The hydrodynamic equations of a phase-separating fluid mixture are derived from the underlying microscopic dynamics of the system. A projection operator method is used in the GENERIC form [H. C. Ottinger, Phys. Rev. E 57, 1416 (1998)]. In this way, the thermodynamic consistency of the final equations is apparent. The microscopic potential is separated into short-and long-range parts, in the spirit of the original work of van der Waals. Explicit expressions for surface tension terms in the hydrodynamic equations are obtained. These terms describe diffuse interfaces in the system. Miscible-immiscible and gas-liquid phase transitions are possible, nonisothermal situations can be studied, and explicit account of cross effects is taken. (C) 2003 American Institute of Physics.