# Navier-Stokes equations of motion in the Lagrangian description

In general, the Navier-Stokes equations of motion are derived in the Eulerian description. I tried to find the Navier-Stokes in the Lagrangian description but was not very successful.

I would be glad if someone could state the Navier-Stokes equation in the Lagrangian description or give me at least a reference where I can find it.

• @AccidentalFourierTransform Lagrangian form is different from the Lagrangian of a system. Lagrangian Navier-Stokes is written following a fluid particle as it moves, as opposed to Eulerian form which tracks the variables at fixed locations in space as the flow moves through them. – tpg2114 Feb 6 '17 at 16:34

It's pretty straight forward to compute, but even easier to locate using search engines -- but here is the mass and momentum equations, you can figure out the energy on your own. The key is using the material, or substantial, derivative:

Mass:

$$\frac{D \rho}{Dt} + \rho \nabla \cdot \vec{u} = 0$$

Momentum:

$$\frac{D\vec{u}}{Dt} = -\frac{1}{\rho} \nabla p - \nabla f_g + \nu \left( \nabla^2 \vec{u} + \frac{1}{3} \nabla \left(\nabla \cdot \vec{u}\right)\right)$$

It is easy to go back and forth between the Eulerian and Lagrangian forms, using the definition of the material derivative. That is left as an exercise.

• +1: Thank you a lot, so it is just the Navier-Stokes equation as we know. – MrYouMath Feb 6 '17 at 17:12
• @MrYouMath Maybe the ones you know -- but I only ever work with the Eulerian formulation, so I never see the Lagrangian form outside of exam questions that make sure I know the difference. – tpg2114 Feb 6 '17 at 17:13
• This is Eulerian description of flow. But the Lagrangian? No offence, but it is far from trivial to switch between the two descriptions, and I seriously doubt whether you can begin with the equations given above and go over to Lagrangian description. See Lagrangian Fluid Dynamics by A. Bennett. – Deep Feb 7 '17 at 4:03