Please explain the meaning of these Boundary conditions [closed]

Question

I am trying to learn Gmsh and Fenics and was looking at an example which shows the application of Boundary conditions on a simple Poisson problem.

Here is the link:

http://fenicsproject.org/documentation/dolfin/1.3.0/python/demo/documented/bcs/python/documentation.html

The example defines the domain Ω to be a model of blood vessels with an aneurysm. It has one inlet vessel and two outlet vessels. They define no slip boundary conditions on the walls of the vessels and aneurysm, that is u=u0=0.0. We let u=u1=1.0 be the Dirichlet condition on the inlet, the outlets will have the prescribed values u=u2=2.0 and u=u3=3.0. In summary, we have:

u=u0=0.0 on ΓD,0 (noslip boundary);
u=u1=1.0 on ΓD,1 (inlet);
u=u2=2.0 on ΓD,2 (outlet 1) ;
u=u3=3.0 on ΓD,3 (outlet 2) ;
f=0.0 (source term)

I can understand the code overall, but I am unable to understand the meaning of these Boundary conditions, i.e., unable to visualize them in a physical sense, intuitively. I' ll appreciate if you can post any other link or better example too which explains the BC phenomena better.

Thank you!

Answer 1

If you look at the documentation,¹ $u$ is described as solving the Poisson equation $(\nabla u,\nabla v) = (f,v)$ for $v\in H^1(\Omega)$. This means that it is unlikely to be a velocity field (it's scalar, after all). I would interpret the documentation insofar as only the geometric domain is meant to model a blood vessel, while the PDE is only a very academic example that isn't supposed to model anything. Hence, there is no further interpretation than that the value of $u$ is prescribed (as different values at different parts of the boundary).

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_{1. which is outdated, by the way -- current is 1.5, and 1.6 is coming up soon}