I know how to solve the diffusion-convection equation for concentration and Stokes equation in FreeFem++, but I have no idea about implementing the force-free condition to obtain **U**. Can anyone give me some suggestions on implementing the force-free condition? Does FreeFem++ have similar examples? I would appreciate any help you give me.

This is simple, put no boundary condition on this part of the boundary.

Dear Prof. Frédéric Hecht,

Thanks for your kind reply. I may not make the question clear. The force-free condition (marked in blue in the above question) is imposed on the particle \Gamma_p (the total hydrodynamic force exerted on the particle is zero). The outer boundary \Gamma_o is the negative translation velocity \mathbf{U} (in the particle frame). Note that \mathbf{U} is unknown and it is determined iteratively by requiring that the total hydrodynamic force (force-free condition) on the particle surface \Gamma_p is zero at all times.