There are a few things to consider:
- Your problem is a first order PDE, so you’ll be over prescribing boundary data with a condition like \frac{\partial \rho}{\partial y} = 0 on some boundary. The link you’ve provided isn’t applicable for your case.
- If the facets marked in red are outlet BCs, and your inlet condition is well behaved, I expect your solution to be symmetric about x = 0.
- It may still possible to prescribe \vec{u} \cdot \vec{n} = 0 on the left boundary, depending on your inlet condition.