# Understanding the influence of terms in Navier Stockes fluid equation.

1. Jul 6, 2010

### Expi

Hi everyone, nobody seemed able to answer on the PDE mathematics forum so I am posting here.

I could use some help to validate (or invalidate) and improve my equations as I am trying to model 2D heat dispersion in a Spa.
Basicaly my model consist of some air inside a cabinet, the cabinet is framed by 2 different insulants each insulant has heat flux on boundaries (a hot source and a cold one).

I model the heat inside insulants by a simple Poisson's equation {$$\nabla$$ (-c$$\nabla$$ T) = 0, no problem here.

But I am really not sure about the equations of air.
I figure I should take 2 things into account, the convection and the conduction.

Which would mean my temperature equation becomes :
{ u $$\nabla$$ T - c $$\Delta$$ T = 0 , where u is my velocity field, I am not too doubtful about this one.

Then I also need to add my velocity field equation. From what I found it should be something like that :
{ $$\rho$$ u $$\nabla$$ u + $$\nabla$$ P =$$\mu$$ $$\Delta$$ u
here $$\rho$$ would be my air density, $$\mu$$ the air viscosity and P the air pressure.

So I have a few questions about that, which could use some physicists insight (yeah I am more the mathematician kind).

First you might have noticed I decided to operate independtly of time, so I ignored the time derivatives in both the heat equation and the velocity field. I am working with constant over time fluxes and looking for a balance state of dispersion. So I think that is the right call. What is your opinion about that ?

I should first explain a bit more the physical conditions of my model. The air is in a range of temperature of 16-39° C (289-312 K). I easely modeled the dispersion without convection. Now my convection is created by 2 gaps in the layers of insulant, one at the bottom (floor - side junction) and the other at the top (lid-side junction). Gap are around 1-2 centimeters wide but all around the spa which is not seen in the 2D model but the ratio is respected imo.

Then 2 questions about my velocity field
1) Concerning the pressure, I don't really know how it is acting inside the cabinet but I hardly believe it would have big fluctuations, meaning I could just ignore the derivative term of pressure. Do you think that's a good estimation?

2) The viscous diffusion term is dependent of $$\mu$$ , which if I am correct is around 15e-6 m²/s in my range of temperature, against a value of $$\rho$$ around 1.15 kg/m$$^{3}$$ so I might also be able to ignore the viscous diffusion. Is it a correct thinking? Honestly I tried my model without it and I am not very satisfied by the results. I would really be grateful if someone could explain me the influence of this term.

Again, all this is really seen in a mathematical pov. I am trying to achieve a computed estimation so I don't need a 0.1 degree preciseness, it's really about knowing where it is hot and where it is cold with a rought estimation of temperature range if possible. But like I said I am not a physicist and could overlook some important effects so if you have anymore suggestion or reflexion please share it, I am always willing to improve the model.