We demonstrate a new approach to assuring fairness in flow-aware networking. Standard routine in flow-aware networking guarantees that each flow receives the same amount of resources. This property can be exploited by dividing a single transmission into a number of flows. We propose and evaluate the per-user fair packet scheduling mechanism, which ensures fairness among users (not flows). We also compare the new mechanism with the one already known from the literature, which is based on altering the admission control routine. Evaluation shows that the new concept of fairness provides better performance and fair resource distribution regardless of the behavior of users.