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PROFESSOR: So while
I've argued that there

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are many ways in which
a room can become

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well mixed,
hopefully well enough

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to apply our well
mixed criterion

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for airborne transmission
through long range aerosols,

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there is one very important
way in which the transmission

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problem is never well mixed.

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And that is taking
into account the source

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of the particles,
which is the exhaling

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of an infected
individual which leads

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to a very high concentration
of particles leaving

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the mouth, which then
ends up being dispersed

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throughout the room.

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And there is a sort of
space and time dependence

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of that process which
we must consider

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in looking at other
possibilities of transmission

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than just the well
mixed background air.

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So in particular,
I've already indicated

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that the flows
generated by breathing

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tend to be at higher
enough Reynolds number

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to generate turbulence.

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And in particular, they generate
a turbulent jet which ends up

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taking the form of a cone.

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Not perfectly but
approximately a cone,

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which means that the
radius of the jet

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is alpha times the
position of the jet here.

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So if I write a
coordinate system,

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a cylindrical
coordinate system where

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z is the direction of the flow
and r is the radial coordinate,

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then that's the cone and
the cone angle is alpha.

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The term alpha also has
a physical interpretation

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as the air entrainment
coefficient.

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And for respiratory
jets in the air,

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this coefficient is
usually around 0.1 to 0.15.

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So that gives you basically
the opening angle of the cone.

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So I will come to
explaining and deriving

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why the jet has the
shape of a cone.

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But let's just assume that.

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And let's continue
with the assumption

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that the flow is at high
Reynolds number which

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means it's dominated by
inertia, by kinetic energy,

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and by the tendency
for the fluid

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to keep wanting to move
in a new direction.

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Now if the fluid
coming out of the mouth

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were just being spit
out at a very high rate,

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you might imagine a very
narrow kind of stream of air.

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But the reason it's widening
is that air is actually

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being entrained, is that
some of the ambient air

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is being sucked in.

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And this is making the jet
have more and more fluid in it.

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But then that fluid is
sharing the momentum.

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It's also spreading and
diffusing the particles.

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And the wider it
gets, the more it

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slows down because
now that momentum

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is being shared through the sort
of turbulent exchange going on.

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So let's do that calculation.

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So we have-- so just
remind you that we

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are in a situation of very
high Reynolds number typically.

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And we will assume
that there is roughly--

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and it's actually
a good assumption--

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roughly a constant
momentum flux.

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What that means if
I take a slice here

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and I look at, essentially,
the kinetic energy

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density or the
momentum per time that

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is crossing a slice of the
jet, that that actually

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should be conserved.

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And so if I write that
momentum flux as capital K

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is the area of the
cross section pi

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r^2, if I look at
a given position r here.

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And then I have the
momentum is the density

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of the fluid, which
is the air, times

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the velocity field,
the velocity.

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And so that's momentum.

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Momentum flux would be
momentum times velocity or rho

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v^2, which is also
kinetic energy density.

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This quantity should
be roughly constant.

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So we can now solve for
the average velocity,

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v bar, which would be square
root of K over pi rho_a times

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1/r, after we
take the square root,

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but then because
we have a cone r

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is alpha z So this
is square root of K

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over pi a 1 over alpha z.

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So we can see the velocity is
decaying like 1 over distance

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from the mouth.

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So the jet is slowing down.

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But it's still, of course,
continued to advance

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as the momentum is being shared
across a larger and larger area

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of entrained, turbulent flow.

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So we can now use this
result to figure out

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what is the rate of
progress of the front.

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So if I call this
z of the front,

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so let's say when I
first start exhaling

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it is kind of a wall
of droplets and I

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can sketch that this flow
is actually full of droplets

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that we're interested
in tracking,

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I'd like to know how those
are first leaving the mouth.

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Well, I can write
that this velocity is,

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at a given position
corresponding to the front,

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is dzf dt.

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And if this is the f, so
I apply this at the front,

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then I can put the zf on the
other side and I have zf dzf dt

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is equal to square root
of K over pi rho_a.

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Then I've got, I guess also,
an alpha or 1 over alpha.

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And then this
expression here can

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be written as 1/2 times the
derivative of zf squared.

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So I can then solve for the
position of the front and I

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find zf is equal to--

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well, let's see what I get.

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I put the 2 on the other side.

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I take a square root.

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So I get 2 over alpha to the
1/2, I get K over pi rho_a

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was a square root.

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And then I take another
square root so I get a 1/4.

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And then I get t to the 1/2.

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Because when I
integrate this equation,

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I get zf squared is all this
stuff times t, starting from t

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equals 0.

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So we find is that, initially,
the jet starts to progress

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like square root of time.

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And so this coefficient
here is some kind of--

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has the units of a diffusivity.

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So it kind of like appears that
the jet is sort of diffusing.

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You could call this
D_effective,

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sort of for the
front of the jet.

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But that doesn't last
forever because some

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point the person
closes their mouth

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and starts breathing in again,
and maybe pulls the fluid

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back a little bit.

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But not too much.

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Because it has momentum and
it keeps moving forward.

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So when somebody breathes, it
starts out looking like that.

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But then at a later time if I
draw this same person again,

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then--

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in fact, actually, let me
draw him with a closed mouth.

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Because he's just
finished, let's say,

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exhaling is getting ready
for his next breath.

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So now this blob of fluid has
kind of worked its way out

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and it's now somewhere out here.

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And this is what we call a puff.

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So if you're smoking,
for example, a cigarette

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you know that you can
create a puff of smoke

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by just releasing a
finite amount of fluid.

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And then it kind
of goes out and it

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makes some interesting
patterns and usually

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is very turbulent unless
you're very careful in trying

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to control it.

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And of course this contains
lots of these droplets

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that we're interested in.

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And so then now we
can briefly ask,

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how fast does the puff move.

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Well, you see here,
the fluid keeps

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moving because it's constantly
being given momentum.

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So if it's a steady jet
as you're breathing,

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you're pushing, pushing,
pushing and so this thing

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keeps moving on the square root
of t, it keeps entraining more.

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But as soon as you stop
giving it more momentum,

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you give it just a finite
amount of momentum,

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then the puff
actually slows down.

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It doesn't keep pushing
ahead as quickly.

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It also doesn't entrain
more air as quickly either.

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And we can do a
very simple argument

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to see what happens
to the scaling.

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So this momentum flux here
was a constant momentum flux

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in the case of a jet.

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But here in the
case of a puff, we

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could, maybe as a
very crude estimate,

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just say that the momentum
flux is something now replaced

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by some kind of constant
value that is maintained only

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for a time, tb, which is
the time of the breath,

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and then averaged over
a longer period times.

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If you think of this as kind of
like an average momentum flux,

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we've injected some momentum
flux but then we took it away.

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And so if I want to find out the
average over a period of time t

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I have to divide by t.

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So you see if I do that then,
I arrive that in the puff case

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that the position of
the front or the puff

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is a lot of these
same constants here,

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but is now scaling as t to the
1/4 because my K has a 1 over t

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and it's raised to 1/4
power, then I end up

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with zf goes like t to 1/4.

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So it slows down.

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So first it was sort
of square root of time.

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Now it's become more
[INAUDIBLE] time.

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And it's almost just
sort of sitting there,

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it's not progressing that much.

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And then a new breath comes.

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And so that's what happens next.

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And especially if one
is speaking or singing,

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then the exhaling is a much
longer and more continuous

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process than the inhaling
which is very sudden.

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So you talk and take a
breath, and you talk.

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And so there's a lot more of
this going on than this, even

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in normal speech and in normal
breathing to some extent

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as well, but especially
when one is speaking.

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And so an interesting
development

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then is that if we have a
person who is speaking, then

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we can generate what has
been called a puff train.

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So we have here is the most
recent new breath getting

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00:10:53,860 --> 00:10:59,140
exhaled and then the
previous one was here, still

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kind of floating around
and the one ahead

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of that is dispersed a bit more.

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But it hasn't progressed as far.

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And so it's a
little bit thinner.

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And if we kind of color
each breath a different way,

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then what we're left
with is something

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00:11:11,950 --> 00:11:17,680
that actually looks an awful lot
like a continuous cone again.

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00:11:17,680 --> 00:11:21,370
So this is a puff train.

208
00:11:21,370 --> 00:11:27,120
This could be, for example,
from speaking or singing.

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00:11:27,120 --> 00:11:31,950
And overall, the behavior
is quite similar to a jet

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00:11:31,950 --> 00:11:36,870
just where the K is replaced by
the time average momentum flux.

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00:11:36,870 --> 00:11:40,320
So I should mention that
this way of thinking here

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was recently introduced
and verified experimentally

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00:11:43,290 --> 00:11:55,020
in a paper by Abkarian and
Howard Stone and collaborators,

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00:11:55,020 --> 00:11:57,910
and also introduced this
notion of the scaling

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00:11:57,910 --> 00:12:00,570
and showed that actually the
puff train has a scaling which

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00:12:00,570 --> 00:12:02,310
is very similar
to the initial jet

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00:12:02,310 --> 00:12:04,460
with the square root of time.

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00:12:04,460 --> 00:12:06,420
And so that's an important
concept come to now.

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00:12:06,420 --> 00:12:08,250
Because we want to ask
now, what if somebody

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00:12:08,250 --> 00:12:11,610
is speaking or even just
breathing continuously

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00:12:11,610 --> 00:12:14,280
in a certain direction and
generating aerosol particles,

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00:12:14,280 --> 00:12:17,100
how does their concentration
evolve in this sort

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00:12:17,100 --> 00:12:21,440
of a respiratory plume or jet.