Solving network flow problems is a fundamental component of traffic engineering and many communications applications, such as content delivery or multi-processor scheduling. While a rich body of work has addressed network flow problems in "deterministic networks" finding flows in "stochastic networks" where performance metrics like bandwidth and delay are uncertain and solely known by a probability distribution based on historical data, has received less attention. The work on stochastic networks has predominantly been directed to developing single-path routing algorithms, instead of addressing multi-path routing or flow problems. In this paper, we study constrained maximum flow problems in stochastic networks, where the delay and bandwidth of links are assumed to follow a log-concave probability distribution, which is the case for many distributions that could represent bandwidth and delay. We formulate the maximum-flow problem in such stochastic networks as a convex optimization problem, with a polynomial (in the input) number of variables. When an additional delay constraint is imposed, we show that the problem becomes NP-hard and we propose an approximation algorithm based on convex optimization. Furthermore, we develop a fast heuristic algorithm that, with a tuning parameter, is able to balance accuracy and speed. In a simulation-based evaluation of our algorithms in terms of success ratio, flow values, and running time, our heuristic is shown to give good results in a short running time.