A rotatable processing bowl assembly adapted for mounting in a centrifuge includes an umbilical tubing system formed of a plurality of individual flexible tubes each of which communicates with the processing bowl. Each tube has an exterior wall and includes a first portion extending adjacently from the bowl and a second portion which is spaced from the bowl. The exterior walls of the tubes are bonded together along their first portions so that the joined exterior walls collectively define the outer periphery of the tubing system along the first portions of the tubes. Preferably, the tubes are also wound to form a predetermined helical pattern along their first portions. This arrangement enhances the tubing system's durability and resistance to fatigue along the bonded together first portions, particularly at higher rotational speeds, and permits direct, facile connection of the free and unattached second portions with external apparatus.