As we know diffussivity of gas is really slow (e.g. $\require{mhchem}\ce{O2}$ and $\ce{H2O}$ in the air are respectively $0.176$ and $0.282 ~\text{cm}^2/\text{s}$).
here is a example to specify how slow it is, to evaporate water filled in a $1~\text{m}$ testing tube without external heating, one requires almost $50$ years!! (Note that the vapor is saturated at the water-air interface so the energy for phase conversion is not a issue here.)
However, if one farts in a room without sensible air flow, the smell can be noticed by him within $10$ seconds. This is definitely a speed that diffusion can not achieve. Does it mean that the air flow in a 'normal' room can be around $1~\text{m/s}$, or the initial far speed from anus gives a strong momentum which I don't think it could be that strong?
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