3
z
z
p
B
w
V
a
z
z
ρ
µ
ρ
∂
∂
−
+
∇
+
∇
=
∂
∂
JG

NAVIER STOKES EQUATION
N
N
N
N
2
1
(
. )
(
.
)
3
body
deformation
pressure
convective
force
stresses
force
local
due to
acceleration
acceleration
compressibility
V
B
p
V
V
V
V
t
ρ
µ
µ
ρ
ρ
∂
− ∇
+
∇
+
∇ ∇
=
+
∇
∂
JG
JG
JG
JG
JG
JG
±²³²´
±²³²´
for
ρ
= const
.
.
0
V
∇
=
JG
2
DV
B
p
V
Dt
ρ
µ
ρ
−∇
+
∇
=
JG
JG
JG
General equation
coordinate independent
Cartesian coord.
x – dir.
2
2
2
2
2
2
x
p
u
u
u
u
u
u
u
B
u
v
w
x
x
y
z
t
x
y
z
ρ
µ
ρ
⎛
⎞
⎛
⎞
∂
∂
∂
∂
∂
∂
∂
∂
−
+
+
+
=
+
+
+
⎜
⎟
⎜
⎟
∂
∂
∂
∂
∂
∂
∂
∂
⎝
⎠
⎝
⎠
±²²²³²²²´
(
)
x
g
if B
g
gravitional acceleration
=
JG
JG
1
1
Force acting on the fluid element as a result of viscous stress distribution
on the surface of element

Note:
Cylindrical coord. pg. 60
•
viscosity is constant
→
•
ρ
= const.
.
0
V
∇
=
JG
isothermal flow. For non-isothermal flows, esp. for liquids,
viscosity is often highly temp. dependent. CAUTION
CONSERVATION OF ENERGY: THE ENERGY EQ.
1st law of Thermodinamics for a system
t
dE
dQ
dW
=
+
3
(
/
)
J
m
work done on system
heat added
increase of energy of the system
N
2
1
.
2
t
gz
E
e
V
g r
ρ
+
⎛
⎞
⎜
⎟
=
+
−
⎜
⎟
⎝
⎠
JG G
3
(
/
)
J
m
x
z
y
r
G
g
JG
E
t
: total energy of the system (per unit volume)
e
: internal enregy per unit mass
: displacement of particle
r
G

moving system
such as
flowing fluid particle
need
Material derivative
:
time rate of change, following the particle
t
DQ
DW
Dt
DE
D
Dt
t
=
+
(J/m
3
.s)
Energy eq. for a flowing fluid
t
De
DV
V
gV
Dt
D
DE
Dt
t
ρ
⎛
⎞
+
−
⎜
⎟
⎠
=
⎝
JGJG
Q & W
in terms of fluid properties
x
q
x
w
dx
x
x
q
q
dx
x
∂
+
∂
x
x
w
w
dx
x
∂
+
∂
• assume heat transfer Q to the
element is given by Fourier ’s
law
x
T
q
k
x
∂
= −
∂
heat flows from positive to the
neg. temp. (
decreasing temperature
gradient
)
(HEAT FLOW)
q
k
T
= − ∇
G
x
y
z

Heat flow (rate) into the element in x - dir. :
Heat flow (rate) out of the element in x - dir. :
The
net
heat transfer
to
the element in x - dir. :
Hence, the net heat transfer
to
the
element
=
x
q dydz
x
x
q
q
dx dydz
x
∂
⎛
⎞
−
+
⎜
⎟
∂
⎝
⎠
W/m
2
x
q
dxdydz
x
∂
−
∂
. .
x
x
x
q
q
q
d
q d
x
y
z
⎛
⎞
∂
∂
∂
−
+
+
∀ = −∇
∀
⎜
⎟
∂
∂
∂
⎝
⎠
G
(
)
.
.
DQ
q
k
T
Dt
= −∇
= ∇
∇
G
[W/m
3
]
neglect internal heat generation
Rate of work done to the element per unit area on the left face
negatif because work is done on the system
(
)
x
xx
xy
xz
W
u
v
w
σ
τ
τ
= −
+
+
surface
direction
derivation
: force on the left face:
(
)
xx
xy
xz
i
j
k dydz
σ
τ
τ
−
+
+
G
G
G

Rate of work done on
the element by this force
(
) (
)
(
)
.
xx
xy
xz
xx
xy
xz
i
j
k
ui
v j
wk dydz
u
v
w
dydz
σ
τ
τ
σ
τ
τ
= −
+
+
+
+
= −
+
+
G
G
G
G
G
G
Similarly, rate of work done by the right face stresses is
x
x
W
W
dx
x
∂
⎛
⎞
= −
+
⎜
⎟
∂
⎝
⎠
• Net
rate of work done on the element
(
)
(
)
(
)
xx
xy
xz
yx
yy
yz
zx
zy
zz
u
v
w
x
u
v
w
y
u
v
w
z
σ
τ
τ
τ
σ
τ
τ
τ
σ
∂
+
+
∂
∂
+
+
+
∂
∂
+
+
+
∂
.

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