We present Herschel/PACS, MMT/Hectospec and XMM-Newton observations of Abell 1835, one of the brightest X-ray clusters on the sky, and the host of a strong cool core. Even though Abell 1835 has a prototypically relaxed X-ray morphology and no signs of ongoing merger activity in strong- and weak-lensing mass maps, it has a complex velocity distribution, suggesting that it is still accreting significant amounts of mass in the form of smaller satellite systems. Indeed, we find strong dynamical segregation of star-forming dusty galaxies from the optically selected cluster population. Most Herschel sources are found close to the virial radius of the cluster, and almost a third appear to be embedded within a filament feeding the cluster from the SW. We find that the most luminous infrared galaxies are likely involved in galaxy-galaxy interactions that may have triggered the current phase of star formation.