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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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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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in different positions relative
to being mouth to mouth
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with the infected person.