Page 456 - Introduction to Continuum Mechanics

440 Integral Formulation of General Principles

We see that this time-dependent integral turns out to be independent of time. This is because
the chosen density and velocity field satisfy the equation of continuity so that, the total mass
of a fixed part of material is a constant.

(f) Total linear momentum is, since v 2 - v-$ = 0,

The fact that P is also a constant is accidental. The given motion happens to be acceleration-
less, which corresponds to no net force acting on the material volume. In general, the linear
momentum for a fixed part of material is a function of time.

7.6 Principle of Linear Momentum
The global principle of linear momentum states that the total force (surface and body
forces) acting on any fixed part of material is equal to the rate of change of linear momentum
of the part. That is, with p denoting density, v velocity, t stress vector, and B body force per
unit mass, the principle states

Now, by using Reynolds Transport Theorem, Eq. (7.4.1), Eq. (7.6.1) can be written as

In words, Eq. (7.6.2) states that
Total force exerted on a fixed part of a material instantaneously in a control volume K c
= time rate of change of total linear momentum inside the control volume + net outflux
of linear momentum through the control surface S c.
Equation (7.6.2) is very useful for obtaining approximate results in many engineering
problems.
Using Eq. (7.4.2) (with T replaced by p \), Eq. (7.6.1) can also be written as