Green’s Dictionary of Slang

1. to suffer a disaster.

1876		‘Dan de Quille’ Big Bonanza (1947) 284: Afferd – don’t git married! [...] If yer git married yer gone up the flume – busted out.
1911		Out West Oct. 241: ‘Up the flume’ was handed down to us by the forty-niners, as was ‘petered out;’ ‘up Salt Creek,’ a synonymous expression, defies research.

2. to be exhausted, to be worn out, to be dead.

1872		‘Mark Twain’ Innocents at Home 333: One of the boys has gone up the flume [...] throwed up the sponge [...] kicked the bucket [...] Why pard, he’s dead!
1876		‘The Days of ’49’ in Lingenfelter et al. Songs of the Amer. West (1968) 559: Poor old Jess [...] To death did at last resign, And in his bloom he went up the flume / In the days of ’49.
1911		(ref. to 1849) J. Lomax Cowboy Songs 9: [as cit. 1876].
a.1968		(ref. to 1849) ‘The Days of Forty-Nine’ in G. Logsdon Whorehouse Bells Were Ringing (1995) 180: He raised a hell of a whine, / And in his bloom went up the flume / In the days of Forty-nine.