Divergence Explained: Velocity & Density in Fluid Flow

[ehrenfest]

My book says that divergence can be understood in the context of fluid flow as the rate at which density flows out of a given region. It says that if F(x,y,z) is the velocity of a fluid, then that is the interpretation of the divergence. I fail to understand where the density comes in when we are only dealing with velocities at first and we only take a spatial derivative. It seems like the interpretation should be the rate of change of velocity i.e. acceleration.

 

[HallsofIvy]

"Acceleration" implies a velocity changing with time. If you have a fluid flowing toward a central drain, for example, the velocity may vary with position but not time. A single molecule of the fluid could move toward the drain with constant velocity while different particles, around the drain, move with the same speed but different velocities.

 

[learningphysics]

Rate of fluid flow is related to volume, and not mass so the density doesn't enter...

Do you see how over a closed volume... the rate at which fluid is leaving the the volume is F.A (dot product of velocity with the area) over the volume... ie the flux of F through the area enclosing the volume...

 

[ehrenfest]

OK and you just make that volume infinitesimal? How does making the volume infinitesimal get you the equation for the divergence? Is