Stochastic thermodynamics of Brownian motion in a flowing fluid

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.