Control of bulk melt crystal growth techniques is desirable for producing semiconductors with the highest purity and ternary alloys with tunable electrical properties. Because these molten materials are electrically conducting, external magnetic fields are often employed to regulate the flow in the melt. However, complicated by the coupled flow, thermal, electromagnetic and chemical physics, such magnetic control is typically empirical or even an educated guess. Two magnetic flow control mechanisms: flow damping by steady magnetic fields, and flow stirring by alternating magnetic fields, are investigated numerically. Magnetic damping during optically-heated float-zone crystal growth is modeled using a spectral collocation method. The Marangoni convection at the free melt-gas interface is suppressed by applying a steady magnetic field, measured by the Hartmann number Ha. Using normal mode linear stability analyses, suppression of detrimental flow instabilities is quantitatively determined in a range applicable to experiments (up to Ha = 300 for Pr = 0.02, and up to Ha = 500 for Pr = 0.001). The hydrodynamic flow instability for small Prandtl number P r float-zone is confirmed by energy analyses. Rotating magnetic field stirring during confined crystal growth in an ampoule is also modeled. Decoupled from the flow field at small magnetic Reynolds number, the electromagnetic field is solved in a finite element solver. At low AC frequencies, the force is only in the azimuthal direction but penetrates deep into the melt. In contrast, the magnetic shielding effect is observed at high alternating current (AC) frequencies, where the external magnetic field penetrates only by a skin depth into the electrically conducting media within the short AC cycle. As a result, the electromagnetic body force is primarily confined to the ampoule surface. At these high AC frequencies the magnetic flux lines are drastically distorted within the melt. The body force is fully three-dimensional and is much stronger than at low AC frequencies, but is confined to near the ampoule surface due to the magnetic shielding effect. These models promote fundamental understanding of flow dynamics regulated by electromagnetic body forces. They provide quantitative guidance for crystal growth to minimize trial and error experimentation that is slow and expensive.