Today we're talking about fluids - my
favourite, my area of expertise. We're
gonna do an experiment and then we're
gonna understand
why
what we saw happened using maths. We're
gonna do some magic; but explained
through maths. - (Brady: Hey duck!)
Here I have a sieve which has some holes in, so when I
pour
the water goes through the sieve, right?
So I'm not cheating. Fill up my pint
glass,
put the sieve on top, and then I'm going
to get this plate
and position it on here. And I'm going to
turn it over, something's going to
leak but hopefully not too much.
And now the magic part is I'm going to
actually lift up this, and remove the
plate, and the water will not fall out.
There we go.
That was off the plate, that doesn't
count. Somehow the water is completely
remaining
inside the glass. We've got the heavier
fluid on top, the more dense water with
air underneath;
and now if I want to make it fall
through - again show you I'm not cheating -
if I move this a little bit
it falls out.
So what was happening there was the
surface tension of the water was holding
the water in the glass and stopping it
from falling out. Let's try the colander;
there's much bigger holes, yeah. If you
compare the two there's definitely much
bigger holes in the bottom.
So you can see I'm not cheating. I need
to put this on top,
and then I turn it over. And now if I can
grab the handle of this slowly and
carefully -
this one's harder.
Oh that was off the plate! There we go,
we've got the water inside the glass
being held inside the glass despite
there being holes in the bottom of the
colander
and yet the water doesn't pass through,
it remains in this precarious position
of the heavier fluid the water above the
lighter fluid here
of the air. And just like before, if I
were to slightly move this
then it all leaks out.
(Amazing!) - Amazing, I agree. So the last
thing I've got is this sort of gauze
kind of thing. And again we can like
pour the water through. So these holes
are quite big, right? Now let's try this,
see if you can get this one to work.
My stopper on top, and my plate. There's
gonna be quite a lot of leakage I think
here.
So I can't do that- I've kind of got it
Ah,
no,
it's not gonna work. I've tried this a
few times, this one doesn't work.
I was trying to build the suspense. So
the the problem here
is the holes are too big, basically is
what's happening. So so we started with
our very fine
mesh on the sieve, so the holes there are
tiny tiny little holes. So we have the
bigger holes for the colander here;
and then
seems like, well let's keep going, we went
bigger to the gauze but this time the
holes were too big.
The surface tension wasn't able to
support
the heavier fluid being on top and the
water fell through. That to me sounds
like we need to investigate this
mathematically. - (It's actually like)
(looking through the window.) - Yeah I told
you, he comes to say hello. So
we- what we had in our experiment was
actually a two layer fluid system. This
is my interface, and this is important,
water on top and then we had air below.
So two different fluids. And we're going
to call the density at the top, rho t; so
we use rho for density and t means top,
and the bottom rho b. So obviously in our
experiment we had rho t, water is more
dense, heavier, than air. Along the
interface we've also got surface tension,
T. And surface tension- the reason I've
drawn it like this is it acts
tangentially, along the tangent of our
shape of our surface. And we want to
figure out now, can we derive a
mathematical expression explaining the
forces and what's going on either side
of this interface? And can we figure out
why we're okay for small holes but past
a certain critical hole size
it actually fell through? Because we're
talking about fluids, we've got
Navier-Stokes equations; I love Navier-Stokes.
But what's really key here is to think
about this interface, which is why I was
keen to label this. Because it's the
conditions on the interface that really
determine what's happening here. So what
we've got is a boundary condition which
we call the dynamic- BC for boundary
condition. And now this tells us that on
the interface we must have a force
balance. - (They just must always balance or)
(they must balance for the water to stay?)
They actually must balance anyway
because - and this comes from Newton's
second law.
Because Newton's second law says
F equals ma; the net force is mass times
acceleration. But our interface, it's like
infinitely thin. Like, I can draw this, but
does it have a mass?
It doesn't have a mass. You can make the
mass of it basically zero. So if this
side of Newton's second law is zero the
force, the net force. has to be zero. So in
order to not contradict
Newton's second law of motion this must
be true. As I said, dynamic boundary
condition, the forces must balance. And
we've got tension, we've got this
difference in density, and of course
we've got gravity pointing us down;
that's going to be the other key force.
And then we have a second boundary
condition which is called the kinematic
boundary condition, and this tells you
that fluid particles on the interface
must remain there. So this is saying, if I
imagine there's sort of being particles
of fluid all bunched up along here, as
time progresses this particle can't jump
out and have another one replace it.
Because the interface is
the boundary, the particle's on the top,
and then if one of them moves, if that
one moved down, your interface would move
down. That's kind of what the kinematic
boundary condition is saying. So there's
some kind of physical intuition,
hopefully it makes some kind of sense.
We're saying that if you are on the
interface then you have to stay there.
And now given these two important
conditions
along the interface, and given
Navier-Stokes, we can actually write down our
equation that's going to control the
shape of the interface because that's
going to be key in determining when the
water can and can't fall through. - (I guess)
(that's the lady duck isn't it?) - I think
yeah, because the other ones are like
green aren't they? The mallard ducks.
(There's the man.) - I told you it was a
pair! The equation that we end up:
omega squared is equal to the density
difference between our fluids times
gravity, so that's like our gravitational
force, plus surface tension multiplied by
2 pi divided by the wavelength squared;
all multiplied by 2 pi over the
wavelength and then the whole thing
divided by the sum of the densities. Now
the densities are given by our setup, T
is surface tension and G is
gravitational acceleration. So the other
things that we've introduced here are
omega
and lambda.
So omega, this is
the frequency of our waves.
So this is, along our interface where we
have this wave-like shape, this is us
saying if I fix a position
then I let time progress I'm looking at
my interface the waves will kind of move
along in time. So you can imagine- let's
take a a fairly gentle still day at the
ocean.
So there are waves along the surface of
the ocean and if you look at a certain
patch there would be a frequency, number
of waves per second.
So this is the frequency of those waves;
and here it's of the frequency the waves
along the interface. The shape of the
wave is represented by lambda which is
the wavelength. So that horizontal
distance there is equal to lambda. Lambda
is the wavelength,
the frequency in time
is omega. And the reason we're talking or
interested in waves
is
if this is going to exist,
if this actual situation with water on
the top and air on the bottom, if this is
going to be possible
and like sit there
stable, to be stable sit there nice and
happy,
you need waves along the interface. It's
not going to be flat, like these
interfaces aren't perfectly flat. There's
going to be some, even if you have to
zoom in to almost microscopic level,
there's some
wiggliness
to that interface. And we want to model
this as though it's a wave, so like um
adding together cos's and sine's and
things. - (Why does it have to be a wave? Why)
(can't it just be some amorphous random-) - So
it possibly could, but something that
we can show mathematically - which I'm not
going to go into - but you can actually
use Fourier series, which is you add
together more and more sine's and cos's
and you can create any shape that you
want. So you can literally have something
that's really spiky, or like a square; you
just add together cos's and sine's in
the right way through this theory of
Fourier series. So you can always use cos and sine
which we think of as waves. So when I say
waves, it's not perfectly like this,
but we're talking about cos and sine,
oscillations,
sort of like the trigonometric graphs.
This tells us that if waves are to exist
then this formula is true. And that would
be stable, that would mean that this
thing sort of holds.
If waves can't exist
then-
as in the water falls through;
then we can't have waves, right? Something
else is happening. It would be- you could
think of it as either some like random
motion, some chaos, some
growth of the wave - whatever's happening
it's not nice and gentle and oscillating.
(The system collapsed?) - Yes, the system
collapsed, the system breaks exactly. And
what we can do now from this equation in
fact is think about when that can happen.
This is a square number, so we want to
square root to find the frequency of our
waves, number of waves per second. If this
thing is negative then I get a frequency
that's an imaginary number, and that's a
clue that the system is breaking or
collapsing and everything- like that's
just that means the water's falling
through. So for waves, which corresponds
to stable, what we need is this thing to
all be positive. Right hand side must be
positive. Because if it is I square root,
I get the frequency, waves can exist. Yeah
they've just settled on my little
patio. - (I like it)
Right, so when is this positive?
Case one. So everything here is pretty
positive actually when you look. So we've
got
G is positive, T is positive, we've got
lambda is a wavelength - it's a
measurement so it's positive. Pi is
positive, density is positive;
everything's positive apart from this
minus sign. So in case one we can
consider the case
where
the bottom density is bigger than the
top density. So this would actually mean
I had water on the bottom, air on the top.
So this is either an actual normal way
of holding a glass of water, so you're
going to drink it, or this is like the
air and the ocean.
This is nice and stable;
and because the heavier fluid is on the
bottom,
this is always positive which is why we
always get waves. So that's why you sort
of- you know, the ocean is in general nice
and well behaved, and why you can have a
glass of water and it's never going to
jump out or anything crazy is going to
happen. It just sits there happily
because the heavier fluid is on the
bottom.
So it's always good to do a sanity check
with these things, because we know, we
know the physics, we have the intuition
as people going about our lives. You want
to check the maths actually fits with
our daily experiences. I wonder what
brought them here because there's no
like pond.
Case two is when we have the one we
really care about here, is where the
heavier fluid is on the top. So now I
look at my formula here for the
frequency.
So this,
rho t is bigger than this,
so this thing is now negative
here. So you've got something negative
plus something positive. And we only care
about the sign of this. So we don't care
about the bottom, that's always positive,
this is always positive; so it's just
this square bracket. So if the right hand
side of our equation here, if that's
going to be negative - right hand side is
negative - then that can only happen when
the square bracket is negative. Rho b
minus rho t times gravity plus tension 2
pi over lambda all squared. So when
that's negative. Since rho t is bigger
than rho b we're going to take this
across the inequality; so we can say that
T 2 pi over lambda squared is less than
G. And now it's rho t minus rho b. This is
now a positive number, top density is
bigger than the bottom, top is water. So
this is now a positive number, this is a
positive number. And then what we want to
do here actually is, T is fixed, it's the
surface tension; so for a certain set of
fluids that has a value. Obviously 2
and pi are fixed, gravity's fixed, the
densities are fixed by the fluids. So
what we can control
here is the wavelength - possibly. So we
want to rearrange this expression for
the wavelength.
So we do that we're going to multiply up,
divide through, square root. For waves to
happen and everything to be okay we need
lambda, the wavelength, less than 2 pi
times the square root of T divided by G
times the density difference; where
importantly in this case the fluid on
top is bigger than on the bottom so
we're square rooting a positive number
which makes sense. So this is it, this is
our condition.
If this holds,
if we have
a short wavelength so their shape is
quite condensed, then
this will actually hold, we get waves. The
water will stay in the glass and will
not fall out as we saw in the experiment.
So now we just want to plug in the
numbers to figure out, why did it work
for some of our holes or not for the
others? So T, the surface tension between
air and water, is 0.07
Newtons per metre. Gravity is 9.81
Newtons per kilogram. Density of the top
fluid is water, is about a thousand
kilograms per metre cubed, for it being
fresh water; and on the bottom is 1.2
kilograms per metre cubed. So they're our
numbers, we plug them in, and we get that
our wavelength has to be less than
1.7 centimetres.
So when we have our glass upside down
with the water and then air below it,
surface tension is able to hold it in
place and allow the interface to exist
with waves, provided the wavelength is
less than 1.7 centimetres. And the way we
control that
is by making our holes
less than 1.7 centimetres. - (So Tom, the)
(holes themselves are kind of physically)
(constraining the wavelength aren't they?) - Exactly,
that's exactly the way to think about it.
So when I had the whole glass it's like
10 centimetres across, so it's possible
there for a 10 centimetre wave to form.
We're not going to see it, but it's
possible. But when we have the holes they
actually restrict, because as the water
can't flow through the hole with a large
wavelength which it needs to. That's why
for the sieve they were really small
like a millimetre or two millimetres, so
definitely less than this value. Then
when we went to the colander with the
medium size, they're about pushing half a
centimetre, maybe 0.3, 0.4? So they're a bit
bigger. And then for the last one, the the
gauze; so they're almost a centimetre.
Which still seems to be less than this,
but this is just giving us an upper
bound. So this is- this is an
approximation right? We've- for instance
very at the very beginning, we assumed
that the fluid was two-dimensional
here. So I've just said it's a 2D slice
through the fluid, so that's a huge
simplification. And there's other
simplifications we've made because- we
know, we can't solve full
Navier-Stokes so we simplify everything. So
because of the approximations this won't
be perfect;
but
it sort of gives us an idea that there's-
there is a critical hole size and if we
go too big with our holes we've got no
chance.
And so it sort of gives us a nice bit of
maths to explain this apparent magic of
the of the floating water.
(I've figured it out, I figured out why)
(they're coming to your office all the)
(time!) - What is it?
(Because of that)
The rubber ducky, yeah!
It's funny.