Let $f:X \to Y$ be a flat, proper, surjective morphism between noetherian schemes. Assume $Y$ is irreducible and smooth over $\mathbb{C}$. Suppose that $X$ is the union of two schemes $X_1$ and $X_2$ both flat, proper and surjective over $Y$. Is there any known condition under which the scheme theoretic intersection, $X_1.X_2$ is flat over $Y$ (for example, if the generic fibers of the morphisms $X_1 \to Y$ and $X_2 \to Y$ do not intersect)?