# 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.