Entropy Generation Analysis due to Permeable Exponential Stretching in Boundary Layer Flows

ABSTRACT Entropy generation represents the irreversible loss of useful energy in thermal–fluid systems, directly reducing efficiency in applications, such as microelectronics cooling, nanofluid‐based heat exchangers, and polymer processing. This study investigates entropy generation in a two‐phase viscoelastic nanofluid flow over a permeable exponentially stretching surface embedded within a porous medium. The model incorporates the effects of porous resistance, viscous dissipation, Brownian motion, and thermophoresis on velocity, temperature, and nanoparticle concentration. The governing nonlinear partial differential equations for mass, momentum, energy, concentration, and entropy are transformed into a system of ordinary differential equations using similarity transformations, converted into an initial value problem via the shooting method with the Newton–Raphson technique, and solved numerically using the Runge–Kutta–Fehlberg method. Numerical simulations reveal that increased permeability and fluid elasticity reduce velocity and skin friction, while Brownian motion and viscous dissipation enhance temperature and entropy generation. Thermophoresis elevates nanoparticle concentration but diminishes wall heat transfer. Overall, entropy generation decreases with higher permeability but rises significantly under strong viscous effects, as confirmed by Bejan number trends. These findings provide practical guidance for optimizing thermal management and nanofluid‐based energy systems by balancing enhanced heat transfer with minimized thermodynamic losses.