3
1
sin
1
2
a
Ur
r
ψ
θ
⎛
⎞
=
−
⎜
⎟
⎝
⎠
See handout

Streamlines are close to
the body!
Recirculation is absent
sphere drags the entire surrounding fluid with it
circulation streamlines
sphere pushes fluid out of the way

Other three-dimensional body shapes
Happel & Brenner,
Low Reynolds number hydrodynamics
(1965)
Disk normal to the freestream:
F=(32/3) Ua
µ
Disk parallel to the freestream:
F=16
Ua
µ
See White (1991) Viscous Fluid Flow

EXACT SOLUTIONS OF NAVIER-STOKES EQS
A)
Parallel Flows: i.e. velocity is unidirectional
B) Similarity solutions
C) Generalized Beltrami Flows-
•
difficulties
- non-linear PDEs
- no general solution is known
0
,
v=w=0
u
≠
N
0
0
3D Flow
V
V
ω
ω
=
⎛
⎞
∇×
×
=
⎜
⎟
⎝
⎠
JG
JG
JG
JG
&
Reference C.Y. Wang (1991) “Exact solutions of the steady-state N-S eqs.”
Ann. Rev. of Fluid Mech
. Vol.23, pp. 159-177.
R.Berker (1963)
Almost all of the particular solutions are for the case of incompressible Newtonien
flow with constant transport properties,
2
2
.
0
p
V
DV
p
V
Dt
DT
c
k
T
Dt
ρ
µ
ρ
∇
=
= −∇
+
∇
=
∇
+ Φ
JG
JG
JG

P: total hydrostatic pressure. i.e. it includes the gravity term for convenience.
or
P=p+ gz
P
p
g
ρ
ρ
∇
= ∇
−
JG
y
x
z
g
JG
g
gk
= −
JG
G
(
)
(
)
1) Convective accel. V.
.
2) Non-linear solutions
V.
does not vanish.
vanishes
∇ →
∇ →
JG
JG
GROUP A:
PARALLEL FLOWS
0
,
v=w=0
u
≠
parallel flow: only one velocity component is different from zero.
i.e. all fluid particles moving in one direction.
N
N
0
0
0
0
u=u(y,z,t)
;
v
0 , w
0
u
v
w
u
x
y
z
x
∂
∂
∂
∂
+
+
=
⇒
=
∂
∂
∂
∂
⇒
≡
≡
y-comp. of momentum eq.
2
2
2
2
2
2
v
v
v
v
p
v
v
v
u
v
w
t
x
y
z
y
x
y
z
ρ
µ
⎛
⎞
⎛
⎞
∂
∂
∂
∂
∂
∂
∂
∂
+
+
+
= −
+
+
+
⎜
⎟
⎜
⎟
∂
∂
∂
∂
∂
∂
∂
∂
⎝
⎠
⎝
⎠
0
p
y
∂
=
∂

z- comp. of mom. eq.
(
)
0
p=p x,t
p
z
∂
=
⇒
∂
x- comp. of mom. eq.
2
2
2
2
2
2
2
2
2
2
(A)
u
u
u
u
p
u
u
u
u
v
w
t
x
y
z
x
u
p
u
u
t
x
x
y
z
y
z
ρ
µ
ρ
µ
⎛
⎞
⎛
⎞
∂
∂
∂
∂
∂
∂
∂
∂
+
+
+
= −
+
+
+
⎜
⎟
⎜
⎟
∂
∂
∂
∂
∂
∂
∂
∂
⎛
⎞
∂
∂
∂
∂
= −
+
+
⎜
⎟
∂
∂
∂
∂
⎝
⎠
⎝
⎠
⎝
⎠
Linear dif. eq. for u(y,z,t)
( )
.
u
0
0
(In general)
t
t
steady
∂
∂
→
=
=
∂
∂
1)
Parallel flow through a straight channel
Couette Flows
a) both plates stationary
x
y

2
2
dp
d u
dx
dy
µ
=
BC's
y=0
u=0
y=a
u=0
→
→
2
1
1
2
2
1
2
1
2
2
dp
0
.
dx
dp
1
dp
dx
2
dx
1
dp
1
dp
0,
0=
c
2
dp
2
dx
dx
2
dx
velocity profile
p
const
y
c
u
c
u
y
y
c
y
c
c
a
a
a
y
y
u
a
a
a
µ
µ
µ
µ
µ
µ
∂
=
⇒
=
∂
∂
⎛
⎞
⎛
⎞
=
+
→
=
+
+
⎜
⎟
⎜
⎟
∂
⎝
⎠
⎝
⎠
⎛
⎞
⎛
⎞
=
+
→
= −
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎡
⎤
⎛
⎞ ⎛
⎞
⎛
⎞
=
−
⎢
⎥
⎜
⎟ ⎜
⎟
⎜
⎟
⎝
⎠ ⎝
⎠
⎝
⎠
⎢
⎣
←
⎥
⎦
⎠
Shear stress distribution
1
1
2
2
yx
dp
dp
dp
y
y
a
a
dx
dx
dx
a
τ
⎛
⎞
⎛
⎞
⎛
⎞⎡
⎤
=
−
=
−
⎜
⎟
⎜
⎟
⎜
⎟
⎢
⎥
⎝
⎠
⎝
⎠
⎝
⎠⎣
⎦

Volume flow rate
3
0
2
.
(
)
12
1
/
12
a
A
l
dp
Q
V d A
u ldy
a
dx
dp
V
Q
A
a
dx
µ
µ
⎛
⎞
=
=
= −
⎜
⎟
⎝
⎠
⎛
⎞
=
= −
⎜
⎟
⎝
⎠
∫
∫
JG
JG
average vel.

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