In this paper, we study the problem of core stability for flow games, introduced by Kalai and Zemel (1982), which arises from the profit distribution problem related to the maximum flow in networks. Based on the characterization of dummy arc (i.e., the arc which satisfies that deleting it does not change the value of maximum flow in the network), we prove that the flow game defined on a simple network has the stable core if and only if there is no dummy arc in the network. We also show that the core largeness, the extendability and the exactness of flow games are equivalent conditions, which strictly imply the stability of the core.