"Since density is constant, we get our incompressibility constraint"...that sounds like circular logic? What is the empirical fact we start with here and what do we conclude from it?

kmcrane

Assuming conservation of mass, we get the first equation. Assuming we're working with a liquid that has constant mass density (e.g., water rather than oil & vinegar) we get the second equation. We didn't start with any assumption on the velocity field (u), so the argument isn't circular---we're just re-expressing our assumption about incompressibility in terms of the velocity.

kmcrane

Another way of thinking about it is: assume you're starting out with a homogeneous fluid that has not been compressed. At this initial moment in time, (\rho) is certainly constant. Now what has to be true about the velocity field that takes us forward to the "next moment in time" in order to remain incompressible? This condition is expressed in terms of the current mass density, which is constant. Again, no circular reasoning there.

"Since density is constant, we get our incompressibility constraint"...that sounds like circular logic? What is the empirical fact we start with here and what do we conclude from it?

Assuming conservation of mass, we get the first equation. Assuming we're working with a liquid that has constant mass density (e.g., water rather than oil & vinegar) we get the second equation. We didn't start with any assumption on the velocity field (u), so the argument isn't circular---we're just re-expressing our assumption about incompressibility in terms of the velocity.

Another way of thinking about it is: assume you're starting out with a homogeneous fluid that has not been compressed. At this initial moment in time, (\rho) is certainly constant. Now what has to be true about the velocity field that takes us forward to the "next moment in time" in order to remain incompressible? This condition is expressed in terms of the current mass density, which is constant. Again, no circular reasoning there.