# Inflow and outflow boundary conditions for advection-diffusion equation

I'm trying to solve this advection-diffusion equation (ADE):

$$\frac{\partial \phi}{\partial t} + \nabla \cdot (-D \nabla \phi + \mathbf{u} \phi) = 0$$

In fact, this ADE framework is coupled to a Navier-Stokes (NS) solver and I get velocity field ($\mathbf{u}$) from NS solver. My question is about ADE boundary conditions on inflow and outflow planes. I postulate that because the flow is directed from inflow to outflow obviously, I put zero concentration ($\phi = 0$) boundary condition on inlet plane because my source of species is located far from inflow boundary. Also, I assumed that the dominant transport mechanism at the outlet plane is convection and as a result of that normal diffusive flux will be vanished at the outlet plane ($-D \frac{\partial \phi}{\partial \vec{n}} = 0$).

I wanted to know how much these assumptions are true and could be justified physically? I should say that the outflow boundary condition is verified by comparing the simulation results with analytical solution of the Graetz problem. Any reference, suggestion, or idea is appreciated.