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.