Ain Sonin and Gareth McKinley
Solutions to Quiz 2
2.25
Fall 2004
Problem 1
The transition from dripping to jetting
See C. Clanet & J. Lasheras, “Transition from Dripping to Jetting”,
J. Fluid Mech
383
, 1999 p307
for more details on this problem!
Figure 1
Figure 2 (snapshot at a single instant in time
t
)
For fluids exiting from typical size orifices, viscous stresses are negligible because a
dimensionless parameter known as the
Ohnesorge number
is small. This is defined as
Oh
=
μ
ρσ
D
. For the case here we thus have
Oh
≈
10
−
3
10
3
(0.07)(0.005)
≈
0.0017
!
(a)
The surface is cylindrical and there is no tangential stress (no viscous effects). Normal to the
jet we have pressure and surface tension acting. The principal radii of curvature for a cylinder
are such that the mean curvature is given by:
2
H
=
1
R
a
+
1
R
b
=
1
(
D
2)
+
1
∞
=
2
D
(1)
hence
p
i
n
=
p
a
+
2
H
σ
( )
n
→
p
i
=
p
a
+
2
D
(2)
The additional axial force arising from surface tension acting along the axial direction of the
jet is
F
z
=
π
D
(remember surface tension is a line force – proportional to length).
(b)
Criterion:
The flux of momentum into the control volume shown must always be greater
than zero
(otherwise a stationary pendant drop will have formed which is attached to the
orifice).
The ‘A form’ of the conservation of linear momentum is most appropriate; this gives
d
P
cv
dt
+
ρ
v
(
vv
c
)
⋅
n
dA
CS
(
t
)
∫
=
F
cv
∑
(3)
g
A
B
U
=
Q
4
D
2
( )
L
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 Fall '05
 GarethMcKinley
 Fluid Dynamics, Mass, pendant drop

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