Page 452 - Introduction to Continuum Mechanics

436 Integral Formulation of General Principles

where V c is the control volume (fixed in space) which instantaneously coincides with the
material volume V m (moving with the continuum), S c is the boundary surface of V c , n is the
outward unit normal vector. We note that the notation D /Dt in front of the integral at the
left hand side of Eqs (7.4.2) emphasizes that the boundary surface of the integral moves with
the material and we are calculating the rate of change by following the material.

Reynold's theorem can be derived in the following two ways:
(A)

Since [see Eq. (3.13.7) 1

therefore, Eq. (i) becomes

This is Eq. (7.4.2).
In terms of Cartesian components, this equation reads, if T is a scalar

If T is a vector, we replace T in Eq. (7.4.2a) with 7} and if T is a second order tensor, we
replace Twith 7^- and so on.

Since

and from the Gauss theorem, we have
f A

so that, with T denoting tensor of all orders (including scalars and vectors)

This is Eq. (7.4.1).