Thinking about breathing out

We breathe in and out all the time, without thinking about at all. But now is the time to think about how we breathe in and out. SARS-CoV-2 leaves an infected person in droplets in their breath, and the evidence is strong that a majority of infections start with someone breathing in the virus.

The easy way to get started here is to see what we breathe out. Now air is transparent so that is not normally possible, but Abkarian and coworkers have basically a fog machine for making little droplets that they can then see, so they can see the patterns your breath makes in surrounding air. And these patterns are cool, go here, watch their movie, and see what your breath does.

As I said up top, I have been breathing out for over 49 years, and I did not know my breath was that cool. But there are more serious points here. The field of view in the movie is about a metre. In effect, social distancing by one or two metres is basically staying out of the danger zone associated with a potentially infected person’s breath.

So how far does your breath travel? There is no simple answer to that question, as Abkarian and co-workers say, the speeds and so distances travelled vary as power laws of time — which means there is no characteristic distance associated with your breath, except that associated with when the speed drops to whatever is the background speed in the room.

The maths needed to see how our breath spreads out is simplest for a short sharp puff of breath. Say you breathe out a puff of air with a momentum

puff momentum =U_B\rho V_B

with U_B = speed of puff, \rho = air mass density and V_B = puff volume

Now we assume that as the puff of air expands and moves away from your mouth, its momentum remains constant. This means as its volume increases, its speed decreases as one over this expanding volume — the density of the air is about constant. If we assume it expands about equally in all three directions (up/down, left/right and forward/back), when it has travelled a distance L, the puff will have expanded to a volume of about the cube of this distance. Then

puff momentum =U_B\rho V_B = U_L\rho L^3

with U_L = speed of puff when it has moved/expanded to a distance L

Rearranging, we get

U_L = \frac{U_BV_B}{L^3} s

So the speed of a puff of air decreases as one over the cube of the distance it has travelled. Let’s put some numbers in. If you breathe out quite forcefully, the speed of your breath is about 1 m/s. If you breathe out a puff of one litre of air, then when it has gone about 1 m, it has expanded by a factor of a thousand, from one litre to a thousand litres (= 1 m3), so its speed has dropped by a factor of a thousand, to 1 mm/s. I think this speed is about as low as your skin can detect. If I breathe out quite forcefully I can feel the breath on a my hand at the end of my extended arm about 1 m away.

But if I breathe out more normally, with perhaps a breath speed of 0.1 m/s, I can feel the breath on my hand about 20 cm away but not 1 m, where it is moving at only about 0.1 mm/s.

Anyway, I am not sure why it took a viral pandemic for me to think about breathing out, but I am glad I did, and in this crazy year we should any small bit of joy where ever we can find it.

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