This study comprehensively analyzes how fractional-order calculus and externally applied magnetic fields synergistically govern the hydrodynamic behavior of electrically conducting Newtonian fluids in divergent geometries by injecting flow through the walls at varying tangential velocities. Using Caputo fractional derivatives, the nonlinear boundary value problem is addressed through a dual approach: an innovative implementation of the Adaptive Fractional Method (AFM) for analytical solutions and a high-precision finite difference scheme for numerical validation. Key observations reveal that enhancement of the Reynolds number (Re) or the fractional differentiation order \(\beta\) suppressing near-wall velocities while amplifying core-region flow magnitudes. In contrast, increased Hartmann numbers (Ha) lead to significant attenuation of peak velocities due to interactions with the Lorentz force, which reveal a dual effect when reaching the Hartmann number of approximately 17: (i) the radial velocity profile begins to show uniformity at the center of the channel, and (ii) the velocity profile indicates the presence of two peak velocities near the channel walls. As well as when the Hartmann number surpasses approximately 26, the memory and non-local effects within the velocity field are mitigated. An increase in Ha results in the expansion of the streaklines. When the Hartmann number is low ( \(Ha \approx 2\) ), an increase in the channel divergence angle leads to a decrease in the maximum velocity and a refinement of the velocity profile. Conversely, at high Hartmann numbers ( \(Ha \ge 7\) ), augmenting the channel divergence angle produces effects on the velocity field akin to those observed with an increase in the Hartmann number, specifically reducing the velocity at the channel’s center and resulting in an M-shaped velocity profile.
