CE 576 Environmental Flows Spring 2012 Homework 2 *10. In three dimensions, the y-momentum equation, where y points up, is ∂ ∂ ∂ ∂ ∂τxy ∂τyy ∂τzy (ρv) + (ρuv) + (ρvv) + (ρwv) = −ρg + + + . ∂t ∂x ∂y ∂z ∂x ∂y ∂z Show that for all ﬂows the y-momentum equation can be written as ρ ∂v ∂v ∂v ∂v +u +v +w ∂t ∂x ∂y ∂z = −ρg + ∂τyy ∂τzy ∂τxy + + . ∂x ∂y ∂z 11. Show that the x-momentum equation in three dimensions is ρ ∂u ∂u ∂u ∂u +u +v +w ∂t ∂x ∂y ∂z = ∂τyx ∂τzx ∂τxx + + . ∂x ∂y ∂z *12. Edwards et al. (2005, Can. J. Fish. Aquat. Sci.) solved a one-dimensional model for the transport of phytoplankton. The model included sinking with a velocity ws , net growth with a rate μ, and turbulent ﬂux qF (say). We will see later that turbulent ﬂuxes are often modeled with a version of Fick’s law: qF = −K ∂F , ∂z where K is called the vertical eddy diﬀusivity. Find this paper and derive equation (1).