I am working on a rectangular domain where $\Gamma_{in}$ is the left hand side inlet boundary and $\Gamma_{out}$ is the outflow boundary, to the right. The top and and bottom are rigid walls with no-slip conditions. The flow equation is a stationary Navier Stokes problem which is solved using Finite Elements method with a non-homogeneous inlet condition $u(y)$. I wanted to know how to correctly implement this boundary into the weak form, or more precisely is there a possibility that we can eliminate face terms which are a result of the weak form.  $\rho(u\cdot\nabla u) - div \, \sigma=\rho f$$div \, \sigma = 0$B.C.: $u=u(y)$ on $\Gamma_{in}$
Non Homogeneous Dirichlet conditions on the inlet for Navier Stokes Problem
finite element;boundary conditions;navier stokes;variational calculus