Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei (
李雨
)
Chapter 2 :
Physical and Mathematical
formulations of Fluid Mechanics
Continuum fluid mechanics
Conservation of mass, momentum (Newton’s 2
nd
law) and energy (1
st
law of thermodynamics)
Use the Lagrangian description to derive the
equations, and the Eulerian description to solve the
problems
Recall the Reynolds transport theorem, which
expresses the time rate of change of properties
within
a volume with fixed mass

Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei (
李雨
)
Reynolds Transport Theorem
(three useful expressions as follows are employed)
( )
( , )
(
)
V
t
d
dF
F
F
t dV
F
dV
F
dV
dt
dt
t
∂
=
+
∇ ⋅
=
+ ∇ ⋅
∂
r
u
u
V(t)
V(t)
( )
( )
V
t
S t
F
dV
t
F
dS
∂
=
+
∂
⋅
n u
(1-27)
(Controlled mass system)
(Lagrangian description)
(Eulerian description) if
V
(
t
)
is chosen as a fixed volume with naterials in/out

Divergence theorem
for
g
=
g
(
x
,
y
,
z
) = any vector function
∂
⋅
=
⋅
∇
D
D
S
V
d
d
n
g
g
D
n
D
∂
domain
boundary

Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei (
李雨
)
Conservation of mass - Continuity equation (1)
The mass of material inside the material volume,
V
(
t
),
=
)
(
)
,
(
t
V
dV
t
m
r
ρ
Conservation of mass implies that
m
=
constant,
0.
dm
dt
=
Applying the Reynolds transport theorem,
( )
( )
( , )
0
V
t
V
t
dm
d
d
t dV
dV
dt
dt
dt
ρ
ρ
ρ
=
=
+
∇ ⋅
=
r
u
(Lagrangian frame)

Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei (
李雨
)
Conservation of mass - Continuity equation (2)
Since
V
(
t
) is arbitrary, it is required that
0,
=
⋅
∇
+
u
ρ
ρ
dt
d
(
t
ρ
ρ
∂
+ ∇ ⋅
=
∂
u)
0
or
(Eulerian frame)

Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei (
李雨
)
Another form
of the Reynolds transport theorem
Replacing
F
(
r
, t) in the Reynolds transport theorem
in (1-27) by
ρ
(
r
, t)
F
(
r
, t),
( )
( )
( )
(
)
(
)
(
)
V
t
V
t
V
t
d
d
F
dF
d
FdV
F
dV
F
F
dV
dt
dt
dt
dt
ρ
ρ
ρ
ρ
ρ
ρ
=
+
∇ ⋅
=
+
+
∇ ⋅
u
u
The sum of the last two terms in the integral is zero according
to the continuity equation. Thus
( )
( )
V
t
V
t
d
dF
FdV
dV
dt
dt
ρ
ρ
=