Page 451 - Introduction to Continuum Mechanics

Reynolds Transport Theorem 435

whereas the integral

is over the material volume V m and the rate of change of this integral, is denoted by

We note that the integrals over the material volume is a special case of the more general
integrals where the boundaries move in some prescribed manner which may or may not be in
accordance with the motion of the material particles on the boundary. In this chapter, the
control volume denoted by V c will always denote a fixed control volume; they are either fixed
with respect to an inertial frame or fixed with respect to a frame moving with respect to the
inertial frame (see Section 7.7).

7,4 Reynolds Transport Theorem
an{ me t
Let T(x, t) be a given scalar or tensor function of spatial coordinates (jcj^^a ) ^ ^ -
V
Examples of T are: density p(x, t), linear momentum p(\, £) (X 0» angular momentum
rX|/>(x,fXM)]etc.
Let

be an integral of T(x, t) over a material volume V m(t). As discussed in the last section, the
material volume V m(i) consists of the same material particles at all time and therefore has
time-dependent boundary surface S m(t) due to the movement of the material.
We wish to evaluate the rate of change of such integrals (e.g., the rate of change of mass,
of linear momentum etc., of a material volume ) and to relate them to physical laws (such as
the conservation of mass, balance of linear momentum etc.)
The Reynolds Transport Theorem states that

or