so that . So far so good.
However, let us consider a simple case in 1D in which the density is of the form and . Both fields satisfy the continuity equation. This is more evident if we use the other form of the continuity equation,
Clearly, for the velocity field that I gave, , and the fluid is incompressible, but as we can see, density changes with time and space. Moreover, at a fixed position (i.e. in a stationary frame of reference), density would change with time.
So, does density depend on the frame of reference? What's the real definition of compressibility in fluid mechanics?