PROFESSOR: So while I've argued that there
are many ways in which a room can become
well mixed, hopefully well enough
to apply our well mixed criterion
for airborne transmission through long range aerosols,
there is one very important way in which the transmission
problem is never well mixed.
And that is taking into account the source
of the particles, which is the exhaling
of an infected individual which leads
to a very high concentration of particles leaving
the mouth, which then ends up being dispersed
throughout the room.
And there is a sort of space and time dependence
of that process which we must consider
in looking at other possibilities of transmission
than just the well mixed background air.
So in particular, I've already indicated
that the flows generated by breathing
tend to be at higher enough Reynolds number
to generate turbulence.
And in particular, they generate a turbulent jet which ends up
taking the form of a cone.
Not perfectly but approximately a cone,
which means that the radius of the jet
is alpha times the position of the jet here.
So if I write a coordinate system,
a cylindrical coordinate system where
z is the direction of the flow and r is the radial coordinate,
then that's the cone and the cone angle is alpha.
The term alpha also has a physical interpretation
as the air entrainment coefficient.
And for respiratory jets in the air,
this coefficient is usually around 0.1 to 0.15.
So that gives you basically the opening angle of the cone.
So I will come to explaining and deriving
why the jet has the shape of a cone.
But let's just assume that.
And let's continue with the assumption
that the flow is at high Reynolds number which
means it's dominated by inertia, by kinetic energy,
and by the tendency for the fluid
to keep wanting to move in a new direction.
Now if the fluid coming out of the mouth
were just being spit out at a very high rate,
you might imagine a very narrow kind of stream of air.
But the reason it's widening is that air is actually
being entrained, is that some of the ambient air
is being sucked in.
And this is making the jet have more and more fluid in it.
But then that fluid is sharing the momentum.
It's also spreading and diffusing the particles.
And the wider it gets, the more it
slows down because now that momentum
is being shared through the sort of turbulent exchange going on.
So let's do that calculation.
So we have-- so just remind you that we
are in a situation of very high Reynolds number typically.
And we will assume that there is roughly--
and it's actually a good assumption--
roughly a constant momentum flux.
What that means if I take a slice here
and I look at, essentially, the kinetic energy
density or the momentum per time that
is crossing a slice of the jet, that that actually
should be conserved.
And so if I write that momentum flux as capital K
is the area of the cross section pi
r^2, if I look at a given position r here.
And then I have the momentum is the density
of the fluid, which is the air, times
the velocity field, the velocity.
And so that's momentum.
Momentum flux would be momentum times velocity or rho
v^2, which is also kinetic energy density.
This quantity should be roughly constant.
So we can now solve for the average velocity,
v bar, which would be square root of K over pi rho_a times
1/r, after we take the square root,
but then because we have a cone r
is alpha z So this is square root of K
over pi a 1 over alpha z.
So we can see the velocity is decaying like 1 over distance
from the mouth.
So the jet is slowing down.
But it's still, of course, continued to advance
as the momentum is being shared across a larger and larger area
of entrained, turbulent flow.
So we can now use this result to figure out
what is the rate of progress of the front.
So if I call this z of the front,
so let's say when I first start exhaling
it is kind of a wall of droplets and I
can sketch that this flow is actually full of droplets
that we're interested in tracking,
I'd like to know how those are first leaving the mouth.
Well, I can write that this velocity is,
at a given position corresponding to the front,
is dzf dt.
And if this is the f, so I apply this at the front,
then I can put the zf on the other side and I have zf dzf dt
is equal to square root of K over pi rho_a.
Then I've got, I guess also, an alpha or 1 over alpha.
And then this expression here can
be written as 1/2 times the derivative of zf squared.
So I can then solve for the position of the front and I
find zf is equal to--
well, let's see what I get.
I put the 2 on the other side.
I take a square root.
So I get 2 over alpha to the 1/2, I get K over pi rho_a
was a square root.
And then I take another square root so I get a 1/4.
And then I get t to the 1/2.
Because when I integrate this equation,
I get zf squared is all this stuff times t, starting from t
equals 0.
So we find is that, initially, the jet starts to progress
like square root of time.
And so this coefficient here is some kind of--
has the units of a diffusivity.
So it kind of like appears that the jet is sort of diffusing.
You could call this D_effective,
sort of for the front of the jet.
But that doesn't last forever because some
point the person closes their mouth
and starts breathing in again, and maybe pulls the fluid
back a little bit.
But not too much.
Because it has momentum and it keeps moving forward.
So when somebody breathes, it starts out looking like that.
But then at a later time if I draw this same person again,
then--
in fact, actually, let me draw him with a closed mouth.
Because he's just finished, let's say,
exhaling is getting ready for his next breath.
So now this blob of fluid has kind of worked its way out
and it's now somewhere out here.
And this is what we call a puff.
So if you're smoking, for example, a cigarette
you know that you can create a puff of smoke
by just releasing a finite amount of fluid.
And then it kind of goes out and it
makes some interesting patterns and usually
is very turbulent unless you're very careful in trying
to control it.
And of course this contains lots of these droplets
that we're interested in.
And so then now we can briefly ask,
how fast does the puff move.
Well, you see here, the fluid keeps
moving because it's constantly being given momentum.
So if it's a steady jet as you're breathing,
you're pushing, pushing, pushing and so this thing
keeps moving on the square root of t, it keeps entraining more.
But as soon as you stop giving it more momentum,
you give it just a finite amount of momentum,
then the puff actually slows down.
It doesn't keep pushing ahead as quickly.
It also doesn't entrain more air as quickly either.
And we can do a very simple argument
to see what happens to the scaling.
So this momentum flux here was a constant momentum flux
in the case of a jet.
But here in the case of a puff, we
could, maybe as a very crude estimate,
just say that the momentum flux is something now replaced
by some kind of constant value that is maintained only
for a time, tb, which is the time of the breath,
and then averaged over a longer period times.
If you think of this as kind of like an average momentum flux,
we've injected some momentum flux but then we took it away.
And so if I want to find out the average over a period of time t
I have to divide by t.
So you see if I do that then, I arrive that in the puff case
that the position of the front or the puff
is a lot of these same constants here,
but is now scaling as t to the 1/4 because my K has a 1 over t
and it's raised to 1/4 power, then I end up
with zf goes like t to 1/4.
So it slows down.
So first it was sort of square root of time.
Now it's become more [INAUDIBLE] time.
And it's almost just sort of sitting there,
it's not progressing that much.
And then a new breath comes.
And so that's what happens next.
And especially if one is speaking or singing,
then the exhaling is a much longer and more continuous
process than the inhaling which is very sudden.
So you talk and take a breath, and you talk.
And so there's a lot more of this going on than this, even
in normal speech and in normal breathing to some extent
as well, but especially when one is speaking.
And so an interesting development
then is that if we have a person who is speaking, then
we can generate what has been called a puff train.
So we have here is the most recent new breath getting
exhaled and then the previous one was here, still
kind of floating around and the one ahead
of that is dispersed a bit more.
But it hasn't progressed as far.
And so it's a little bit thinner.
And if we kind of color each breath a different way,
then what we're left with is something
that actually looks an awful lot like a continuous cone again.
So this is a puff train.
This could be, for example, from speaking or singing.
And overall, the behavior is quite similar to a jet
just where the K is replaced by the time average momentum flux.
So I should mention that this way of thinking here
was recently introduced and verified experimentally
in a paper by Abkarian and Howard Stone and collaborators,
and also introduced this notion of the scaling
and showed that actually the puff train has a scaling which
is very similar to the initial jet
with the square root of time.
And so that's an important concept come to now.
Because we want to ask now, what if somebody
is speaking or even just breathing continuously
in a certain direction and generating aerosol particles,
how does their concentration evolve in this sort
of a respiratory plume or jet.