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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)

Navier Stokes (Incompressible)

[tex]-v\Delta u+\triangledown p=0 \! in\! \Omega[/tex]

[tex]\triangledown \cdot u=0 in \Omega[/tex]

Boundary conditions:

[tex]u=u_d on \Gamma_{1,w}[/tex]

[tex]u=0 on \Gamma[/tex]

[tex]u\cdot n=0[/tex] and [tex]t\cdot (-pI+v(\triangledown u + \triangledown^T))n = 0 on \Gamma_f[/tex]

[tex]pn-n\cdot v\triangledown u = 0 on \Gamma_{2,w}[/tex]

**What I tried:**

Multiply both sides by test function w,

[tex]\int_\Omega-v\Delta u w+ \int_\Omega\triangledown p w=0[/tex]

Using green's theorem,

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

**So I have two questions after this step:**

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

[tex]\int_{\delta \Omega} vw(\triangledown u \cdot n) = \int_{\Gamma_{2,w}}wpn[/tex]

2. What do I do with the [tex]\int_\Omega\triangledown p w[/tex] term?

**Next Heat Equation:**

[tex]-\triangledown \cdot (k \triangledown \theta) + \rho C u \cdot \triangledown \theta = 0 \Omega[/tex]

Boundary conditions:

[tex]\theta = \theta_f on \Gamma_f[/tex]

[tex]n\cdot (k\triangledown \theta)=0 on \Gamma[/tex]

[tex]\theta = \theta_0 on \Gamma_w[/tex]

I have one question for this part

I have one question for this part

I have solved the heat equation without the term [tex]\rho C u \cdot \triangledown \theta[/tex]

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

How do I multiply by test function [tex]w[/tex] and use the green's identity in this case?

Thanks for reading