Microorganisms navigate and divide on surfaces to form multicellular structures called biofilms, the most widespread survival strategy found in the bacterial world. One common assumption is that cellular components guide the spatial architecture and arrangement of multiple species in a biofilm. However, bacteria must contend with mechanical forces generated through contact with surfaces and under fluid flow, whose contributions to colonization patterns are poorly understood. Here, we show how the balance between motility and flow promotes the emergence of morphological patterns in Caulobacter crescentus biofilms. By modeling transport of single cells by flow and Brownian-like swimming, we show that the emergence of these patterns is guided by an effective Peclet number. By analogy with transport phenomena we show that, counter-intuitively, fluid flow represses mixing of distinct clonal lineages, thereby affecting the interaction landscapes between biofilm-dwelling bacteria. This demonstrates that hydrodynamics influence species interaction and evolution within surface-associated communities.