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PROFESSOR: So as
a technical aside,

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let me go through and sketch
the derivation of the structure

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of a turbulent jet, in
particular the conical shape

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that we have when the
flow is turbulent.

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So in order to study
the mean flow profile

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we began with the
Navier-Stokes equations

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which describe the momentum
conservation and mass

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conservation or continuity of
an incomprehensible so-called

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Newtonian fluid.

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So this is a complicated
set of equations.

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In particular, we have
this nonlinear term here,

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which is the inertial term.

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And we've already said that
we had our high Reynolds number

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and turbulence results
because the inertia is

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very strong compared to the
viscous term which is here.

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So that's the divergence
of the viscous

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or the viscous
forces on the fluid.

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So these two terms we
know are important.

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They have to balance and the
inertia is particularly strong

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and it is what leads to the very
complicated flows that we see.

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So you can solve these equations
numerically on a computer

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and generate simulations that
look a lot like experiments

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on turbulent jets.

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What I'd like to do here is
just to derive by simple scaling

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arguments what sort of the
structure of the solutions

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could look like.

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So these two terms,
as we just indicated,

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are the ones that are most
likely to balance in the time

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average flow.

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So let's consider
a time averaged

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steady flow which has a
velocity component of v_z

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that depends on r and z.

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So it's basically
something like this,

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which is basically
expanding, but has

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a certain sort of localization
of the flow in the middle.

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And it's smooth because
we're averaging over all

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the complexity of the jets.

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So the jet looks something like
this with all kinds of vortices

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and eddies that
are getting bigger

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as it goes as you're
entraining more

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and more air from the outside.

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So we're going to look
at the time average flow.

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And we're also going
to, importantly,

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assume that we have
an eddy viscosity.

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So the kinematic viscosity,
nu in the equations

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as I've written them here,
represents the diffusion

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momentum.

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If a parcel of fluid is moving
with a certain momentum,

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it has a chance of
passing that momentum

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to the neighboring fluid
and moving it along with it.

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And that is accomplished
through viscous stresses.

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So the eddy viscosity
basically assumes

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that that diffusion
process from momentum

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happens at the scale of the
largest eddy in the flow.

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And so we've talked about the
assumption of eddy diffusivity.

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But for eddy viscosity,
what I'll write

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is the eddy viscosity
is a typical velocity

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which is v_z times a length
scale which is delta.

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So what I'm saying
here with this

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is that if I go out to a certain
position z and ask myself,

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what is the sort of width
of the jet at site z,

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then there's all kinds of
eddies but the largest eddy

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is kind of at that scale.

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And so if I write
down an eddy viscosity

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it's going to be these
sort of average velocity

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there are times that scale.

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So that's going to
be the eddy velocity.

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And I'm going to replace--

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so when I do my
time averaging, I'm

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going to replace the
microscopic viscosity

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of the fluid,
kinematic viscosity,

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with the eddy viscosity.

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So that's an important
modification.

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And so if I do that.

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If I do this time averaging
and look at the eddy viscosity,

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then I take these two
terms and balance them,

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I'm going to get v_z bar,
so that's my average v_z.

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I'm looking at the z
component of momentum

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here of that first
Navier-Stokes equation.

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And I get v_z dot derivative of
v_z with respect to z plus v_r.

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And there's also an r
component of velocity.

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So there is also
some velocity in fact

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which is coming
in from the sides.

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But I'm just going be interested
in this term here v_z dr.

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And I'm going to balance this
against the eddy viscosity,

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the eddy--

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or nu eddy I should say, sorry,
nu eddy is eddy viscosity--

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times and then the Laplacian
in is 1 r d/dr r dv_z dr. So

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that's just the Laplacian in
its cylindrical coordinates.

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And now I'm going to make
the assumption that this ve

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scales as v bar z times delta.

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And so now I'm going
to do a scaling

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analysis on this equation.

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And so what we see is we
have v_z over z times--

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and then at least
for that first so--

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I should say these
two terms will

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be of comparable size
because of incompressibility--

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the second equation.

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I won't go through
the details of that.

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And we'll just do a
scaling argument

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balancing these two terms.

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So if I look at v_z
divided by z times v_z

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so that's a scaling
of those two terms,

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I can balance that
against v_z delta times--

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and the scale for r is delta.

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So I have 1/delta
for the 1/r,

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1/delta for
the derivative times

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delta * (1/delta) * v_z.

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So there's a lot there.

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But notice the v_z's all cancel.

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And we're left with a
bunch of deltas here.

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And how many, because
of the eddy viscosity,

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we are left with--

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all of this is
just 1 over delta.

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This is 1 over z.

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And so we find here
the delta scales as z.

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So in other words, we
have a conical shape.

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So the boundary
of their thickness

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is a constant times z.

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And what we write is that
delta is equal to alpha z

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specifically.

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And we define the turbulent
entrainment coefficient alpha

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that way.

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And then once we've
done that, we've

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already shown that
from the momentum flux

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that v_z scales as the square
root of k/rho_air

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times 1 over delta.

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So this basically now gives
me the scaling of the problem.

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In fact, there is a
similarity solution

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for the shape this profile
that one could solve for.

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And it has the form
that, for example, the v_z

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is square root of--

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because delta is
proportional to z.

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So it's the square
root of k over rho

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a z times some function
of r over alpha z.

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And then there's a
similar expression

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for the other
velocity component.

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And the function F looks very
much as I've sketched here.

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It's essentially a Gaussian
type profile or a bell curve

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that kind of localizes the
velocity across this distance

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delta.

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The second thing that
we're interested in

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is the mean concentration.

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And that would be a
concentration of, let's say,

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virions contained in
infectious aerosol droplets.

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So there's a mean
concentration profile

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in the jet assuming
that we're injecting

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a fluid of a constant
concentration

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at the source of the jet.

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So again, we can do some
scaling arguments here.

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So if we ask ourselves,
what is the mass flow

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rate through a slice or actually
the volumetric flow rate,

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use me, that is what is called a
Q and it'll just be an average.

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This will be the
average velocity

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times the cross-sectional
area at a given position.

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So this is scaling like--

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so area scaled is
like delta squared.

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And then the velocity scales
in this way is 1 over delta.

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So this ends up scaling as k
over rho a times just delta.

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So the volumetric flow
rate is increasing

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with r and that's a sign that we
are actually in training fluid

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as I indicated.

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This is not just the
fluid we're injecting

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but it's moving forward and
it's sucking more fluid in.

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And all that fluid
is kind of becoming

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part of the turbulent
jet as it grows.

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Now if we ask-- so this is
our flow rate, volumetric flow

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rate, but we can
also ask ourselves,

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what is the flux
of concentration

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of virions per unit volume.

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Well, that would be the
average concentration

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times the average flow
rate because flow rate

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is volume per time
and concentration

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is number per volume.

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So this is a total
number per time.

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And this we will assume
should be a constant

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because as you can
see from this picture,

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if we're injecting a bunch
of concentration of let's say

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droplets here, they will spread
out in the turbulent flow

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but they don't really have
a good mechanism to get out

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of the turbulent flow.

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The turbulent flow is
sucking fluid into the plume

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and so the particles
are just kind of well

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mixed in that plume and we
could assume they have a roughly

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constant concentration.

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And so, if that's the case,
and in fact, this constant

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would be lambda Q if we're
thinking of, for example,

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infect--

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c is the concentration
infection quanta.

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Then lambda Q is the rate of
admission of infection quanta

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from the mouth.

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We've already talked
about that quantity.

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And this is now telling me how
the concentration infection

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quanta decays with time.

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And so we find if
we substitute now

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is that the concentration
infection quanta at a position

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z scales as so I have
to divide by Q so I

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get the inverse of this.

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So I get square root
of rho a over k.

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And then I have
lambda Q over alpha z.

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So this tells me that if I
plot as a function of distance

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from the mouth in the
direction of the jet,

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the concentration
of infection quanta

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that are carried by virions
in aerosol droplets,

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then somewhere here I have,
let's say, at z equals 0,

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is the mouth where I'm exhaling.

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And the concentration
there is actually cQ.

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In fact that is
something we've talked

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about before, which is that's
the key disease parameter--

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the concentration of infection
quanta in the exhaled breath

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00:11:48,140 --> 00:11:49,430
of an infected person.

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So we know at the mouth,
that's what we start with.

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And what the turbulent
theory is telling us

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is how that concentration
is decaying with time

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and is decaying like 1 over z.

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And so that tells us sort of
our relative risk of infection

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00:12:03,660 --> 00:12:07,350
in different positions relative
to being mouth to mouth

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00:12:07,350 --> 00:12:09,620
with the infected person.