Temporal interaction networks capture the history of activities between entities along a timeline. At each interaction, some quantity of data (money, information, kbytes, etc.) flows from one vertex of the network to another. Flow-based analysis can reveal important information. For instance, financial intelligent units (FIUs) are interested in finding subgraphs in transactions networks with significant flow of money transfers. In this paper, we introduce the flow computation problem in an interaction network or a subgraph thereof. We propose and study two models of flow computation, one based on a greedy flow transfer assumption and one that finds the maximum possible flow. We show that the greedy flow computation problem can be easily solved by a single scan of the interactions in time order. For the harder maximum flow problem, we propose graph precomputation and simplification approaches that can greatly reduce its complexity in practice. As an application of flow computation, we formulate and solve the problem of flow pattern search, where, given a graph pattern, the objective is to find its instances and their flows in a large interaction network. We evaluate our algorithms using real datasets. The results show that the techniques proposed in this paper can greatly reduce the cost of flow computation and pattern enumeration.