A formulation of the theorem of minimum entropy production is developed to include the case in which mechanical irreversibility due to viscous forces occurs. We have thus obtained a generalization of the dissipation function of Lord Rayleigh as well as a generalization of Helmholtz' theorem, which states that the dissipation function is a minimum in the stationary state. This generalization refers to the slow movement of a viscous incompressible fluid at nonuniform temperature.