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

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).

• Why do you think diffusion is described by the NS equation? Have you looked into Fick's laws of diffusion? Nov 20 '20 at 12:37
• I have looked into Fick's laws of diffusion, may I ask if you think that would be better? I wanted to try NS equations first because the fluid I am measuring is compressible. I also wanted to have a vector field as a solution. Other than that, it was really just arbitrary (probably not a good idea). I'll read up on it more. Are there any sources you would recommend? Nov 20 '20 at 13:15
• Wait, I just realized that I cannot have a vector field for a single dimension. Nov 20 '20 at 13:49
• Yes you can, they will just be 1D vectors :) Nov 20 '20 at 13:53
• Oh right! So the vector will just be arrows pointing away from the point source on one line, then the magnitude will determine the arrow's length? I suppose I can still use the NS equation since it's only one-dimensional which will make the solution much easier. That aside, may I ask for any sources you would recommend for Fick's laws of diffusion? Nov 21 '20 at 11:40

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.

• I see, alright. Is it advisable to resort to Fick's laws of diffusion instead, leaving NS equation to be supplementary? The only variables I can work with is time, distance from point source, and concentration in g dm^-3. Nov 20 '20 at 13:23

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:

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

• Cool! I'll use this one for sure. I'm thinking of adding an R value of greater than one since I am using a diffuser as a point source. Is there a way to determine a value for it? Nov 22 '20 at 2:25
• What do you understand by $R$-value?
– Gert
Nov 22 '20 at 12:51
• I understood it as a measure of how source-y a source is if positive. Ethanol was constantly diffused throughout the experiment from a diffuser. Would it make sense to still add it since I don't really know how to obtain a value for it? Nov 23 '20 at 11:58
• If you're constantly adding EtOH, you'll need a source term in there. See the Wiki entry I linked to.
– Gert
Nov 23 '20 at 12:03