A bearing system including a foil bearing assembly positioned within the cylindrical bore of the sleeve. The foil bearing includes a top foil layer and an outer layer. The bearing system includes a driveshaft including a recess axially aligned with the top foil layer. The recess defines a cavity having a volume and includes a leading end at a first circumferential position and a trailing end at a second circumferential position. The volume of the cavity decreases from the leading end to the trailing end.