SADI numerical solution of the heat transfer by Hartmann laminar flow inside a channel subjected to a constant temperature

This work investigates the thermal behavior of a fluid flowing in a planar channel whose outer walls are kept at a fixed temperature while being exposed to a homogeneous magnetic field. The primary purpose is to examine how the governing parameters affect both the thermal distribution and the Nusselt number. For this purpose, a generalized Graetz-type problem is considered by including the influence of a magnetic field applied perpendicular to the flow direction, together with viscous dissipation, Joule heating, and axial heat conduction. The numerical treatment is carried out using the Splines Alternating Direction Implicit (SADI) technique. One of the main advantages of this approach is its ability to compute spatial gradient terms directly without relying on classical finite-difference approximations, which improves the accuracy of gradient boundary conditions and facilitates the treatment of complex geometries. The study also evaluates the effects of the controlling parameters on both axial and bulk temperature distributions as well as on the heat transfer characteristics represented by the Nusselt number. This methodology provides an accurate and efficient tool for simulating heat transfer phenomena in magnetohydrodynamic flows.