Reynolds transport theorem and the Sputnik crisis

Do you find science difficult? You can put a bit of the blame on the Sputnik satellite mission. Why?

I have been teaching Reynolds transport theorem (RTT) for more than a decade as part of the courses related to fluid mechanics and hydrology. I always find students struggling to understand the concept. A couple of years back, I started wondering if there was any real value in teaching the theorem since I didn’t see anyone using RTT to tell what was not already known.

To those who are unaware, RTT can be used to derive the mass, momentum, and energy conservation laws. However, the three laws were derived much before RTT was formulated. What purpose does then the RTT serve? We need to understand the historical context of RTT.

In 1957, the USSR launched the first artificial satellite, Sputnik 1, creating fear of a communist takeover of the USA. American educators and policymakers thought the Russians had an edge due to their superior mathematical skills. The RTT was thus introduced by Shames in 1962.

What is the difficulty with RTT? RTT is an attempt to do what is commonly termed formal mathematics. Formal mathematics attempts to minimize the emphasis on observations. You will be referred to some axiom that is hard to relate to your real-world experience. That is exactly why students find RTT difficult. If this is not clear, let me cite the popular example 1+1=2. You can put two apples side by side and prove the statement. But formal mathematics would not accept it. This is why Russel took about 370 pages to provide proof, which anyway is not valid for all cases.

So, if you are finding a concept difficult to understand. Don’t put all the blame on yourself. Read the papers that provide a clear picture of something without introducing mathematics unnecessarily. In the context of fluid mechanics, you can just imagine a bucket draining water to understand the mass balance principle.

For more details on RTT, see: Gray, D. (2008, June). Why the Balance Principle Should Replace the Reynolds Transport Theorem. In 2008 Annual Conference & Exposition.