Introduction to Viscosity Video Lecture | Fluid Mechanics for Civil Engineering - Civil Engineering (CE)

OK, one more property I would like to spend a
little bit of time talking about is called viscosity.
And that's because it's going to be important.
The property itself is important. But also its
implications on how we analyze fluid flow are
very important. And we're going to see
those throughout the course. So it's worthwhile
to spend a little bit of time on it. I think we all
kind of have
the basic idea about what visosity is in terms of
this idea of resistance to flow.
So for example, honey, we'd say has a high
viscosity because it flows very slowly when
we try to pour it out of a container. Whereas
water pours very quickly.
So we can say that has a lower viscosity.
We can maybe think about that a little more
precisely for the purposes of this course.
So I'm going to draw, here, a simple geometry.
So these are two plates.
Two flat plates that are separated by some
distance, H. The bottom plate is stationary.
And the top plate is free to move.
And in between these plates is some fluid.
You can imagine whatever you want. Water is a
good picture to have in your mind.
And so the top plate is free to move. And it's
moving at some velocity, V. And I've drawn the
coordinate system such that the
forward direction is x, and y is positive from the
bottom plate upward.
So the x component of velocity v,x is capital V.
And so the question is: What is happening to
the fluid in between these plates?
I've drawn some idea of what might be
happening.
But we can think about that and and see if that
makes sense. So the first step is to make some
observations.
So the first observation is that the fluid at the
two boundaries, the upper and lower plates,
moves at the same velocity as the boundaries.
So for example, at the lower plate which is
stationary, the velocity of that plate is zero. So it
makes sense that the velocity of the fluid
immediately adjacent to the plate would also be
zero.
Similarly at the top. This plate is being pulled so
that it's moving in the x direction with a velocity
V.
So the fluid that's immediately in contact with
that plate is probably also moving with that
same velocity.
And so this is kind of intitutive. But this is what
we call the no slip condition.
It seems intuitive. But there's details about
special kinds of fluids, or special kinds of flows,
where this boundary condition may not be 100%
valid.
I think there's even some controversy
about that even today but for the purposes of
this course
we can assume the no-slip boundary condition
is valid. So at these two boundaries the fluid is
moving at the same
velocity as the boundary. So the atoms and
molecules that are right adjacent to the
boundary
are going to be moving at the same velocity of
the boundary.
OK, so then the next observation
we can make is that, because we have these
two boundary conditions, zero and V.
The fluid velocity inside this gap has to change
from zero at the bottom to V at the top.
So there has to be some change in velocity
to go between between these two boundary
conditions.
I haven't said what the characteristic of the
change is, but it has to change from zero V.
Another observation we can make is that, for
many fluids, if this gap, H, is small
then this variation in velocity can be
approximated as a linear variation.
So, in other words, the velocity increases
linearly as we go up in y from the bottom to the
top plate.
And we can write an equation for this. If the
velocity of the top plate is V at y = H.
And the velocity is zero at y = 0.
Then the equation for this velocity profile v,x(y)
is y * (V / H). So you can see this satisfies the
boundary conditions. At y = 0, the velocity is
zero.
And at y = H, these H's cancel so the velocity
equals V.
So this equation expresses a linear variation in
velocity, a linear increase as we go from the
bottom plate to the top plate.
So this is an example of what we call laminar
flow.
So in this kind of flow, these fluid paths are
straight lines.
There is no mixing and there is no tangential
component to the motion. And so
one way to think about it is like a deck of cards.
If you slide a deck of cards, each card moves
forward at a velocity that maybe looks
like this. The top card will move with the velocity
of your hand and the bottom card will be stable.
So this is what we call lamianr flow. The
opposite case is what we call turbulent flow.
Where there is less order
or more disorder to the motion. You have not
only motion the forward direction,
but you have motion in the tangential direction,
lateral to the flow.
And this motion has complex characteristics
that are hard to describe.
So this is an example of laminar flow that we
have shown here.
OK, so given this simple description of this flow
field that we've shown here.
How exactly does one layer that's moving in this
laminar fllow, how does it pull the others with it?
What is the relationship between the motion of
one layer of fluid and the motion of the other
layers?
OK, so I'll go the next slide here, and redraw
the picture that I had on the previous slide so we
can refer to it again. We have two plates
separated by a distance H.
And we said if H is small then we can
approximate this velocity profile as a linear one,
going from zero at the stationary bottom plate
to V at the top plate that's free to move. And we
want to know,
or try to figure out, how how these different
layers can interact in a way that allows one
layer to pull another layer with it.
OK, so let's make another observation here.
So this observation is an important one. And this
observation states that the sheer stress
in the fluid
is linearly related to velocity gradients.
And a velocity gradient. Remember what a
gradient is. It's a slope.
So the velocity gradient is the slope of this
velocity profile.
We can write that mathematically in the form of
an equation. And I'll do that up here .
And then talk about it. So this equation says the
shear stress.
And I have written it as tau,yx. And I'll explain
that in a minute.
The shear stress is equal to the velocity
gradient, dv,x/dy.
So that's the slope of this line basically.
Because it's a linear function.
But if it's not linear, it is the local slope of this
velocity profile.
Times some constant, eta, the Greek letter eta.
And this constant is called the viscosity
coefficient.
And this equation is called Newton's law of
viscosity. So it's one of those relationships, like
Hooke's law for a spring, F = k * x.
Force is related to the deformation here, for a
fluid. We're saying that the shear stress
is related to velocity gradients. And that
proportionality constant is the coefficient of
viscosity.
So we thought already about
viscosity qualitatively as resistance to flow.
But this is a more precise definition that we can
use in thinking more precisely about what's
going on in the flow.
And how the fluid responds to deformations.
Because remember that, in the first video, we
said that
one of the main characteristics of fluids is that
they deform
in response to forces or stresses. And so this
equation,
Newton's law of viscosity, expresses exactly
that.
So a couple of things here to notice. If this
visocisty coeficient is constant, then the fluid is
said to be Newtonian.
So often if it is a constant, then the Greek letter
mu is used to represent the viscosity.
So this is a material property, just like density.
Or thermal conductivity. Any property like that.
The coefficient of viscosity is a material property
that you can look up in a table.
Now, for many other kinds of fluids the viscosity
coefficient may not necessarily be constant.
And it is often the case that this coefficient
depends on the rate of deformation. Or this
velocity gradient.
And if that's the case, then the fluid is said to be
non-Newtonian. So an example is a lot
polymer solutions, polymer melts, surfactants.
A lot of things like that are examples of non-
Newtonian fluids. And a lot of times they exhibit
what's called shear-thinning behavior.
So as you shear them at a greater rate, as you
apply a deformation at a faster rate. Then they
exhibit less resitance to flow.
The viscosity goes down. So that's just one
example. But any property where
the fluid's viscosity changes or depends on the
rate of deformation
is an example of a non-Newtonian fluid.
OK, now what did I mean when I wrote tau,yx
here? That is important to clarify.
So here I'm blowing up a plane in this flow.
And I have two different things drawn here. So
this n is what's called a normal vector. So it is a
unit vector that's normal to the surface.
And then I have shown the velocity v,x which is
going in the x direction.
So tau,yx. When I write this, it means the shear
stress on a surface whose normal vector is
oriented in the y direction.
Due to a flow or force that's acting in the x
direction.
So that's what these two indices mean.
Because notice that if I need to describe the
state of stress, I need to know
not only the direction of the applied force. But I
also need to know the orientation of the surface.
Because both of those quantities
are necessary for me to describe the state of
stress.
So that's why I need two indices here to
describe it.
Let me write that down here. So tau,yx is the
shear stress on the surface with normal in the y
direction,
due to a force or flow in the x direction.
That's important keep in mind because we are
going to use that
when we get to the conservation of momentum.
So this is an example of a tensor quantity. We
need nine components to describe the state of
stress.
Because it depends on both the orientation of
the surface and the orientation of the force.
Both of which are vector quantities. Now that we
have some idea how to think about viscosity in
terms of the relationship
between stress and deformation. It's useful to
think more deeply about what exactly does this
coefficient represent.
Thinking back to our qualitative description of
viscosity in terms of the resistance to flow.
We can imagine that this viscosity coefficient
expresses interaction between fluid layers that
can transfer momentum between them.
So what kind of interactions can have this
capability?
If we go back to our discussion about liquids,
solids, and gases. Remember in the case of
liquids.
You have this situation where you have these
weak cohesive interactions. You can imagine
if you apply a deformation to a liquid, these
weak interactions will break, but they may
locally continually break and re-form
during the course of the flow. So this is a good
way to think about the kinds of interactions that
can transfer momentum
between fluid layers. At least for short periods of
time, when these interactions form then
momentum will be able to be transfered
between neighboring layers, or neighboring
molecules, in the fluid.
Another thing that we thought about, in terms of
gases, was this idea of random motion or
interactions by collisions.
You can imagine in a gas, that you have this
random component of motion.
So if a gas molecule from one layer comes into
contact with a molecule in another layer. There
is a collision. And then some energy,
or momentum, is transferred between those
layers as a result of these collisions that
continually take place between layers.
So these kinds of pictures are qualitative. They
are useful to have in our heads to imagine what
is going on.
But they also do explain what we observe in
terms of the temperature dependence of
viscosity.
So for liquids, we observe that as we increase
the temperature, the viscosity goes down.
This makes sense because at high
temperatures, these weak interactions between
molecules become even weaker.
Because the thermal motion will become greater
in the individual molecules.
So that further acts against these weak
cohesive interactions.
So a typical example is pancake syrup. The
syrup is often served heated up, and when it's
heated up, it flows easily, almost like water.
So that is an example of this kind of property.
So as you increase temperature, the viscosity
goes down because these cohesive
interactions between molecules in a liquid
become weaker.
In a gas, that's more representative of the
second case. So in gases, the molecules are
interacting more by collisions.
So for gases we observe that the viscosity
increases as the temperature increases.
This makes sense because as you increase the
temperature, that is representative of the internal
energy of the system.
So you would expect to have more collisions,
and more energetic collisions, as you increase
the temperature.
So there would be more opportunity or capacity
to transfer momentum between different layers
of the fluid. And that is what's observed
The viscosity goes up with temperature for
gases. So liquids and gases actually show
opposite behavior in general.
Opposite depdence on temperature.
And we can explain that based on these ideas
about how individual atoms or molecules within
the fluid
would interact and be able to transfer
momentum between layers.
So this what we mean, really fundamentally,
when we're talking about viscosity.