Let $G$ be a graph, and $\lambda$ the smallest integer for which $G$ has a nowhere-zero $\lambda$-flow, i.e., an integer $\lambda$ for which $G$ admits a nowhere-zero $\lambda$-flow, but it does not admit a $(\lambda -1)$-flow. We denote the minimum flow number of $G$ by $\Lambda \left(G\right)$. In this paper we show that if $G$ and $H$ are two arbitrary graphs and $G$ has no isolated vertex, then $\Lambda (G\vee H)\le 3$ except two cases: (i) One of the graphs $G$ and $H$ is $K_2$ and the other is $1$-regular. (ii) $H=K_1$ and $G$ is a graph with at least one isolated vertex or a component whose every block is an...