Laws of fluid motion: from applications to the theory and back to applications

The popular undergraduate engineering fluid mechanics textbooks introduce the conservation of mass, the conservation of energy, and the linear momentum equations based on the Reynolds Transport Theorem (RTT). RTT provides the equivalence of the time rate of change of a property of a fixed-mass fluid element in a flow to the time rate of change of the same property within a control volume.

There are several shortcomings in the presentation of RTT in the textbooks:

1. The term “control volume” is not sufficiently clarified, and the purpose of transforming the laws from a fixed-mass system to a control volume is not illustrated with real-world, exciting applications.
   
   The whole presentation spans tens of pages as different special control volumes, from stationary to deforming ones, are dealt with separately, confusing students and young instructors.
   
   Using “D/Dt” to denote the total derivative is awkward and unnecessary as students do not have prior exposure to this notation in the calculus course, and it can be replaced by the familiar notation “d/dt”.
   
   Not all example problems are relevant and exciting. Some need a more explicit connection to a real-world engineering application.
   
   The smoothness assumption necessary to take the time derivative inside the integral over the control volume and to transform the integral form to the differential one is not highlighted.

Alternatively, RTT can be presented more clearly and concisely for the most general case that takes only a few pages. The motivation for the control volume version of the laws of fluid motion must include the relevant, exciting applications, including pumps, turbines, rockets, and propulsion. The analysis and discussion of these applications are often left to the latter chapters of the textbook and not addressed in the first course on fluid dynamics. For maximum student engagement and learning, the essence of these applications (not necessarily their detailed, thorough analysis) should be discussed and analyzed when the laws of fluid motion are first introduced.