  We present a mapping between a Schr\"odinger equation with a shifted
non-linear potential and the Navier-Stokes equation. Following a generalization
of the Madelung transformations, we show that the inclusion of the Bohm quantum
potential plus the laplacian of the phase field in the non-linear term leads to
continuity and momentum equations for a dissipative incompressible
Navier-Stokes fluid. An alternative solution, built using a complex quantum
diffusion, is also discussed. The present models may capture dissipative
effects in quantum fluids, such as Bose-Einstein condensates, as well as
facilitate the formulation of quantum algorithms for classical dissipative
fluids.
