Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 19

Consider at time t a moving sample of material occupying a fixed volume V bounded by a closed surface S. In the absence of mass source and sink terms, the total mass of material within the region V is fixed so that on setting ϕ = ρ in (2.27) the global form of the mass balance equation for the medium may be written as

(2.28)

where ρ is the mass density. The term on the left-hand side of (2.28) is the rate of change of the total mass in the fixed region V. The term on the right-hand side is the rate at which the mass of the medium is transported across the surface S into the region V. On using the divergence theorem, the mass balance equation may be written as

(2.29)

As (2.29) must be valid for any region V of the system, the following local form of the mass balance equation for the medium must be satisfied at all points in the system for all times t

(2.30)

On using (2.22) and the identity (2.26), the continuity equation (2.30) may be written in the equivalent form

(2.31)