Question

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 + kdz/dt

Show that then dρ/dt= ∂ρ/∂t + v (dot) ∇ρ.
. . Combine this equation with

∇ (dot) v + dρ/dt=0 to
to get

ρ∇(dot) v +dρ/dt =0

Answer 1

(Physically, dρ/dt is the rate of change of density with time as we follow the fluid along a streamline; is the corresponding rate at a fixed point.) For a steady state (that is, time-independent), , but dρ/dt is not necessarily zero. For an incompressible fluid, dρ/dt = 0; show that then . (Note that incompressible does not necessarily mean constant density since dρ/dt = 0 does not imply either time or space independence of ρ; consider for example, a flow of water mixed with blobs of oil.)

how to approach thisquestion?
i think its under divergence theorem.
pls explain in detail.....

Answer 2

SInce ρ depends on 4 varaibles, you can write the general equation:
dρ = ∂ρ/∂x dx + ∂ρ/∂y dy + ∂ρ/∂z dz+ ∂ρ/∂t dt

Then
dρ/dt = ∂ρ/∂x dx/dt + ∂ρ/∂y dy/dt + ∂ρ/∂z dz/dt + ∂ρ/∂t dt/dt

Simplifying gives
dρ/dt = ∂ρ/∂x vx + ∂ρ/∂y vy + ∂ρ/∂z vz + ∂ρ/∂t
dρ/dt = \(\vec v . \vec \nabla \)ρ + ∂ρ/∂t QED

Answer 3

Thank you....
i didnt get the last line. can you pls rewrite it?

Answer 4

Here it is:
dρ/dt = v (dot) ∇ρ + ∂ρ/∂t QED

Answer 5

dρ/dt= ∂ρ/∂t + v (dot) ∇ρ.
∇ (dot) v + dρ/dt=0

how can i combine these two?

Answer 6

(Vx+Vy+Vz) + dρ/dt=0

Answer 7

Careful: this equation is wrong: ∇ (dot) v + dρ/dt=0

You should combine the equation we derived with the local conservation of mass:
∇ (dot) (ρv) + ∂ρ/∂t = 0 which is the correct equation.

Answer 8

(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 not necessarily zero. For an incompressible fluid, dρ/dt = 0; show that then ∇ (dot) (v)=0 . (Note that incompressible does not necessarily mean constant density since dρ/dt = 0 does not imply either time or space independence of ρ; consider for example, a flow of water mixed with blobs of oil.)

this is correct, sme parts were missing

Answer 9

Thank you very much..
thats correct, i made a mistake.

still i dnt get this combining,
in boas it is ∇ (dot) (v) + ∂ρ/∂t = 0
mmm...:(