Irreversible capture of an advecting tracer ($C_t+U{C}_x=D_{\mathrm{eff}}{C}_{xx}-\lambda C,S_t=\lambda C$)

A first-order irreversible transfer from fluid concentration $C$ to stored concentration $S$, measured per matching fluid storage volume, obeys the displayed balances. A sustained inlet can produce a stationary decaying fluid profile, but the stored amount then grows rather than becoming stationary. A finite pulse instead has $C\to 0$ and a finite final deposit. Decay, desorption or finite adsorption capacity is additional physics, not a consequence of the linear capture term alone.