forces (e.g. suction or imposed flow) to stabilize the boundary layer. This study provides actionable strategies for optimizing thermal regulation and flow control in such systems. Originality/value This work advances the understudied area of MHD flow with combined viscous and Joule heating effects on nonlinear permeable sheets. Novel contributions include the identification of critical thresholds for dual solutions, stability characterization and the application of asymptotic methods to resolve complex ordinary differential systems. The results offer a framework for enhancing efficiency in thermal-fluid systems reliant on conductive media.