Conservation of linear momentum is a little bit different then conservation of mass.
The main thing that you will recognize this is newtons law, it is the sum of the forces
equals the mass times the acceleration.
So what forces are we talking about.
Well we have the normal forces, and we use sigma to indicate them, and we have the tangential
forces, and we use tau to indicate that.
So lets take a look on one plane of a differential Q, and our coordinate system is going to be
such that this is z, this is x, and this is y.
The first thing that we are going to look at is this normal force that comes out of
the plane, normal to the plane.
We call this force sigma sub xx, and if you look at the two subscripts the way they are
defined.
Is the first subscript is the direction of the normal to the plane.
So if you look at it.
It is in the x-direction.
Now lets look at the same plane, but lets look at the tangential forces.
So our tangential forces would go this way along the plane, and this way along the plane.
So lets name them.
Lets call this one number 1, and this one number 2.
If you recall the firsts subscript is the direction normal to the plane.
So that is going to be x.
The second subscript is the direction of the force, or stress.
So for normal forces those two subscripts are going to be the same, but if we look at
number 1.
The direction id in Z.
If we look at number 2 again it is in the normal is in the x plane, but the stress is
in the y plane.
You might look at these and say why are these stresses positive, when they are in the negative
direction...
The reason for that is that we consider the normal to the plane as the positive direction.
So what that means in this particular case is we shift our coordinate system such that
the positive coordinate system looks like this.
This is x, this is y, and this is z.
So we basically rotated the axis.
Now we can look at a differential element using the different forces, and using a Taylor
series expansion, and what we get in differential form is the surface forces.
So this is our normal.
This is one of our tangential forces.
This is the second tangential force and this is all multiplied by the volume of our differential
element.
We can do the same thing with the y forces.
Here however it will be sigma yy, tau xy, tau zy.
We will also would have the surface forces in the z direction.
In addition the other kind of force we have are body forces.
These body forces we generally assume are gravity.
So now when we sum up our differential forces we have both our surface forces and our body
forces.
Notice those arrows, that indicates that these are vectors, which means we have them in the
x-direction, the y-direction, and the z-direction.
Now using newtons second law, and I am just going to look at it in the x direction.
The sum of these forces in the x-direction have to be the mass times the acceleration
in the x-direction.
What this leads us to is the equations of motion, and the equations of motions are fundamental
equations that we use in fluids.
We just simplify them for our particular needs.
So this is in the x-direction, rho,gx, plus our normal force, plus our tangential forces.
Remember there are 2 of them.
Tau zx/dz, and this has to equal rho, du/dt, plus u, du/dx, plus v, du/dy, plus w, du/dz.
So this part right here are the sum of the forces in the x direction, and this part right
here is the acceleration in the x-direction.
Where this is the local acceleration, and these are the convective term.
Since it is a vector we would have to do it in all 3 directions.
So for example in the y direction plus our other forces is going to equal rho.
Now it is dv/dt, plus u*dv/dx, plus v*dv/dy, plus w*dv/dz), and when you put this all together
all three of these in the x-direction, y-direction, and z-direction we get the equation of motion.