In this study, a hypothetical fast spectrum Liquid Fuel Molten Salt Reactor (LFMSR) core is modeled using the multiphysics C++ code GeN-Foam (General Nuclear Foam). GeN-Foam is based on OpenFOAM, a C++ opensource library for solution of continuum mechanics problems. The code utilizes a unified fine/coarse mesh approach, modeling different physics such as neutron kinetics, thermal-hydraulics based on porous fluid equations, and structural thermal-mechanics. A steady state analysis of a simplified two-dimensional (2D) LFMSR model has been performed assuming rotational symmetry to cross verify the code with the commercial ANSYS Computational Fluid Dynamics (CFD) code Fluent. The calculations showed very good agreement between the two codes allowing progression to a three-dimensional (3D) model simulation. A coupled neutron kinetics and CFD steady state analysis of a right-cylindrical 3D LFMSR core has been performed modeling one quarter of the core while using symmetry boundaries to reduce the computational time. Mixed uranium and plutonium chloride fuel has been selected in this preliminary study. Both 2D and 3D simulations showed appearance of recirculation zones within the right-cylinder core. These zones can be a challenge for LFMSR control and materials. A new hyperboloid design is proposed to remove recirculation zones, which is based on eight symmetrical loops. An Unprotected Loss of Flow accident (ULOF), in which the pump head is instantaneously reduced to zero, has been selected to demonstrate the safety characteristics of the reactor in one of the most challenging possible situations for LFMSR. The obtained results (e.g., reduced total precursors concentration at the core inlet and reduction of the core nominal power following the transient) confirm that GeN-Foam is capable of performing coupled LFMSR transient analysis and can be used for design analysis and optimization. Although the current design needs further assessment and development, it shows encouraging performance under ULOF conditions paving the way to the next step in the optimization process.