href="#ulink_2a4bcde2-c305-5db1-bdb6-95d1b181fc94">2.22) that
where use has been made of the following identity
On using the divergence theorem, the identity (2.25) may then be written as
The first term on the right-hand side accounts for any changes of the property ϕ locally at points within the region V, whereas the second term accounts for the mean advection of the property across the bounding surface S where n is the outward unit normal to the surface S bounding the region V. The important identities (2.22) and (2.27) are used repeatedly in the following analysis.
2.6 Continuity Equation
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
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
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
On using (2.22) and the identity (2.26), the continuity equation (2.30) may be written in the equivalent form
2.7 Equations of Motion and Equilibrium
The global form of the linear momentum balance equation for a fixed region V bounded by the closed surface S having outward unit normal n is written as
where σ is the stress tensor and where b is the body force per unit mass acting on the medium. Such body forces usually arise from the effects of gravity. On using (2.27) and the divergence theorem, the linear momentum balance equation (2.32) may be written as
As relation (2.33) must be satisfied for any region V of the system, it follows that the local form of the linear momentum balance equation has the form
As
it follows from (2.34) on using (2.22) that
On using (2.31), relation (2.36) reduces to the well-known equation of motion