  We study stochastic thermodynamics of over-damped Brownian motion in a
flowing fluid. Unlike some previous works, we treat the effects of the flow
field as a non-conservational driving force acting on the Brownian particle.
This allows us to apply the theoretical formalism developed in a recent work
for general non-conservative Langevin dynamics. We define heat and work both at
the trajectory level and at the ensemble level, and prove the second law of
thermodynamics explicitly. The entropy production (EP) is decomposed into a
housekeeping part and an excess part, both of which are non-negative at the
ensemble level. Fluctuation theorems are derived for the housekeeping work, the
excess work, and the total work, which are further verified using numerical
simulations. A comparison between our theory and an earlier theory by Speck et.
al. is also carried out.
