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STUDENT: Have you ever
wondered how rivers flow,
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or how fluid travels
inside pipes?
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Watch this video and
you will learn how.
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If you apply Newton's
second law to fluid motion,
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and at the same time, consider
the viscosity of the fluid,
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you will get the equations you
need to describe fluid flow.
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The equations that
you get this way
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are called the
Navier-Stokes equations.
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This is the general form of
the Navier-Stokes equations.
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Solving this equation in
this form is not easy.
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Interestingly,
mathematicians have not yet
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proven that solutions exist
to the Navier-Stokes equations
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in three dimensions.
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To make our life easy,
let's consider motion
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for fluid in one dimension.
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The Navier-Stokes
equations then become this.
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As you can see, I've broken
down the previous problem
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in three dimensions into just
one dimension, the x dimension.
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The main body force that
is Fx acting on a fluid
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will be the force
due to gravity.
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If the fluid is flowing
at an angle, theta,
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to the horizontal,
then the component
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of gravity in the
direction of fluid motion
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is going to be g sine theta.
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We now have an equation
that looks much
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easier than a general form.
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As you can see, this
is just a restatement
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of Newton's second law, with
the viscosity term included.
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On the left side
of this equation,
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you get the net force, acting
on a unit volume of the fluid.
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The three terms on the
right side of the equation,
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which are governed by the
viscosity, the pressure
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gradient, and the force
due to gravity together,
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give the net force
acting on a unit volume.
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As you can see from
this term here,
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this is a second order
differential equation.
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To solve this, we will have to
impose two boundary conditions.
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For the river problem, we
will consider the velocity
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at the bottom of
the river to be 0.
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This is true because the
river bed imposes resistance
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to the fluid layer just above
it, preventing its motion.
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The velocity of the
river reaches a maximum
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towards the top of the river.
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So we get the condition that
dVx over dy equals 0 at the top.
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For the pipe problem,
using the same argument,
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the velocity at the
rim of the pipe is 0.
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And at the center of
the pipe, dVx over dr
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equals 0, where
r in this case is
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the distance from the center
of the pipe to the point we
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are looking at.
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Now that we have our
differential equation
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in one dimension and the two
boundary conditions needed
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to solve it, there is nothing
left to do but to solve.
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We can do this
using Mathematica.
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I was able to solve this
equation using Mathematica.
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In the pipe, the velocity at
a distance, r, from the center
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is given by this formula.
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And in the river, the velocity
the height, y, in the riverbed
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is given by this formula.
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If you want to know the average
velocity inside the pipe
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or inside the river,
all you have to do
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is add up each individual
velocity at all possible radii,
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or heights, you can think of.
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And then divide that sum by the
area that you added it over.
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It's just integrating
and dividing.
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Mathematica gave
me this result. You
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can see that the average
velocity of the fluid in a pipe
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and in a river increases
if you increase
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the density of the
fluid, or if you increase
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the inclination of the pipe
or the slope of the river
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to the horizontal.
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Also, if you increase the
negative pressure gradient,
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the velocity increases.
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A lower viscosity
of the fluid will
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increases the average velocity.
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This is all intuitive,
but wouldn't it
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be cool if we could
actually model
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how the velocity profile
looks like inside a pipe
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and inside a river?
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I'll do this using Mathematica.
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I will start with
the river problem.
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First, I will create
a grid of coordinates.
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For each coordinate on this
grid, I will assign a vector.
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This vector will correspond
to the velocity at that point.
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We know how this velocity
looks like because we
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solved the differential
equation for velocity.
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Once we have vectors for
each of these points,
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you'll get a picture
that looks like this.
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As you can see, at the bottom
of the river, where y equals 0,
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you have 0 velocity.
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As you increase y, the
velocity increases.
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Let's look at just one
slice of the previous image.
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We'll see what happens
when you change parameters.
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When you change mu, the
average velocity decreases.
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Think of these dark
blue dots as fixed.
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These are the reference frame.
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With higher mu, you
have lower velocities.
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You can do the same
for other parameters.
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I used a similar method
to model the velocity
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profile inside a pipe.
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I first created a
circular grid, and then
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I assigned each point
on the grid a vector.
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As you can see, on the rim of
the pipe, you have 0 velocity.
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And towards the center,
this dark blue line
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here, you have the
highest velocity.
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[MUSIC PLAYING]
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I hope you now
understand how fluids
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flow in pipes and
rivers, and that you
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will be able to apply
Navier-Stokes equations
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to solve similar problems.
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Thank you for watching.
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[MUSIC PLAYING]