The velocity field in an incompressible flow is $\boldsymbol{v}=\alpha x y \hat{\boldsymbol{\imath}}+v_{y} \hat{\boldsymbol{\jmath}}+\beta \widehat{\boldsymbol{k}}$, where $\hat{\boldsymbol{\imath}}, \hat{\boldsymbol{J}}$ and $\widehat{\boldsymbol{k}}$ are unit-vectors in the $(x, y, z)$ Cartesian coordinate system. Given that $\alpha$ and $\beta$ are constants, and $v_{y}=0$ at $y=0$, the correct expression for $v_{y}$ is
- $\frac{-\alpha x y}{2}$
- $\frac{-\alpha y^{2}}{2}$
- $\frac{\alpha y^{2}}{2}$
- $\frac{\alpha x y}{2}$