Fractional MHD analysis of Oldroyd–B fluid with variable thermal conductivity and thermoelectric coupling

Fractional models have emerged as powerful tools for capturing the intricate physical dynamics of systems where memory effects play a crucial role. This study investigates the unsteady magnetohydrodynamic (MHD) flow and heat transfer of a fractional Oldroyd–B fluid between parallel plates. The model incorporates Caputo fractional derivatives ( 0 < α , β , γ < 1 ) to capture viscoelastic memory effects, along with variable thermal conductivity, fractional Joule heating, and thermo-electrical coupling via the Seebeck effect. A hybrid numerical scheme—combining Galerkin finite elements for spatial discretization with the L1 finite-difference approximation for fractional derivatives—is developed and validated through rigorous error analysis, demonstrating second-order spatial convergence in L 2 and H 1 norms. Parametric analysis demonstrates that the transverse magnetic field significantly retards the flow, increasing skin friction by 22.58%, while increasing the variable thermal conductivity parameter enhances the wall heat transfer rate by 39.80%. In contrast, the Seebeck effect and electric body force are observed to assist the flow, thereby reducing wall shear stress, whereas internal heat generation diminishes the local Nusselt number. Additionally, the fractional relaxation and retardation parameters exhibit opposing effects on the velocity boundary layer thickness. These findings provide precise theoretical benchmarks for optimizing thermal management in electromagnetic viscoelastic processing.