1 1 2HydrostaticEqEditedVideo-en

PROFESSOR: So if we're dealing with a material that
can't support a shear stress, then eventually,

it's going to rearrange itself to find this condition that we
called hydrostatic equilibrium.
So a good example would be if I take a material and I deform it
and let it flow for a while, you'll eventually
come to a point where, if it's a Newtonian fluid as we just
described, then it'll find a new equilibrium shape where it'll
form a material shape where it no longer has any shear
stress acting on it.
So this would be an example with a water bottle,
but even if you take something like a hand sanitizer
or a material like that, it will still also flow.
And you can see that it takes some time now
for the viscous stresses in the material to rearrange.
But eventually, it comes to the same equilibrium shape
that we deal with in this material.
So although viscosity will set the timescale at which this
happens, eventually, when the system comes to equilibrium,
there are no shear stresses acting on the material.
And so we will think about the system
as having achieved its hydrostatic equilibrium.
And so we might think about a container.
And our material will form or find a shape
where eventually there are no shear stresses.
And so the only balance of forces is given by gravity.
The body force is balanced by the normal stresses,

or the normal force per area, which is a stress acting
at the boundary.
And so hydrostatics that we're going

to be interested in the next couple of classes, going
to be thinking about, what is the distribution of forces
that do this?
But we'll essentially have a gravitational body
force acting on this.
And then somewhere distributed over this,
we'll find that there is locally normal force acting everywhere
on the bottom of the container that is supporting
the weight of the fluid.
Now, that's easy if we're dealing
with a material like this where maybe we have a rigid boundary.
It becomes less easy to think about if we're dealing with,
let's say, thinking about a balloon.
So let's say that we take a water balloon
and we fill it with water.
When it comes to equilibrium and there's no longer any flow,
it may be sitting on a surface that would look like this.
But the material inside will still
rearrange itself so that it flows
to find an equilibrium shape.
And again, there will be a normal force
per unit area acting on the material at every single point
on the boundary.
And the pressures will be larger at the bottom
and progressively smaller at the top.
So we're going to have to deal with thinking about how shapes
come to equilibrium like that.
This leads to a corollary.
When we said that the materials were either solid,
like the container that this is in,
or liquid or fluid that flows inside here
is that there's actually two different kinds of fluids.
So if we think about fluids, there
are two different classes that we have to think about.
One of them are liquids, and the second one are essentially
gases or vapor, which are clearly very different
in the way that they respond and the kind of equilibrium
shapes that they take.
So liquids-- essentially the volume
changes very little upon deformation.
And so as it arranges itself, it assumes the shape

of the container it's in.
But its volume doesn't change.

And instead, it'll come to equilibrium
with a free surface.
So just as I've drawn in either of these examples over here,

there's a headspace or a space above the material where
there is no liquid.
But this interface here is going to be characterized
by a property that is called the surface tension, which
limits the deformation or constrains
the shape of the material.
And we're going to call that surface tension sigma.
And as we'll see later in the course,
that has units of Newtons per meter.
And so that's going to be an important force.
That's a force that depends on the surface
area that's available.
But essentially, the volume of liquid in here is the same--
always the same.
So when I bring this to equilibrium,
no matter what orientation is, you
can see that eventually the material comes to equilibrium
with a free surface.
By contrast, if we deal with gases or vapor or plasma,
then it is strongly coupled--
the response of the system is strongly
coupled to the temperature and pressure of the system.
So this is a strongly coupled system to the state,
or the thermodynamic state, which is set by the pressure
and temperature.
And our fluid expands to fill the entire container
with no free surface.
So frequently, you'll hear people
talk about the thermodynamics of a strongly coupled system,
like a gas or a vapor.
And alternatively, if we're dealing with a liquid,
then it changes very little and so
people will often call this a weakly
coupled thermodynamic system.
And this will be much easier for us
to deal with because we don't have to worry about changes
in the density as a function of the external conditions
on the system.
So here's a very fluid-centric view of the world
as put together by Keith Moffatt, professor
of fluid mechanics at the Department of Applied Math
and Theoretical Physics in Cambridge
in the UK, which kind of shows the idea that fluid mechanics
is something that you might be interested
in many, many different branches of engineering.
So by background, I'm a chemical engineer.
And I'm very interested in non-Newtonian fluids,
as we just talked about.
And that's frequently discussed under the field of rheology.
And there's many, many other kinds of materials,
including blood and multi-phase materials where
chemical engineers and biological engineers
are interested in fluids.
For mechanical engineers and aerospace engineers, of course,
people are very interested in design of wings and lifting
surfaces, minimizing drag, minimizing noise.
And here, we frequently have to think about transitions
to supersonic flows.
If you're interested in geophysics or in geology,
then you're frequently interested in transport
in the ocean or pollution and smoke.
Again, it still comes down to fluid mechanics.
Even if you're more interested in mathematics and differential
equations, turbulence and thinking about how
to analyze the equations that we're going to develop-- which
are very complicated, nonlinear, partial differential equations,
there's a lot of interest and challenge
still for people who are interested in mathematics.
And then for civil engineers, people
are interested in bridge design or erosion or wave transport.
Civil and nautical engineers will find many, many interests
in fluid mechanics.
So this is just a way of saying that it doesn't matter
what branch of engineering or science
that you're interested in, there's
some challenges for you in the field of fluid mechanics.

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PROFESSOR: So if we're dealing with a material that

can't support a shear stress, then eventually,

And so the only balance of forces is given by gravity.

The body force is balanced by the normal stresses,

And so hydrostatics that we're going

And so as it arranges itself, it assumes the shape

But its volume doesn't change.

So just as I've drawn in either of these examples over here,

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