# Weak form of Navier-Stokes and heat transfer coupling(COMSOL)

I am asked to simulate a 2-D coupled problem in COMSOL(Navier stokes with Heat transfer) of a simple room.

I'm not sure if COMSOL already has preexisting physics for navier stokes and heat tranfer that I could use directly but I am provided with two differential equations and boundary conditions.

So what I did was to try to solve for the weak formulation and to add two partial differential equations into COMSOL. However, I am stuck in the equations converting into the weak form and I hope you could help me.

Question:

Navier Stokes (Incompressible)

$$-v\Delta u+\triangledown p=0 \! in\! \Omega$$

$$\triangledown \cdot u=0 in \Omega$$

Boundary conditions:

$$u=u_d on \Gamma_{1,w}$$

$$u=0 on \Gamma$$

$$u\cdot n=0$$ and $$t\cdot (-pI+v(\triangledown u + \triangledown^T))n = 0 on \Gamma_f$$

$$pn-n\cdot v\triangledown u = 0 on \Gamma_{2,w}$$

What I tried:

Multiply both sides by test function w,

$$\int_\Omega-v\Delta u w+ \int_\Omega\triangledown p w=0$$

Using green's theorem,

$$\int_\Omega\Delta w\cdot\Delta u-\int_{\delta \Omega} vw(\triangledown u \cdot n)+ \int_\Omega\triangledown p w=0$$

So I have two questions after this step:

1. Splitting the $$\int_{\delta \Omega} vw(\triangledown u \cdot n)$$ term and simplifying with B.C. (Not sure if I'm doing it right)

$$\int_{\delta \Omega} vw(\triangledown u \cdot n) = \int_{\Gamma_{2,w}}wpn$$

2. What do I do with the $$\int_\Omega\triangledown p w$$ term?

Next Heat Equation:

$$-\triangledown \cdot (k \triangledown \theta) + \rho C u \cdot \triangledown \theta = 0 \Omega$$

Boundary conditions:

$$\theta = \theta_f on \Gamma_f$$

$$n\cdot (k\triangledown \theta)=0 on \Gamma$$

$$\theta = \theta_0 on \Gamma_w$$

I have one question for this part

I have solved the heat equation without the term $$\rho C u \cdot \triangledown \theta$$

but with the term $$\rho C u \cdot \triangledown \theta$$ I am not sure what to do with it due to it having u in it.

How do I multiply by test function $$w$$ and use the green's identity in this case?

Thanks for reading

## Answers and Replies

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?