Ever wonder how we designed airplanes, chemical reactors, and heat exchangers before computers? Unlike what many believe, it wasn't just trial and error.
Turns out that it's the same answer to why I wanted to go to grad school, and it's also what drove me to working with fracture mechanics. Grad school was a masterclass into understanding the foundations of engineering with a (heavy) hands-on component.
For me it starts with the Navier-Stokes equations. How (Newtonian) fluids move. The general case is not solvable analytically. Laminar flow over a flat plate? Impossible too. Or so we thought.
Ludwig Prandtl's insight was simple in his 1904 paper. He embraced the no-slip condition, that fluid velocity at a solid surface is zero, and realized that viscous effects are confined to a thin layer near the surface: the boundary layer. Outside it, inviscid theory holds. Inside it, viscous terms matter. Essentially he separated the problem into two, one where the plate mattered, and one that didn't see the plate. Almost too simple of an insight, but nobody else had seen it before him.
With dimensional analysis guided by this insight he simplified the set of Partial Differential Equations (Navier-Stokes, 2D) significantly within that thin region into a more tractable set of equations.
It got him promoted to teach at Göttingen. But he still didn't have specific results. His insight simplified the equations but didn't solve them.
He was aware of his mathematical limitations. So he threw the problem at his PhD students. One of them, Blasius, took it to the next level.
Blasius applied a similarity transformation (very nontrivial) to reduce the boundary layer equations into a single third-order nonlinear ODE. Infinitely more tractable than the original setting. The self-similarity argument allowed for a pen-and-paper solution. The problem was no longer geometry-dependent, just like the FAD Level 2 curve from API 579 (2021) 😉.
After several pages of series expansions computed entirely by hand, he became the first person to determine the skin friction coefficient of a flat plate, cementing the foundations of aerodynamics. You can see his (translated) work here: Blasius's PhD thesis
But Blasius didn't become a rockstar to the level of Prandtl. What I think happened is that solving math problems is simply not enough.
In academia, insight often gets all the glory, but outside of it, execution is everything. Today, bridging that gap between theoretical insight and practical execution is exactly what we do in Fitness-for-Service even if we don't think every day about J-integrals, traction vectors, or conservation equations.
So this is why I went to grad school. Now I know where pen and paper simplifications come from (like Tada/Paris/Irwin for fracture, the Kirsch solution for a hole in a plate, or the famous Bird/Stewart/Lightfoot for Chem Eng), under which assumptions they hold and why.
Understanding is what generalizes. It's what allowed me to get involved in Fitness-for-Service and understand fundamentals of fracture mechanics. Different fields based on the same principles.
AI these days seems like a silver bullet. Turns out that there is no replacement to understanding other than doing the groundwork. And this will only matter more with time, not less. Because we as humans will always value another human who can explain things we didn't know, with empathy and, many times, pure altruism. Machines can't do that.
Some people think a PhD is too narrow. For me, it was the ultimate tool for learning how to generalize. How many of you ended up working exactly in what you went to school for? How much of it came down to understanding the fundamentals?