Background information: In this assignment we will use what we have learned in the course to analyze a data set measured in the Hydrodynamic Laboratory at the Mathematical Institute. The experiment was done in a tube, the tube has a circular cross section radius 5 cm. The measurements were made with "Particle Imaging Velocimetry" (PIV). This enables us to measure the velocity field. Note tah we Only measures the speed components $u$ in the $x$-direction and $v$ in the $y$-direction. The The full velocity field is $\vec{v}=u\vec{i}+v\vec{j}+w\vec{k}$, but we do not measure it speed component in the $z$-direction
Problem: We can accurately assume that both the gas and the liquid are incompressible because the flow is significantly slower than the sound speed in air and in water. *Explain the consequences of this for the divergence of $v$ and what we are in In that case, say the speed component $w$ that we have not measured
Approach: Since the water and gas are incompressible, it means that the water and the gas do not change densist by compression. Since the water and gas do not change the densities, so is the density the same. Since the density is constant, we can follow from the continuity equation that the (Current field/Stream field?) is divergence free. We can say that if the flow rate in the water and in the gas is less than $330$ m / s in air, and $1500$ m / s in water, so we can say that it is divergence free. Since the flow in our Example is slower than the sound speed in water and in air, we can say that we have divergence-free. Since the velocity field is divergence-free, then $w$ is a constant.
Is this correct? Or do I need to explain further?
This seems correct to me. To be especially precise, maybe you could say that the flow speed is much less than the speed of sound, so the model is justified.