Help to define sedimentation on a surface

Question

Hello.
I am considering the problem of dust transfer and I want to work on its sedimentation on the surface.
It is necessary for me that the dust fell on the surface, but did not leave it.

I need help with define sedimentation on a surface.
Consider a simple example

du0 = u.dx(0)
a = (dt1 * u * v + du0 * x_w * v) * dx
L = (S + dt1 * u_e) * v * dx

where u is concentration of dust, dt1 is time step (1/sec), u_e is value of u on previous iteration, x_w is speed, S - source.

I think, that u on boundary (Ub) is:
Or

But i don't know, how i can realize this on Fenics.

What i have without boundary: https://youtu.be/22synVetBoY
What i want: https://youtu.be/78-NQt-SUcg

Please help me to add boundary! Freelance is being considered

Comments

Your problem regards modelling, not fenics. For us to help you, we need at least the model you are trying to implement and what you expect. Regards.

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I added more information, can you help me?

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There is something wrong with that formulation, because $\nabla u$ is a vector, and $dt$ is a scalar differential, so that integral would give a vectorized $u_b$. Now, if you want to impose a boundary condition depending also on $u$, I would recommend incorporating it weakly (Nitsche's trick) so that you can use the nonlinear solver. For now, the condition you want to implement does not make sense. Check it and let me know when you know how it should be. Best regards!