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anonymous
 one year ago
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, timeindependent), ∂ρ/∂t=0, but dρ/dt is
anonymous
 one year ago
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, timeindependent), ∂ρ/∂t=0, but dρ/dt is

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0not necessarily zero. For an incompressible fluid, dp/dt=0; show that then ∇ (dot) v = 0. (Note that incompressible does not necessarily mean constant density since dp/dt = 0 does not imply either time or space independence of ρ; consider, for example, a flow of water mixed with blobs of oil.)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I got this far. dρ/dt = ∂ρ/∂x dx/dt + ∂ρ/∂y dy/dt + ∂ρ/∂z dz/dt + ∂ρ/∂t dt/dt \[\frac{ d \rho }{ dt } = \frac{∂\rho}{∂x}v _{x} +\frac{∂\rho}{∂y}v _{y} +\frac{∂\rho}{∂z}v _{z} +\frac{∂\rho}{∂t}\] \[\frac{d \rho}{dt} = (v _{x} + v _{y} + v _{z}) \cdot (\frac{∂\rho}{∂x} +\frac{∂\rho}{∂y} +\frac{∂\rho}{∂z}) +\frac{∂\rho}{∂t}\] \[\frac{d \rho}{dt} = \vec{v} \cdot \vec{∇} \rho +\frac{∂\rho}{∂t} \]

BAdhi
 one year ago
Best ResponseYou've already chosen the best response.0jst wandering isnt \[\nabla\rho = \left\langle\frac{\partial \rho}{\partial x},\frac{\partial\rho}{\partial y},\frac{\partial \rho}{\partial z},\frac{\partial \rho}{\partial t}\right\rangle \] ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0No, time is the parameter of the function. So the x, y and z components are all functions of the one variable t.

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.0firstly to clarify that \(\vec v\) is \(\vec v\) at all times in this question, and and not \(\vec V\) where \(\vec V = \rho \vec v\) ....as is used in a common version of the continuity equation: \[\nabla ·\vec V + \frac{\partial ρ}{\partial t} = 0.\]
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