The most common boundary condition condition for fluid mechanics is to know the concentration at the boundaries. This arises primarily as the no slip condition. The no slip condition means that the fluid velocity at the boundary is equal to the boundary's velocity. This holds true both for fluid-solid boundaries (which might be zero like at point(s) 1) as well as fluid-fluid boundaries. (where the two fluids have the same velocity, like at point 2).
We can specify a constant flux at the boundary when we know the value of the stress there (i.e., knowing the momentum flux). The most common place that this would arise would be as a continuity of stress condition at the boundary between two fluids. (At point 2 notice that the slopes of the velocity profiles have the same sign, but not necessarily the same value of the slope, because the viscosities may be different)
A special case of this is for the case of where we assume that
Not common in fluid mechanics.
Identify reasonable boundary conditions in a fluid mechanics problems (explain when each is most useful)