• anonymous
Suppose the density ρ of a fluid varies from point to point as well as with time, that is, ρ = ρ(x,y,z,t). If we follow the fluid along a streamline, then x, y, z are functions of t such that the fluid velocity is V = i dx/dt + j dy/dt + k dz/dt. Show that then dρ/dt = ∂ρ/∂t + v (dot) ∇ρ. Combine this equation with ∇ (dot) v + ∂ρ/∂t = 0 to get ρ∇ (dot) v + dρ/dt = 0. (Physically, dρ/dt is the rate of change of density with time as we follow the fluid along a streamline; ∂ρ/∂t is the corresponding rate at a fixed point.) For a steady state (that is, time-independent), ∂ρ/∂t=0, but dρ/dt is
Physics

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