Hart, 20, was called up from Lehigh Valley of the American Hockey League on Monday.
The Flyers will be the first NHL team to use six starting goalies in the first 35 games of a season. They're the 14th team to use six starters at any point in a season, the first since the Buffalo Sabres and Edmonton Oilers in 2013-14.
Scott Gordon will work his first game as coach for the Flyers, who have lost four straight (0-3-1). He was promoted from Lehigh Valley on Monday, replacing Dave Hakstol.
The Red Wings have lost three straight games (0-2-1) and are 2-4-2 in their past eight.