Dear Sir.!

It cannot be applied.

Try to search about the Boltzmann Transport Equation (BTE). This is the approach you will have to use to get into your problem when regarding with heat conduction and energy transfer at the nanoscale.

The Boltzmann Transport Equation is practically the equivalent to the Navier-Stokes equations but in the limit of Near-equilibrium transport.

I don't know 'till what extent you will be able to approach to your particular problem with this equation, since it has n't been solved yet for many general cases, but please give it a look.

Boltzmann Transport Equation at nanoscale fluid, or nanoscale liquids .

Hope you will find help with this!

Regards !

By the way, another approach is : if your system behaves somehow stochastically, you will necessarly have to get deep into Stochastic Thermodynamics.

I don't understand myself the framework of Stochastic Thermodynamics but for someone who masters the mathematics of the Navier-Stokes equations may be understandable. Also it is a field of study which has there for nearly 20 years.

I'll attach this paper about stochastic thermodynamics. It could happen the approach your are looking be here.

I would take a look first into the BTE though .

Best Regards !

Thanks a lot , I would recommend to read the following paper "Temperature profiles and heat fluxes observed in molecular dynamics simulations of force-driven liquid flows".

Best Regards