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.

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