# What is the velocity in the Navier-Stokes equation?

I have been looking at the Navier-Stokes equation, and can't seem to find anywhere a clear description of what velocity it represents.

From what I have read it could be any of the following:

• The 'flow velocity'.
• The velocity of an individual particle (I think this is very unlikely).
• The mean velocity of particles near the point of consideration.

Which of these (if any) is correct and why? (A source would be helpful)

• The $\mathbf{v}$ is the bulk flow velocity of a fluid element (i.e., a "blob" of the given fluid). – honeste_vivere Jun 20 '16 at 18:31
• @honeste_vivere would this not be the same as my third option? – Quantum spaghettification Jun 20 '16 at 18:33
• Kind of but not really... You treat it as an ensemble of fluid elements, not individual particles (which would be kinetic theory). The Navier-Stokes equation is a fluid approximation derived from kinetic theory by taking ensemble spatial averages. It's the bulk flow velocity you find after you "fluidize" the equations, if you will. Does that help? – honeste_vivere Jun 20 '16 at 18:36
• @honeste_vivere Sorry, please can you specify what you mean by 'ensemble spatial averages'. I know what an ensemble average is, but what do you mean by 'spatial', what are we averaging and over what (space?)? – Quantum spaghettification Jun 20 '16 at 18:48
• An ensemble average requires a quantity that you specify over which one averages. For example, one can do a spatial (yes over space) or temporal (i.e., similar to but not the same as a time-average) ensemble average. There are also energy ensemble averages (e.g., related to the partition function in statistical mechanics). – honeste_vivere Jun 20 '16 at 22:19

You are correct, it is the velocity of a small volume of fluid centered at the point, that is a macroscopic motion, but it is also the result of the average velocity of the particles in that volume.

A bit of 1, a bit of 3...

The technical name is flow velocity, as correctly stated in the Wikipedia article about NS equations.

But one could ask what "flow velocity" means. From the Wikipedia article:

flow velocity [...] is a vector field which is used to mathematically describe the motion of a continuum.

Although correct, this definition is somewhat obscure. So let's take a closer look.

When describing the motion of a fluid, we theoretically have two options: the first is to label every single particle in the fluid and solve Newton's equations

$$\frac{d^2}{dt^2}\pmb x_n(t)=\pmb F_n(t) \ \ \ \ n=1,\dots,N$$

with the initial conditions

$$\pmb x_n(t_0)=\pmb x_{n,0}$$

where $N$ is the number of particles in the fluid. If we were able to solve these equations, we could express the time evolution of the system through a set of functions $\pmb U_n$ giving the position of particle $n$ at time $t$:

$$\pmb x_n(t)=\pmb U_n(\pmb x_{n,0} \ ,t)$$

This viewpoint is known as the Lagrangian viewpoint. The problem of this approach is that even if it could work theoretically, it is completely useless if we want our theory to be somewhat predictive. The reason is that of course we cannot know the microscopic initial conditions of our sysstem, let alone solve Newton's equations.

We therefore choose another approach, known as the Eulerian viewpoint. We choose a point $\pmb x$ in space and measure the velocity of the fluid around that point (in a small control volume). In practice, we would take a really small paddle wheel and put it in $\pmb x$, then measure the speed of its rotation to deduce the velocity of the fluid in that point of space.

If we take an infinite number of infinitely small paddle wheels, we will obtain a vector field for velocity, giving us the velocity of the fluid in every point of space:

$$\pmb u(\pmb x, t)$$

This is what we call "flow velocity" and what appears in the NS equations.

The two viewpoints are related by the following expression:

$$\pmb u(\pmb U (\pmb x_0, t), t) = \frac{\partial \pmb U}{\partial t} (\pmb x_0, t)$$

because it can be seen that both sides describe the velocity of the parcel labeled $\pmb x_0$ at time $t$.