Melting Heat Transfer in Ferromagnetic Carreau Fluid Flow Under a Magnetic Dipole: Critical Eckert Number and Entropy Generation

This study investigates melting heat transfer in the unsteady two-dimensional boundary-layer flow of a ferromagnetic Carreau fluid over a linearly stretching sheet under the influence of a magnetic dipole, with emphasis on thermodynamic consistency and interfacial stability. The thermal model incorporates a Stefan condition, the full magnetocaloric work term, and the exact Carreau differential viscosity. A relaxing stretching law is employed to obtain a well-posed far-field formulation, while entropy-generation and Bejan numbers quantify local irreversibility. The resulting nonlinear system is solved using adaptive collocation with an asymptotic far-field condition that removes the algebraic truncation error of the temperature tail. It is also independently validated through a mesh-free physics-informed extreme learning machine, which is in agreement with up to nine significant figures on a common truncated domain and with the recovery of the classical Crane and Grubka–Bobba limits. The similarity reduction is further tested against a marching solution of the parent equations in exact non-similar variables. For a static dipole with negligible induced currents, Joule heating vanishes, and the magnetic field influences irreversibility indirectly through changes in velocity and temperature gradients. A multi-start search reveals a second solution branch with near-wall flow reversal; a spectral solution of the linear temporal-stability problem shows that this branch is unstable, while the first branch is stable over all parameter ranges examined. A critical Eckert number is identified, above which magnetocaloric cooling reverses the interfacial temperature gradient and prevents sustained melting, thereby defining the physically admissible regime for a sustained melting interface. The locally similar model predicts this threshold at Ec = 0.352 in the reference state, whereas the parent-equation solution places it at 0.498, so the critical value is reported as a prediction of the locally similar model with an explicitly quantified non-similar correction. For a self-consistent phase-change ferrofluid (Mn–Zn ferrite in n-octadecane), the threshold corresponds to draw speeds of about 2 m s−1 per kelvin of Curie-point margin.