The conservation of momentum is a fundamental concept of physics

along with the

conservation of energy

and the

conservation of mass.

Momentum is defined to be the mass of an object multiplied by the velocity

of the object.

The conservation of momentum states that,

within some problem domain, the amount of momentum remains constant;

momentum is neither created nor destroyed, but only changed

through the action of forces as described by Newton’s

laws of motion.

Dealing with momentum is more difficult than dealing with mass and energy because

momentum is a

vector quantity

having both a magnitude and a direction. Momentum is conserved in all three

physical directions at the same time. It is even more difficult when dealing with a

gas

because forces in one direction can affect the momentum in another direction

because of the collisions of many molecules.

On this slide, we will present a very, very simplified flow problem

where properties only change in one direction.

The problem is further simplified by considering a steady flow which does

not change with time and by limiting the forces to only those

associated with the

pressure.

Be aware that real flow problems are much

more complex

than this simple

example.

Let us consider the flow of a gas through a domain in which flow properties

only change in one direction, which we will call “x”. The gas enters the domain

at station 1 with some velocity u and some pressure p

and exits at station 2 with a different value

of velocity and pressure. For simplicity, we will assume that

the density r remains constant within the domain

and that the area A through which the

gas flows also remains constant. The location of stations 1 and 2 are separated

by a distance called del x. (Delta is the little triangle on the slide

and is the Greek letter “d”. Mathematicians often use this symbol to denote

a change or variation of a quantity. The web print font does not support

the Greek letters, so we will just call it “del”.)

A change with distance is referred to as a gradient

to avoid confusion with a change with time which is called a rate.

The velocity gradient is indicated by del u / del x;

the change in velocity per

change in distance. So at station 2, the velocity is given by the velocity

at 1 plus the gradient times the distance.

u2 = u1 + (del u / del x) * del x

A similar expression gives the pressure

at the exit:

p2 = p1 + (del p / del x) * del x

Newton’s

second law of motion states

that force F is equal to the change in momentum with

respect to time. For an object with constant mass

m this reduces to the mass times acceleration a.

An acceleration is a change in velocity with a change in time

(del u / del t). Then:

F = m * a = m * (del u / del t)

The force in this problem

comes from the pressure gradient. Since pressure is a force per unit area,

the net force on our fluid domain is the pressure times the area at the

exit minus the pressure times the area at the entrance.

F = – [(p * A)2 – (p * A)1] = m * [(u2 – u1) / del t]

The minus sign at

the beginning of this expression is used because gases move from a region

of high pressure to a region of low pressure; if the pressure increases with

x, the velocity will decrease. Substituting for our expressions for velocity

and pressure:

– [{(p + (del p / del x) * del x} * A) – (p * A)] = m * [(u + (del u / del x) * del x – u) /

del t]

Simplify:

– (del p / del x) * del x * A = m * (del u / del x) * del x / del t

Noting that (del x / del t) is the

velocity and that the mass is the density r times the volume (area times del x):

– (del p / del x) * del x * A = r * del x * A * (del u / del x) * u

Simplify:

– (del p / del x) = r * u * (del u / del x)

The del p / del x and del u / del x

represent the pressure and velocity gradients.

If we shrink our domain down to differential sizes, these gradients become differentials:

– dp/dx = r * u * du/dx

This is a one dimensional, steady form of

Euler’s Equation.

It is interesting to note that the pressure drop

of a fluid (the term on the left) is proportional to both the value of the

velocity and the gradient of the velocity.

A solution of this momentum equation gives us the form of the

dynamic pressure

that appears in

Bernoulli’s Equation.

Activities:

Guided Tours

Basic Fluid Dynamics Equations:

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