Page 455 - Introduction to Continuum Mechanics

Principle of Conservation of Mass 439

(b)

Thus, the equation of continuity is satisfied.
(c) The total mass inside the control volume at time t is

and the rate at which the mass is increasing inside the control volume at time t is

Le, the mass is decreasing.
(d) Since v*i - v$ = 0, there is neither inflow nor outflow through the lateral surface of the
control volume. Through the end face jcj = 1, the rate of inflow (mass influx) is

On the other hand, the mass outflux through the end face Xi = 3, is

Thus, the net mass influx is

which is the same as Eq. (vi).
(e) Hie particles which were at xi = 1 and x± = 3 when t = 0 have the material coordinate
X\ — \ and Xi = 3 respectively. Thus, the total mass at time t is