This paper concerns maximal flows on ℤ2 traveling from a convex set to infinity, the flows being restricted by a random capacity. For every compact convex set A, we prove that the maximal flow Φ(nA) between nA and infinity is such that Φ(nA)/n almost surely converges to the integral of a deterministic function over the boundary of A. The limit can also be interpreted as the optimum of a deterministic continuous max-flow problem. We derive some properties of the infinite cluster in supercritical Bernoulli percolation.