A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

14 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
Since the above equality holds for an arbitrary domain Ω t , we obtain the pointwise
equation of continuity
∂ρ
∂t + ∂ρv x
∂x
+ ∂ρv y
∂y
+ ∂ρv z
= 0, (2.17)
∂z
which can be written concisely as
2.2.3 The Reynolds transport theorem
∂ρ
+ ∇ · (ρv) = 0.
∂t
In this subsection, we will use the material derivative (2.16) as well as the equation of
continuity (2.17) to derive the celebrated Reynolds transport theorem. By (2.16) we find
that
∫
∫
d
( ∂(ρvz )
ρv z dΩ =
+ ∂ρv zv x
+ ∂ρv zv y
dt Ω t Ω t ∂t ∂x ∂y
+ ∂ρv )
zv z
dΩ.
∂z
Then using the equation of continuity (2.17), we find that the integrand of the right
side of the above equation is equal to
∂ρ
∂t v z + ρ ∂v (
z
∂t + v ∂ρvx
z
∂x
+ ∂ρv y
∂y
( ∂ρ
=v z
∂t + ∂ρv x
∂x
+ ∂ρv y
∂y
+ ∂ρv )
z
+ ρ
∂z
( ∂vz
=ρ
∂t + v x
Hence, we have
∂v z
∂x + v y
∂v z
∂y + v z
∂v
) z
.
∂z
+ ∂ρv z
∂z
( ∂vz
∂t + v x
)
∂v z
+ ρv x
∂x + ρv ∂v z
y
∂y + ρv ∂v z
z
∂z
)
∂v z
∂x + v ∂v z
y
∂y + v z
∫
∫
d
( ∂vz
ρv z dΩ = ρ
dt Ω t Ω t ∂t + v ∂v z
x
∂x + v ∂v z
y
∂y + v ∂v
) z
z dΩ. (2.18)
∂z
Eq. (2.18) is the Reynolds transport theorem, which is usually written concisely as
∫
∫
d
ρv z dΩ = ρ Dv z
dt Ω t Ω t Dt dΩ,
where Dv z /Dt is the total derivative of v z , and is given by
D
Dt v z(x, y, z, t) = ∂v z
∂t + v ∂v z
x
∂x + v ∂v z
y
∂y + v ∂v z
z
∂z .
We note that the above is independent of any coordinate system and depends only on
the rules of calculus and the assumptions of continuity of mass in a time dependent
volume of particles.
∂v z
∂z