5.4.1.1 Conservation of Mass
Consider an arbitrary volume V fixed in space, bounded by surface Ω. The rate of
change of the mass within the volume V is (Malvern 1969)
d
dt
Z v
ρdV ¼
Z v
∂ρ
∂t
dV
ð5:36Þ
where ρ is the density (mass per unit volume). If no mass is created or destroyed
inside V, this quantity must be equal to the rate of the material flow into the volume
V through its surface Ω (Malvern 1969):
Z v
∂ρ
∂t
dV ¼
Z Ω
ρv Á dΩ
ð5:37Þ
where v is the velocity and dΩ is a vector with magnitude |dΩ| normal to the surface
and counted positive from the inside to the outside. The quantities ρ and v are all
functions of time and of space coordinates. Applying Gauss’s theorem to the surface
integral in Eq. (5.37), we obtain
∂ρ
∂t
¼ Àdiv ρν
ð5:38Þ
Equation (5.38) is valid for an arbitrary volume V, which expresses the fact that
the total mass is conserved, i.e., that the total mass in any volume element of the
system can only change if matter flows into (or out of) the volume element. This
equation has the form of a so-called balanced equation: the local change of the
density is equal to the negative divergence of the flow of mass. The continuity
equation in the vector form of Eq. (5.38) is independent of any choice of coordinates.
The conservation of mass equation can also be written in an alternative form by
introducing the substantial time derivative (Mazur and De Groot 1962):
d
dt
¼
∂
∂t
þ ν Á grad
ð5:39Þ
With the help of Eq. (5.39), Eq. (5.38) becomes
dρ
dt
¼ Àρ div v
ð5:40Þ
With the specific volume v ¼ ρ
À1 , Eq. (5.40) may also be written as
5.4 Thermodynamic Fundamental Equation in Thermo-mechanical Problems
217
Consider an arbitrary volume V fixed in space, bounded by surface Ω. The rate of
change of the mass within the volume V is (Malvern 1969)
d
dt
Z v
ρdV ¼
Z v
∂ρ
∂t
dV
ð5:36Þ
where ρ is the density (mass per unit volume). If no mass is created or destroyed
inside V, this quantity must be equal to the rate of the material flow into the volume
V through its surface Ω (Malvern 1969):
Z v
∂ρ
∂t
dV ¼
Z Ω
ρv Á dΩ
ð5:37Þ
where v is the velocity and dΩ is a vector with magnitude |dΩ| normal to the surface
and counted positive from the inside to the outside. The quantities ρ and v are all
functions of time and of space coordinates. Applying Gauss’s theorem to the surface
integral in Eq. (5.37), we obtain
∂ρ
∂t
¼ Àdiv ρν
ð5:38Þ
Equation (5.38) is valid for an arbitrary volume V, which expresses the fact that
the total mass is conserved, i.e., that the total mass in any volume element of the
system can only change if matter flows into (or out of) the volume element. This
equation has the form of a so-called balanced equation: the local change of the
density is equal to the negative divergence of the flow of mass. The continuity
equation in the vector form of Eq. (5.38) is independent of any choice of coordinates.
The conservation of mass equation can also be written in an alternative form by
introducing the substantial time derivative (Mazur and De Groot 1962):
d
dt
¼
∂
∂t
þ ν Á grad
ð5:39Þ
With the help of Eq. (5.39), Eq. (5.38) becomes
dρ
dt
¼ Àρ div v
ð5:40Þ
With the specific volume v ¼ ρ
À1 , Eq. (5.40) may also be written as
5.4 Thermodynamic Fundamental Equation in Thermo-mechanical Problems
217