# Coupling three equations for a homogenization problem with the Boussinesq approximation
**URL:** https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936
**Category:** General Discussion
**Created:** [May 25, 2025, 4:52pm UTC](https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936 "2025-05-25T16:52:13Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@taha](https://community.freefem.org/u/taha)
#### Post date: [May 25, 2025, 4:52pm UTC](https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936/1 "2025-05-25T16:52:13Z")
Hello everyone, I’m trying to solve a steady flow with the Boussinesq equation. I have three equations with no nonlinear terms, but I still have this error: ‘End of String could not be found.’ Thank you so much for your help.
[boussinesq.edp](https://community.freefem.org/uploads/short-url/mMDdKIqIMn60DmyKvn9gHJtFFVV.edp) (1.8 KB)
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 26, 2025, 3:16am UTC](https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936/2 "2025-05-26T03:16:40Z")
Hello, here is your code with several modifications. I hope it will help you.
[boussinesq\_eqn.edp](https://community.freefem.org/uploads/short-url/sIMj34Gynetig3ikAyQNYobhbJA.edp) (1.9 KB)
If you give me your schemes ( i.e, set of equations that you implemented here) and exact solution, i can check order of convergences.
Thanks in advance!.
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### Author:  [@taha](https://community.freefem.org/u/taha)
#### Post date: [May 26, 2025, 5:39am UTC](https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936/3 "2025-05-26T05:39:26Z")
thank you so much for you answer ,my equations are basically ΔP + a ∂T/∂y = 0
2. μv = K(∇P + aTe\_z)
3. v·∇T = ∇·(D∇T)
and in the weak form i get
4. ∫\_Ω ∇P·∇Q dΩ = a∫\_Ω (∂\_y T)Q dΩ
5. velocity :
v\_x = (K/μ) ∂P/∂x
v\_y = (K/μ) (∂P/∂y + aT)
6. then temperature:
∫\_Ω v·∇T ϕ dΩ + ∫\_Ω D∇T·∇ϕ dΩ = 0. and thanks again for your help
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 26, 2025, 6:58am UTC](https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936/4 "2025-05-26T06:58:26Z")
Can you please share it’s details reference (that is paper)??.
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### Author:  [@taha](https://community.freefem.org/u/taha)
#### Post date: [May 26, 2025, 7:13am UTC](https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936/5 "2025-05-26T07:13:25Z")
i’m sorry,i just derived them by myself
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 26, 2025, 7:26am UTC](https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936/6 "2025-05-26T07:26:34Z")
Actually, i need exact solutions to check order of convergence. I asked for that reason. It is fine you give me exact solution and source function.
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### Author:  [@taha](https://community.freefem.org/u/taha)
#### Post date: [May 26, 2025, 7:34am UTC](https://community.freefem.org/t/coupling-three-equations-for-a-homogenization-problem-with-the-boussinesq-approximation/3936/7 "2025-05-26T07:34:17Z")
oh yeah i know,but there is no exact solution for this probleme(as much as i know),but i think there is a problem in the converging,because T never converge with my dirichlet conditions