Here's a thing I don't understand about fluid dynamics.
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?
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?