Why is it that at $y=0$ (at the wall), we have $v=0$ (vertical component of velocity)? Obviously $v$ cannot be negative there as there is no flow through the wall, however how do fluid particles move off the wall if they don't have positive velocity?
Individual molecules can collide with the wall and move to it and away from it, but we're talking about a continuous model of fluid motion here. If $v$ was positive at some point $(x_0,0)$ on the wall (and continuous, so positive in some rectangle $R = \{(x,y): |x - x_0| < \epsilon, 0 \le y < \delta\}$, then (taking $\delta$ sufficiently small relative to $\epsilon$) all the fluid would soon flow out of $R$, leaving a vacuum.