Question: Do people smell their own farts before others do?
Short answer: Usually.
Long answer: This question comes to us from a reader who asked about the veracity of the old adage, “Whoever smelt it, dealt it.” The SIDI (Smelt It/Dealt It) statement effectively claims that the first person to smell a fart is always the person who farted.
Knowledge of one’s own fart smell is a general instance of the philosophical concern with self-awareness, first expressed by the Greeks in the 5th century BCE. This ancient concern with self-knowledge likely included farts, as suggested by this mosaic from a Roman convent, which features the Greek phrase “Know thyself” underneath a reclining skeleton.
Whether the skeleton is farting is still a matter of debate among historians.
In any case, the SIDI claim conforms to the intuition that people should be in a good position to smell their own farts, simply by proximity to the source. At the same time, this intuition is rather in contradiction to another commonly held belief, namely that farts exit the anus with some implausible velocity, usually on the order of 10 ft/s. In that case, one would expect the fart smell to be long gone before the farter could smell it.
Now, we know that the second belief is wrong. In fact, the human body has virtually no ability to propel farts more than a few inches, and even then, most of what is propelled is not the least bit smelly. As a result, the endogenous speed of fart smell is extremely slow.
This means that the invasion of fart smell into the surrounding environment is almost entirely at the mercy of air flow in the vicinity of the farter. In indoor environments, this air flow typically has a velocity of around 0.1 m/s, which is about 30 times faster than the rate of diffusion of fart smell. Outdoors, air flow becomes even more important, because wind velocity is often in excess of 1 m/s.
With these facts, we can derive a general formulation of the conditions under which the SIDI proposition holds. One simply has to consider the following geometry:
In the schematic above, θ is the angle between the vectors describing the direction of air flow and the anonasal axis of the farter, and xan is the corresponding distance. The quantity xp is the distance to the nearest person, and Φ is angle between the vectors describing the direction of airflow and the direction between the two people. Let’s call the speed of air flow va and the speed of isotropic fart diffusion vf.
It follows that the person who dealt the fart will smell it first, if and only if:
Here the cosine captures the projection of the air flow vector onto the vector connecting the release point of the fart to the nose of the receiver. The lefthand side of the expression is then the time it will take for the fart to reach the nose of the other person, and the righthand side is the time to reach the nose of the farter. Flatologists refer to this expression as the Cohen-Stephen Inequality.
Rearranging terms makes it clear that the critical factor is the ratio of the two distances xp and xan:
We measured the anonasal distance and angle for an average observer, our intern Cletus. While Cletus was in a sitting position, we obtained values of xan = 0.59 m and θ = 82o. If we assume a typical value of va = 0.1 m/s, we can simulate the conditions required for the SIDI proposal to hold:
In these figures, we have placed the farter at the origin, and simulated conditions for an observer located at relative different positions (xp, Φ). The lefthand column indicates the typical results for indoor conditions, assuming air flow is parallel to the ground. The righthand column shows the same results for typical outdoor conditions.
We see that SIDI can be violated for observers downwind from the farter, especially when the farter is standing. Observers who are upwind from the farter will almost never be the first to smell the fart.
It should be noted that these conditions are almost optimal for violations of the SIDI conjecture, since they place the axis of air flow nearly orthogonal to the anonasal axis of the farter. As a result, θ approaches 90o, so that the cosθ tends toward zero, inflating the righthand side of the Cohen-Stephen Inequality. Under most other conditions, SIDI will hold, and the farter will be the first to experience the smell.