Incompressible flow / The number of fluid elements per unit volume

Question

If we have a steady isothermal incompressible flow (with non-zero velocity field), the density must be constant and uniform, right? so:

Can we conclude that the number of fluid elements(fluid parcels/very tiny fluid particles, from which the flow is made of) in the(per) unit volume is the same? To rephrase it, is the number of fluid particles in the area where the flow is very slow and the area with higher velocity (per unit volume) same?

The problem here is that if we replace the flow with infinite number of discrete fluid parcels, we can clearly see that the number of parcels does not remain uniform if we have the velocity gradient. But if we assume that the number of parcels are not uniform, how can we say that the density is constant and uniform?

Answer 1

Let's say that the flow is in the $x$ direction, and that the flow velocity $v_x$ is not constant due to changes of the pipe cross area.

The mass element is $\rho \Delta x \Delta y \Delta z = \rho v_x \Delta t \Delta y \Delta z$, where $\Delta t$ is a given time interval.

As $\rho$ is constant by hypotesis, and mass must be conserved, for the same time interval, $\Delta x$ must increase if the product $\Delta y \Delta z$ for example decreases.

We can say that the volume element changes its length, if we compare the same time interval at different points of the pipe.

Answer 2

If the fluid is incompressible and isothermal, pressures may change with changing flow rate per Bernoulli's principle. But it does not compress, so the density of individual areas does not change with changing flow rates.