OK, in this video we're going to begin a new topic dealing with the conservation of momentum.
So previously we talked about another
important conservation law, the conservation of mass. So here we're going to
use the same basic ideas, but apply them to momentum.
And in this case we are talking about linear momentum.
As opposed to angular momentum.
We can also write a conservation law for angular momentum, and sometimes that's important
when you're talking about characterizing the deformations of fluid elements in more complex fluids.
But for us, the conservation of linear momentum is a starting point, and it gives us some very important results
that will help us to analyze many of the flow fields of interest to us in this course.
And so again, going back to the definition of a conservation law. We can apply
the general definition to the case of linear momentum.
And so we can state that the rate of change of momentum in the control volume, CV, is equal
to the rate of inflow of momentum into the control volume,
minus the rate of outflow away from the control volume, plus the sum of the forces that are acting on the control volume.
So this statement, this general conservation law,
is actually just a statement of Newton's second law of motion
which states that the sum of the forces is equal to the time rate of change of momentum.
Normally we think of momentum as mass times velocity, m*v. So normally we consider cases
in physics, remember, where mass is considered to remain constant.
So the time rate of change of momentum becomes mass times the time rate of change of velocity, which is m*a.
So F = m*a is an expression of Newton's second law that may be more familiar to us.
But the more general case is that the sum of forces is the time rate of change momentum. But in addition
on the control volume, because we are dealing with flow, we need to account for inflow and outflow of momentum
through the control volume.
So in order to do that you may remember from some of the earlier videos. We talked
about the basic definition viscosity and shear stress.
The idea is that we need to be able to describe the forces that are acting
on the surface of a fluid element. The surface of a fluid element. We can imagine a surface,
or differential sized piece of fluid, inside the larger volume. We want to be able to
describe forces or stresses that are acting on this fluid element.
And the observation we can make is to notice that the orientation of both the force and the
surface of the fluid element are important in order to fully describe
these interactions. And you may remember that we can describe the orientation of surfaces in terms of a normal vector, n.
Which is a unit vector that's normal to the surface. So the way to think about it is
you know when you are driving a car and you put your hand out the window.
The surface of your hand is up and down, so it's perpendicular to the ground.
You can really feel the wind force because it's hitting your hand straight on.
But if you turn your hand sideways so it's parallel to the ground.
Then the wind is going over the surface of your hand so it's more of a shearing force. The orientation of the surface
relative to the direction of the force is what we need to describe.
In order to talk about the stresses in the fluid.
And so both of these quantities, both the force and the surface orientation, are vector quantities.
The force has a magnitude and direction, and the surface is described by a unit vector.
To give us a way to locate its orientation in space.
So given that that's the case, we need to keep track of two quantities, each of which has three components.
So forces, or stresses that are associated with these forces, and surface orientations
are going to be described by a different quantity called a tensor.
So a tensor has nine components. And so that includes
the information from both of these vector quantities.
It's a quantity that is able to represent information from both of these vector quantities.
So remember that when we were talking about viscosity.
The shear stress, we write as the greek letter tau.
So tau,yx indicates that a stress is acting on a surface that is oriented with a normal vector in the y direction
due to a force acting in the x direction. So that's what this index notation means.
It's the stress on a surface whose normal is in the y direction, due to a force in the x direction.
So the first index represents the surface orientation, and the second represents the direction of the force.
And remember, stress is force per unit area. So pressure, we think of pressure as force per unit area. That's a special case
where the force is acting normal to the surface.
OK, so a tensor I'm going to denote with two squiggle lines under it. A vector remember just has one.
So a stress tensor includes nine components because it represents all possible combinations of the force direction
and the surface orientation.
So all these components in this matrix are represented by this quantity tau with the two lines underneath.
And this is what we call the stress tensor.
So instead of three components, like a vector, it has nine components.
Let me draw a cube here to show some things about how we talk about these stresses.
So this is a simple shape that has surfaces
that happen to be oriented in each of these directions.
Relative to a Cartesian coordinate system x, y, and z. Which the axes are shown here.
So I can write on this face, tau,xx.
So this is the stress on the surface whose normal is in the x direction
due to a force in the x direction, tau,xx.
So this is a normal stress. This would be similar to what we think of as pressure.
OK, I could also write another component here so, tau,xy. So again this is a stress on a surface whose normal is in the x direction.
So the same surface we were talking about. Due to a force that's in the y-direction.
So this would represent tau,xy. So it's a shearing force. It's acting parallel to the surface.
And similarly I can write a third component tau,xz. So this is the stress
again on a surface whose normal is in the x direction
due to a force in the z direction. So that's another component that is a shear stress. So you can see how these indices help me
orient or describe these different stresses. And how they relate to both the orientation of the surface that I'm talking about.
And the direction of the force that's applied.
So I can do the same thing in the other directions. So on the top face
This face has a normal in the y direction. So the normal stress is tau,yy. And then I have two shearing components, tau,yx and tau,yz.
That would be due to forces acting in the x and z directions.
And on this front face, this has a normal in the z direction.
So I have components tau,zz, which is the normal stress.
And then shear stress components tau,zy and tau,zx due to the forces that are acting in the y and x directions respectively.
Okay, so this sort of shows the
orientations of these stresses and how they relate to both the surface orientation
and the the direction in which a force is applied.
OK, so now let's let's look at one plane in this shape and look at what
forces would be associated with a particular direction.
So I'm just going to draw the xy plane.
We have some point x,y,z and I'm just going to look at the xy projection of this.
And so I can draw a box around this point
with sides delta x and delta y.
And z is coming into or out of the screen here.
Now with this surface arrangement we can think about looking at the
forces acting on these surfaces. In the same way that we used to talk about the conservation of mass. So at this surface here
a force in the x direction would produce a normal stress.
And that normal stress at this surface I could represent as tau,xx, which would be mapped to this point x,y,z in space.
Plus the differential change in tau,xx with respect to x times the distance from the
center to this face, which is (delta x) / 2.
So mapping the value of the normal stress
at this point out to this surface that's a differential distance away.
Similarly I can make the same analysis on the left hand face, so tau,xx, the value associated with the point x,y,z.
Mapped out to this left hand face. So plus the partial derivative of tau,xx with respect to x.
times the distance minus (delta x) / 2 from the center to the left hand face.
So now I can look at the top surface. So the surface has a normal vector in the y direction.
And I'm considering forces acting in the x direction.
So the component of interest here is tau,yx
plus the rate of change of tau,yx with respect to y times the distance
from the surface to the center, (delta y) / 2.
And on the bottom face I have the point value tau,yx plus
the rate of change of tau,yx with respect to y
times the distance from the center to the bottom face, minus (delta y) / 2.
So now I can also think about this front face
which would be on the top of this surface.
So here again the normal vector to the surface is in the z direction.
And I'm talking about forces in the x direction. So I have tau,zx
plus the rate of change of tau,zx with respect to z times the distance from the center
to the top face of (delta z) / 2. Or to the front face.
And then behind here.
It's hard to represent because I just drew a 2D projection.
But if you can imagine the back side of this surface.
Would be tau,zx plus the partial derivative of tau,zx with respect to z times the distance minus (delta z) / 2.
OK, so these are all the components associated with the force acting in the x direction
on this kind of control volume.
Now notice that I took some liberties assigning directions to these arrows. Some of these are pointing left to right.
And some are pointing right to left. Why is that?
Okay so these arrows that are pointing in the negative x direction.
These reflect the sign convention that we have when we talk about stresses. And that sign
convention is such that the stress is considered positive
when it's exerted by a fluid of greater first subscript in the stress tensor notation,
on a fluid of lesser first subscript in the stress tensor notation. So let me show you what I'm talking about here.
The point is that we need some convention to define what's positive and what's negative. And so this is
what is generally accepted as the convention to describe that.
So let me show you what I'm talking about. So again this is kind of a simplified view of this 2D projection of this element.
If I look at just the components on the x face. So I have tau,xx on the right hand side and tau,xx on the left hand side.
So let's look at the right hand side. So my control volume is here, and I've sort of
shaded it in to indicate that. So the stress on this surface of the control volume is exerted by the fluid out here.
So the fluid out here that's acting on this control volume
has a position that's at a greater value of x.
It's further out in the x direction than the position of this surface.
So this fluid out here at greater x
is acting on a surface of lesser x.
So therefore by convention this would be a positive stress.
And similarly let's look at the surface on the left hand side. So the fluid over here on the far left is acting on the surface that is
a little bit further to the right. So by our convention then this fluid of lesser x is acting on a surface of greater x.
So then by our convention this would be considered a negative stress.
So this is just a convention that people agreed on. We have to define positive and negative somehow.
And this is the way that is generally accepted to describe the stresses.
So once everyone agreees that's how we're going to do it, then that makes it easier to
proceed and come up with a common set of equations to describe these kinds of things.