  The stability of the flow under the magnetic force is one of the classical
problems in fluid mechanics. In this paper, the flow in a rectangular duct with
different Hartmann (Ha) number is simulated. The finite volume method and the
SIMPLE algorithm are used to solve a system of equations and the energy
gradient theory is then used to study the (associated) stability of
magnetohydrodynamics (MHD). The flow stability of MHD flow for different
Hartmann (Ha) number, from Ha=1 to 40, at the fixed Reynolds number, Re=190 are
investigated. The simulation is validated firstly against the simulation in
literature. The results show that, with the increasing Ha number, the
centerline velocity of the rectangular duct with MHD flow decreases and the
absolute value of the gradient of total mechanical energy along the streamwise
direction increases. The maximum of K appears near the wall in both coordinate
axis of the duct. According to the energy gradient theory, this position of the
maximum of K would initiate flow instability (if any) than the other positions.
The higher the Hartmann number is, the smaller the K value becomes, which means
that the fluid becomes more stable in the presence of higher magnetic force. As
the Hartmann number increases, the K value in the parallel layer decreases more
significantly than in the Hartmann layer. The most dangerous position of
instability tends to migrate towards wall of the duct as the Hartmann number
increases. Thus, with the energy gradient theory, the stability or instability
in the rectangular duct can be controlled by modulating the magnetic force.
