Thread: Fluid Mechanics-Isn't the volume flux for an incompressible fluid always zero?

In fluid mechanics, when we need to find the volum flux we use ʃ_sq.nds.
By Gauss Divergence thm ʃ_sq.nds = ʃ_vdivqdv
For incompressible fluids divq=0
Then shoudn't the volum flux for an incompressible fluid be zero always??

Looking forward for an explanation

Hi uthpi!

What makes you think volume flux for an incompressible fluid isn't zero?
Because it is.

Actually, I'd like to think of it the other way around.

Suppose we take a small volume at a fixed point.
And suppose we can say that at any moment in time the amount fluid that goes in is equal to the amount that goes out.
Then that means 2 things:
1. The fluid is incompressible, or rather it is never compressed or decompressed.
2. The divergence is zero (as a result of the definition of divergence).

As for the definition of divergence, in words it is (abbreviated from wikipedia):
"the divergence represents the outward flux from an infinitesimal volume"

Or more formally (also from wikipedia):
"the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p."