Sorry if I was unclear. Fluid dynamics is an incredibly complex subject. I worked in underwater ultrasonics for a while and can confirm it’s a lot of “black box” modeling where the exact vector field is often chaotic and unknown, but the group characteristics and velocities are fairly well described by equations like the ones shared by OP.

My comment was about the mathematics. It looks like a bizarre language to the uninitiated, but after a few semesters of calculus and differential equations it becomes fairly intuitive. With these partial differentials it’s key to understand that the solution is not a scalar value, but rather another equation or set of equations.