In this study, the heat transfer pattern and flow bifurcation in the fluid is observed by the application of low intensity magnetic field in the gradually constricted cavity. The natural convection flow solver with Lorentz force and Boussinesq approximation as a source term is developed in the open-source CFD platform OpenFOAM. The Lorentz force in the flow is altered by varying the Hartmann number of Ha = 0 – 100, however the buoyancy force is kept constant in the flow at fixed Rayleigh number of Ra = 106. The orientation of magnetic field is exposed to be in the y-direction (By). The significance of using the By magnetic field with its various intensity in the constricted enclosure on the heat transfer and flow pattern is reported. It is perceived that the transverse magnetic field (By) and its varying intensity regulates the heat transfer with multiple convection rings. The detail study on the isotherms, streamlines, and the time average Nusselt number is reported.