I think these results must apply to a limited set of circumstances. For
instance, viscosity is fixed (water) and depth is shallow enough to avoid
turbulence, as in a skin boards on an ocean beach or a sheet of paper sliding
across a desk on a thin film of air. Once the depth is great enough for
turbulent flow you should get a whole new ball of wax and I would expect the
simple formula to fail.

-- Charlie Springer