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_{s}__q__.__n__ds.

By Gauss Divergence thm ʃ_{s}q.__n__ds = ʃ_{v}div__q__dv

For incompressible fluids div__q__=0

Then shoudn't the volum flux for an incompressible fluid be zero always??

Looking forward for an explanation

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

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."