r
z
v
v
v
v
θ
θ
=
=
≠
Equations of motion of incompressible Newtonian Fluids in
cylindrical
and
spherical
coordinates: See appendix B,
Viscous Fluid Flow
, F.
White

Continuity equation:
(
)
(
)
(
)
( )
1
1
0
0
6
r
z
V
rv
v
v
r
r
r
z
θ
θ
θ
θ
∂
∂
∂
∂
+
+
=
=
∂
∂
∂
∂
( )
steady flow,
0
axisymmetric flow
0
cylinder infinitely long
0
unidirectional motion
v
t
z
v
r
θ
θ
θ
∂
=
∂
∂
=
∂
∂
=
∂
=
Momentum eq. in r - direction
( )
2
1
v
dp
dr
r
θ
ρ
=
balance between centrifugal force & force
which is produced by the induced pressure
field.
0
p
as
r
dp
dr
>
/
/
p
streamline

( )
2
2
0
2
d v
v
d
direction
dr
dr
r
θ
θ
θ
⎛
⎞
−
→
+
=
⎜
⎟
⎝
⎠
( )
2
viscous dissipation term
k
equation
0=
3
r
dv
v
d
dT
Energy
r
dr
dr
dr
r
θ
θ
µ
⎛
⎞
⎛
⎞
+
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
±²²³²²´
BC’s
1
1 1
1
1
2
2 2
2
r=r
v
&
T=T
p=p
r=r
v
&
T=T
At
r
At
r
θ
θ
ω
ω
=
=
Integrate eq. (2)
(
)
(
)
1
2
1
1
1
2
2
0
1
2
/
dv
v
dv
v
d
c
dr
dr
r
dr
r
d
d
r
rv
c
rv
c r
rv
c
c
r dr
dr
v
cr
c
r
θ
θ
θ
θ
θ
θ
θ
θ
⎡
⎤
+
=
⇒
+
=
⎢
⎥
⎣
⎦
=
=
→
=
⇒
=
+
=
+

(
)
2
2
2
2
2
2
1
1
1
2
r
r
z
r
r
V
V
V
V
V V
V
v
v
t
r
r
r
z
V
V
V
r
r
z
r
r
V
p
g
r
r
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
ρ
θ
µ
ρ
θ
θ
θ
∂
∂
⎛
⎞
⎜
⎟
∂
∂
∂
∂
⎛
⎞
+
+
+
+
⎜
⎟
∂
∂
∂
∂
⎝
⎠
⎡
⎤
∂
∂
∂
∂
=
+
+
+
−
+
⎢
⎥
∂
∂
∂
∂
⎣
⎦
∂
∂
⎝
⎠
1. steady
2.
ρ
= const.
3.
Fully developed in z-dir.
4. axisymmetric in
θ
-dir.
5. v
r
= v
z
= 0
(
)
(
)
(
)
1
1
1
1
0
rV
c
rV
c r
r
r
r
rV
r
r
r
θ
θ
θ
∂
∂
=
→
=
∂
∂
∂
⎛
⎞
=
⎜
⎝
⎠
∂
⎟
∂
∂
Momentum eq. in
θ
direction

1
2
2
2
V
/
2
r
rV
c
cr
c
r
c
θ
θ
=
+
=
+
→
Energy eq.
2
0
(3)
dV
V
k d
dT
r
r dr
dr
dr
r
θ
θ
µ
⎛
⎞
⎛
⎞
=
+
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
BC’s
1
1 1
1
1
2
2 2
2
r=r
v
&
T=T
p=p
r=r
v
&
T=T
At
r
At
r
θ
θ
ω
ω
=
=
using the B.C’s
(
)
2
2
2
2
2
1
2
1
2
2
1
1
1
2
2
2
2
2
1
2
1
,
c
r r
r
r
c
r
r
r
r
ω
ω
ω
ω
−
−
=
=
−
−

(
)
(
)
( )
2
2
2
2
2
1
2 2
1 1
2
1
2
2
2
1
1
( )
4
r r
V
r
r
r
r
r
r
r
θ
ω
ω
ω
ω
⎡
⎤
=
−
−
−
⎢
⎥
−
⎣
⎦
Eq. (1)
determines the radial pressure distribution resulting from the motion.
p=p(r)
←
obtain this distribution
!
Having found v
θ
(r) , it is substituted into eq. (3)
to find temperature distribution
(
)
(
)
(
)
(
)
(
)
2
4
2
2
2
1
1
1
1
1
4
4
2
2
1
2
1
2
1
2
1
1
/
ln
/
ln
/
Pr
1
1
ln
/
ln
/
r
r
r
r
r
T
T
r
Ec
T
T
r
r
r
r
r
r
r
ω
ω
⎡
⎤
−
⎛
⎞
−
=
−
−
+
⎢
⎥
⎜
⎟
−
−
⎝
⎠
⎣
⎦
where
(
)
2
2
1
1
2
1
Pr
r
Ec
k T
T
µ
ω
=
−
Brinkman number
expressing the temp. rise due to dissipation
(
)
(
)
1
1
2
1
2
1
ln
/
PrEc=0
heat conduction solution
ln
/
r
r
T
T
if
T
T
r
r
−
=
−

SPECIAL CASES :
i)
case when r
1
→
0
i.e.
in the limit as the inner cylinder vanishes
(
α
=0)=r
1
/r
2
1
2
.(4)
V (
0)
rigid-body rotation
Eq
r
r
θ
ω
→
=
2 2
r
ω
i.e.
fluid rotates inside the outer cylinder as a rigid body
2
V
ω
ω
= ∇×
=
JG
JG
ii)
single cylinder rotating in an infinite fluid
2
2
(
,
0)
r
ω
→ ∞
=
1 1
r
ω
2
1
1
2
(
)
r
V
r
r
θ
ω
→ ∞ =
flow
v .
.
0

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