We use a fluid mechanics model to analyze glass fluid flow during the welding of bulk metallic glasses with reactive multilayer foils acting as local heat sources. The resulting welded joints were shear tested, and fracture surfaces were analyzed by optical microscopy. Fracture surfaces of failed metallic glass joints show distinct regions of metal-metal veins that indicate effective metallurgical bonding. We observe a monotonic increase in the failure strength of the joints with the fraction of the joint composed of such veins. For the strongest joint tested (shear strength of 420 MPa), nearly 60% of the fracture surface is comprised of metal-metal veins. We have developed a qualitative fluid mechanics explanation of the welding process, in which shear stresses (due to pressure applied during joining) push the reactive foil from the joint interface and create the metal-metal veins. The welding process is more effective at higher joining pressure and greater foil thickness, leading to increased joint strength.