I've seen the derivations for both viscous and inviscid momentum balance in fluid flow in courses, but I'm curious as to where the original derivation for the equations now referred to Euler equations or Navier-Stokes equations is. I would love to see the papers or lecture notes, even if they're in a different language.
Does anyone know what the papers are called/how to find them? I couldn't find them despite quite a bit of Googling.
According to the book Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl by Olivier Darrigol, the derivation of Euler's equation by Euler and of Navier-Stokes by Stokes are essentially the modern methods: that is, one considers the balance of forces on an infinitesimal volume element. (see pages 23-25 and 135-139).
The main difference is that vector calculus notation was invented only much later, so these equations were written in a more complicated form.
You can find Euler's paper here. The paper is originally in Latin, however at that page you'll find a German translation. See also here.
You can find Stokes' paper (Trans. Camb. Phil. Soc., vol 8, pp 287-305) here.
Dr. Taha here covers Navier's derivation briefly at 58 minutes into this video. There are references given to the fluid mechanics history book which contains it. Navier's derivation is super cool, for it considered two fluid particles and equated virtual work between the two repelling each other.