# Coupling Navier-Stokes and stochastic models for particle tracking in micro-scale free convection?

I have been using a commercially available software to simulate laminar free convection in a specific small domain (let's use channel w/ heated lower wall as an example). The scale is approx 50-100 micrometers and a Knudsen # < 0.01 is feasible so I've been using the NS equations and a finite element mesh to do so.

I would like to demonstrate that the convection observed is (or can be if the temperature gradient is large enough) strong enough to overcome molecular thermal diffusion by plotting the trajectory of a small particle, say ~1 micrometer diameter in the flow. Considering particles this size, the Knudsen number is getting larger, and the continuum approximation is becoming questionable. I thought this meant the NS would no longer be appropriate and I'd have to switch to a stochastic model.

After a massive amount of Google searching, I'm still left with two (related) questions:

1. Why is it justified for computational models of particles in nano- and micro-scale flows to assume continuity and solve general NS equations (rather than eg a langevin or monte carlo stochastic model) when the scale is on the order of the mean free path?? What am I missing?

2. Is there an obvious way to couple the bulk convective flow from NS with random thermal motion for the purposes of my particle trajectory problem? I've found examples online of the addition of brownian motion to the navier stokes equations:

.. as well as focusing on a stochastic model and just incorporating the mean or bulk flow (eg convection) into it:

but I can't figure out which of these would be the simplest or most appropriate way for me, and they all seem far more complex than I'm able to implement in the time I have (this is a "spare time" project for me outside my full time job). I just want to demonstrate some random diffusive motion of a single particle while it follows the bulk convective flow, I was naively hoping this would be as simple as adding a random component to the velocity u in the NS.

Apologies if this is an inappropriate or vague question. Any help or advice would be greatly appreciated.

Thank you.

## 1 Answer

To address your questions:

1) If the length scale is of the order of the mean-free path (i.e. the Knudsen number is approaching 1), then I don't think the continuum assumption would be appropriate. Ultimately, that is what the Knudsen number is for - determining when the continuum assumption breaks down. If that is the case, you might be better off using a different (stochastic) analysis tool.

2) If the continuum assumption holds, then I think this 'random particle' approach to determining whether thermal convection or diffusion dominates may not be the best approach.

Firstly, are you aware of the Rayleigh number, which is another dimensionless number that compares the buoyant effects on the flow to the viscous effects? As the Wikipedia article states, when it is above a certain critical value for the flow, convective heat transfer should dominate. It looks like you would need to estimate the critical Ra number for your flow. It's not particularly precise, but it might provide you with a fairly quick answer, without having to do any simulation.

Another option you could consider to compare the convective and diffusive heat transfer would be to run a couple of trial simulations in your analysis tool: one fully convective and then another one with the same geometry and boundary conditions, with heat transfer enabled, but no flow (i.e. turn buoyancy effects off). Then, you can compare the rate of heat transfer to a certain target surface in both simulations. If the convective transfer dominates, then the heat transfer rate in the first case should be significantly larger than the second.

So, I think there are some easier things you can try, without having to get into trying to simulate random particle motion.

• Hi, Time4Tea! I agree on the Knudsen number, but I've seen several papers in the literature that continue using NS even when Kn should be high. I haven't figured out why they are able to do this. This is where my problem comes in, the continuum approx. holds for the bulk convective flow, but not if I want to consider a particularly small particle within it that will likely be experiencing random brownian motion. It's coupling these two scales that's leading to my confusion. – E. Howard Sep 1 '18 at 21:46