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

[GreenBerg]

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

 

[mighty2000]

my post and any help would be appreciated.
Thank you for reaching out and sharing your progress and questions with us. I am a scientist with experience in solving coupled problems in COMSOL and I would be happy to assist you.

Firstly, I would like to confirm that COMSOL does indeed have preexisting physics for Navier-Stokes and heat transfer, so you do not need to manually add the equations into the software. You can simply select the relevant physics modules and input your equations and boundary conditions.

In regards to your specific questions, here are some suggestions:

1. For the term \int_{\delta \Omega} vw(\triangledown u \cdot n), you are correct in splitting it into \int_{\Gamma_{2,w}}wpn. However, you also need to consider the other boundary conditions, such as u=0 on \Gamma. You can also use the divergence theorem to simplify the term further.

2. For the term \int_\Omega\triangledown p w, you can use integration by parts to convert it into \int_\Omega p\triangledown \cdot w - \int_{\delta \Omega}w\cdot pn. You can then apply the boundary conditions to simplify the term.

For the heat equation, you can use the same approach of multiplying by a test function w and using the divergence theorem to simplify the term involving u. You can also use integration by parts to handle the term \rho C u \cdot \triangledown \theta.

I hope these suggestions will help you in solving your coupled problem in COMSOL. If you have any further questions or need more clarification, please do not hesitate to ask.

Best of luck with your simulation!