Equations describing the velocity, temperature, and vorticity distributions in the melt of the melt extraction process are derived. It is shown that the melt extraction process can only produce ribbon with a thickness greater than that of the viscous shear layer in the melt. From the minimum quench rate required to produce amorphous metal, a maximum amorphous ribbon thickness and a minimum roller velocity are found. A maximum roller velocity is determined from a balance of opposing centrifugal forces and melt pressure and adhesion forces exerted on the ribbon.