u
v
w
x
y
z
∂
∂
∂
+
+
=
∂
∂
∂
particles of constant volume,
but shape of volume can
change.

STREAM FUNCTION
ψ
2-D , steady flow : continuity
(
)
(
)
0
u
v
x
y
ρ
ρ
∂
∂
+
=
∂
∂
steady compressible or unsteady incompressible
Define stream
ψ
function such that:
,
u
v
y
x
ψ
ψ
ρ
ρ
∂
∂
=
= −
∂
∂
2
2
0
x y
y x
ψ
ψ
∂
∂
−
=
∂ ∂
∂ ∂
continuity identically satisfied.
ψ
: first & second order der. exist & continuous
Advantage
• Continuity eq. discarded
• # of unknowns (dependent variables) reduces by one.
Disadvantage
• Remaining velocity derivatives are increased by one order???

( ,
)
d
=
.
x
y
x y
dx
dy
vdx
udy
V ds
dm
ψ
ψ
ψ
ψ
ψ
ρ
ρ
ρ
∂
∂
=
+
= −
+
=
=
∂
∂
JG
G
¸
(
)
0
dm
=
¸
Physical significance of
ψ
•
Lines of constant
ψ
(d
ψ
=0) are lines across which mass flow
They are stream lines of flow
is zero
ds
dxi
dy j
=
+
G
G
G
Along AB x=const.
d s
dyi
=
G
G
0
.
B
B
B
B
A
A
A
A
y
y
y
B
A
y
y
y
V
ui
v j
m
V ds
udy
dy
d
y
ψ
ψ
ψ
ρ
ρ
ψ
ψ
ψ
=
+
=
∂
=
=
=
=
=
−
∂
∫
∫
∫
∫
JG
G
G
JG
G
¸
ρ
=const.