This work examines the basic issue of magnetic fluid flow in a lid-driven cavity under the influence of a constant localized magnetic field. The magnetohydrodynamics (MHD) principles are consistent with the mathematical treatment of the problem. The magnetic fluid is viewed as a homogeneous Newtonian fluid that exhibits magnetization and is electrically non-conducting. The continuous partial differential equations are discretized using the second-order finite volume method (FVM) for the numerical solution using the free and open-source C++ program OpenFOAM version 10. Solver ferroSimpleFoam is created as a modification and development of the built-in solution simpleFoam. This solver is changed into a new program that may be used to study the dynamics of ferrofluid flow under the influence of an external magnetic field. Results pertaining to velocity show a significant influence of the magnetic field on the flow field.