Brenner Equations of Motion

Howard Brenner made a fascinating observation about fluid mechanics: the derivation of the equations of motion rely on identifying the [Eulerian] mass velocity with the [Lagrangian] volume velocity, despite the fact the former is kinematical whereas the latter is dynamical. We can experimentally test this hypothesis, and Brenner did that!

• Howard Brenner,
  "Fluid mechanics in fluids at rest".
  Physical Review E 86, no.1 (2012): 016307. Eprint

Unsurprisingly, these quantities are different. We end up with three sets of equations:

1. Conservation of mass: \(\displaystyle\frac{\mathrm{D}}{\mathrm{D}t}\rho+\vec{u}_{m}\cdot\vec{\nabla}\rho=0\)
2. Euler's equations: \(\displaystyle\rho\frac{\mathrm{D}}{\mathrm{D}t}\vec{u}_{v} =-\vec{\nabla}p\)
3. Constitutive equation: \(\vec{u}_{v}-\vec{u}_{m}=K\nabla p\).

For incompressible fluids with small pressure differences, this behaves the same as the Euler equations of motion.

But for compressible fluids with large pressure differences, the behaviour is completely different.

• Eduard Feireisl and Alexis Vasseur,
  "New perspectives in fluid dynamics: Mathematical analysis of a model proposed by Howard Brenner."
  In New directions in mathematical fluid mechanics, Adv. Math. Fluid Mech (2009): 153-179. Preprint.
• Howard Brenner,
  "Fluid mechanics in fluids at rest".
  Physical Review E 86, no.1 (2012): 016307. Eprint
• Howard Brenner, Nishanth Dongari, Jason M. Reese,
  "A molecular dynamics test of the Navier-Stokes-Fourier paradigm for compressible gaseous continua".
  arXiv:1301.1716, 12 pages
• Howard Brenner,
  "Unsolved Problems in Fluid Mechanics".
  PDF.
• Howard Brenner,
  "Beyond the no-slip boundary condition".
  http://hdl.handle.net/1721.1/68655