How can I apply Dirichlet boundary conditions weekly for viscoelastic Fluids?

variational formulation
1

Hello,

I am trying to implement the 3 Field GLS Method for viscoelastic Oldroyd B Fluids (original Paper: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.8041 ) in Fenics.

I want to compute the flow past a circular cylinder, which is a standard benchmark problem for two dimensional viscoelastic flow simulation. Therefore I have to set the following no slip boundary condition for the velocity u and the extra (deviatoric) stress tensor T at the cylinder wall

u = 0

(T*n)*n= 0

I have figured out, that i can apply these Dirichlet boundary conditions weakly by using Nitsche Method, but I struggle to derive the Nitsche’s terms.

Can someone help me with that issue or knows another way to set these boundary conditions in Fenics?

Thanks in advance!

2

This problem is not simple.

The linear case is rather straightforward e.g. here.

The nonlinear case is more difficult, but I have made efforts in this (unpublished preprint) work with examples for FEniCS. This builds on very similar work published here.

You may also consider @dokken’s work using strong imposition with multi-point constraints here https://github.com/jorgensd/dolfinx_mpc. Stokes example here.