According to wikipedia's explanation of the derivation of Navier-Stokes, in a Newtonian fluid you make an assumption of proportionality between stress and strain, and "viscosity" is precisely the proportionality constant.
However, it seems like you could just start from a Lagrangian setup by saying that my generalized "position vector" at each point in time is a measure-preserving bijection f from space to itself --- that is, a record of where a test particle at time t=0 ends up. The kinetic energy of a time-varying f is just ∫ (1/2)ρ(f_t)^2 dV for a volume element dV and fluid density ρ. Assume ρ to be a constant (and therefore the fluid to be incompressible) if you like.
This looks to me like it should give a determinate behavior to the system, given enough boundary conditions. But why didn't the viscosity coefficient show up? Is it some trick of smoke, mirrors, and unitless constants?