Understanding the influence of terms in Navier Stockes fluid equation.

[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.

Thank you for your time and answers.
Have a good day.

 

[eljose79]

Hello, I am a physicist and I would be happy to provide some insight on your equations for modeling 2D heat dispersion in a spa.

Firstly, it is important to consider the physical conditions of your model. You mentioned that you are working with constant over time fluxes and looking for a balance state of dispersion. In this case, it is appropriate to ignore the time derivatives in both the heat equation and the velocity field. This is because you are interested in the steady state solution where the temperature and velocity fields are not changing with time.

Regarding your velocity field equation, it is important to consider both convection and conduction in the air. Therefore, your equation should include both terms: u\nabla T - c\Delta T = 0. The velocity field equation you have provided is correct, however, it is important to note that the pressure term should also have a gradient operator on it. This is because pressure is a scalar quantity and the gradient of a scalar is a vector.

In terms of your questions about the pressure and viscous diffusion terms, it is difficult to say whether they can be ignored without more information about your specific model. However, it is generally safe to assume that pressure fluctuations will be small in a confined space such as a spa. As for the viscous diffusion term, it is important to include it in your model as it can have a significant influence on the velocity field and subsequently on the temperature field.

In conclusion, it seems that your equations are on the right track, but it may be beneficial to do some further research and possibly consult with a physicist or engineer who specializes in fluid dynamics to ensure that all relevant physical effects are taken into account in your model. I hope this helps and good luck with your research!