Page 454 - Introduction to Continuum Mechanics

438 integral Formulation of General Principles

This equation states that the time rate at which mass is increasing inside a control volume = the
mass influx (Le,, net rate of mass inflow ) through the control surface.

Substitutingp for Tin Eq. (7.4.3), we obtain from Eq. (7.5.2b)

This equation is to be valid for all K c, therefore, we must have

This equation can also be written as

This is the equation of continuity derived in Section 3.15.

Example 7.5.1

Given the motion

and the density field

(a) Obtain the velocity field.
(b) Check that the equation of continuity is satisfied.
(c) Compute the total mass and the rate of increase of mass inside a cylindrical control volume
of cross-sectional area ,4 and having as its end faces the plane *i = 1 and x\ = 3.
(d) Compute the net rate of inflow of mass into the control volume of part(c).
(e) Find the total mass at time t of the material which at the reference time (t = 0) was in the
control volume of (c).
(f) Compute the total linear momentum for the fixed part of material considered in part (e)
Solution, (a)