How can I use the Navier-Stokes equation to model the diffusion of a gas with the data I gathered?

Question

I am currently a senior high school student taking IB and I chose to write an extended essay in physics. The essay is about the diffusion of a gas with the research questions:

"How does the diffusion of a gas vary with distance from its origin?"

"What factors affect the variation?"

I was already able to gather data for this but I'm having difficulty with relating it to the compressible Navier-Stokes equation. My data only has 1 dimension of movement so my data is a concentration vs time for increasing distances from the point source. The data I gathered is either natural-logarithmic or follows an inverse law of 1/t.

I would like to ask the following:

1. Is it feasible for me to do so? Considering that my data is low-dimensional.

I would really appreciate any answers to the questions above and possibly even an explanation of how the Navier-Stokes equation might explain diffusion (I am still trying to read up on it but I can't seem to find the right source to help me learn).

Answer 1

The short answer for 1: you can't.

Long answer: While the NS are exact equations, getting useful information about transport in air strictly from the equations is unfeasible. The reason is that the NS equations are nonlinear partial differential equations that cannot be solved in closed form. To describe the motion of a tracer in a fluid we necessarily have to rely on a statistical description of the turbulent eddies that cannot be derived directly from the NS equations (and are ultimately based on experiments or numerical simulations). Since fluid turbulence is random it, however, makes sense to assume that the large scale motion we should get is similar to diffusion: what we call diffusion (say, in biology) is the random motion of particles due to random kicks by molecules. In a gas what we get is the random motion of particles as a result of random motion of eddies. Under a few assumptions, we can find that the motion of a tracer $\phi$ (the density of the gas) is governed by the diffusion equation: $$\partial_t \phi = \kappa\nabla^2 \phi$$ where $\kappa$ is some constant, depending on the problem at hand (e.g. typical velocity of air flow, temprature, viscousity, shape of the room) that in principle could be calculated from the NS equations, but in practice can only be obtained from experiment/simulation.

As a recommended reading, I would advise to start with the Feynman lectures chapters on the relation between randomness and diffusion . This is a good explanation as to why, for example, diffusive behaviour results in the size of a "blob" of gas growing as $\sqrt{t}$. You should probably also check the lecture on the NS equations. Diffusion by turbulence is a little advanced, but is covered in many textbooks. A favorite of mine is chapter 10 of "Atmospheric and Oceanic Fluid Dynamics" by Vallis.

Answer 2

Forget about Feynman lectures, what you need here is the convection-diffusion PDE:

$$\frac{\partial c}{\partial t}=\nabla(D\nabla c)-\nabla(\mathbf{v}c)$$

This is essentially a version of the transport equation.

Assuming $D$ and $\mathbf{v}$ to be constant and for the $\text{1D}$ case we have:

$$c_t=Dc_{xx}+v c_x$$