Let \(\mathcal{S}\) be the half-cylinder \(x^{2}+y^{2}=1, x \geq 0,0 \leq z \leq 1\). Assume that \(\mathbf{F}\) is a horizontal vector field (the \(z\)-component is zero) such that \(\mathbf{F}(0, y, z)=z y^{2} \mathbf{i}\). Let \(\mathcal{W}\) be the solid region enclosed by \(\mathcal{S}\), and assume that
\[
\iiint_{\mathcal{W}} \operatorname{div}(\mathbf{F}) d V=4
\]
Find the flux of \(\mathbf{F}\) through the curved side of \(\mathcal{S}\).