Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 22

Consider now the special case when the heat source per unit mass r = 0 and the body force is derivable from a scalar potential function ζ as follows

(2.46)

The effects of the Earth’s gravitational field can then be taken into account. The substitution of (2.46) into (2.41) leads to the relation

(2.47)

As

(2.48)

it then follows from (2.47) that the local form (2.44) for the total energy balance equation may also be expressed

(2.49)

The energy balance equation may, therefore, be written in the more compact form

(2.50)

where the total energy per unit volume χ is given by

(2.51)

which is the sum of the internal energy, potential energy and kinetic energy per unit volume, and where the total energy flux Jχ is given by

(2.52)

On using (2.51), relation (2.52) may also be written in the form

(2.53)

which identifies the mechanisms whereby energy can enter or leave the system, namely advection (the term χv), as heat (the term h) and as external work (the term −σ.v).

The energy balance equation (2.50) implies energy conservation as the energy within any region V always remains fixed during any nonequilibrium process provided that there is no energy flow across the surface S bounding the region V. This result is easily established by integrating (2.50) over the region V, and then making use of the divergence theorem.