# 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?