Page 224 - Introduction to Continuum Mechanics

Entropy inequality 203

where q s is the rate of heat input (known simply as the heat supply) per unit mass by external
sources. In invariant notation, Eq. (4.14.2a) is written

4.15 Entropy Inequality

Let q(x, i) denote the entropy per unit mass for the continuum. Then the entropy in a volume
dV of the material is pydV, where p is density. The rate of increase of entropy following the
volume of material as it is moving is

which is equal topdl^j^-, because —(pdV) = 0 in accordance with the conservation of mass
D Yl
principle. Thus, per unit volume, the rate of increase of entropy is given by p -rp

The entropy inequality law states that

where © is the absolute temperature, q is the heat flux vector and q s is the heat supply.