Transcript DINAMICA DE PUENTES LIQUIDOS CASI INESTABLES

EXPERIMENTAL ANALYSIS OF THE
BREAKAGE OF A LIQUID BRIDGE
UNDER MICROGRAVITY CONDITIONS
I. Martínez, J.M. Perales
Universidad Politécnica de Madrid, Spain
COSPAR 2010
Thursday, 22nd July 2010
Spacelab-D2, Experiment “STACO”, Run 2
Experiment description
• Experiment performed on board Spacelab D-2 (1993) with AFPM.
• Fluid used: silicone oil of n=10 cSt, r=920 kg/m3, s=0.020 N/m.
• Supports: two 30 mm in diameter circular coaxial disks made of
aluminium black-anodized, with a 30º dove-tail cut back.
• Nominal shape: cylindrical liquid column with L=85 mm length.
• Diffuse white background illumination (9·8 leds).
• Ref.:
• Martínez, I., Perales, J.M., Meseguer, J., Stability of long liquid
columns (SL-D2-FPM-STACO), in Scientific Results of the
German Spacelab Mission D-2, Ed. Sahm, P.R., Keller, M.H.,
Schiewe, B, WPF, pp. 220-231, 1995.
• Martínez, I.,
http://webserver.dmt.upm.es/~isidoro/lc1/SL/ST2_118_13_30_41stretch_xvid.avi
Automated image edging
Stability diagram for unloaded liquid bridges
Stretching evolution at constant volume from A to B
Experimental values 09/07/2010
(length, volume, stability)
Event
Liquid column idle
Start of stretching
Stability limit
End of stretching
Breaking (last bridge)
GMT [s]
118:13:28:23
118:13:30:39
118:13:31:02
118:13:31:07
118:13:31:16
t [s]
L [mm] L/(2R) =/1

85.0
2.83
0.098
36.5
85.0
2.83
0.098
14.5
90.9
3.03
0.034
9.5
93.9
3.13
0.004
0
93.9
3.13
0.004
vV/(R2L)1
0.005
0.005
0.070
0.099
0.099
100
L [mm]
90
80
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
-45
-40
-35
-30
-25
-20
time t [s]
-15
-10
-5
0
64
62
V [cm3]
60
-50
0.1
s
0
-0.1
-50
1
sv/2


0
0.046
0.046
Non-dimensionalization
R0  15 mm
Unit length
Unit time
Slenderness
r R03
920  0.0153
t0 

 0.394 s
s
0.020

L
85

 2.83
2 R0 2 15
Reduced slenderness


2.83
1 
 1  0.098


Excess liquid volume
v
V
59.8

1

 1  0.005
R02 L
 52  8.5
Column shapes and their stability (nondimensional)
• Equilibrium shapes with v<<1 (linearized) and ~π
v
z
r  z   1  1  cos 
2
 
  z  
• Dynamic shapes (first eigenfunction)
v t  
z 
z
r  z, t   1 
1

cos

a
t
sin



2 
 t  
 t 
• Stability limit
v
 3 2



 a  0
2
 4
Fitting the liquid shape with 1-, 2-, 3-terms
r  z, t   1 
v t  
z 
z
1

cos

a
t
sin




2 
  t  
 t 
ri  zi   ri ,mean  b  t  cos
2 zi
2 zi
 a  t  sin
zi ,bottom  zi ,top
zi ,bottom  zi ,top
100
100
50
50
A1=1.9 px, B1=3.5 px
0
0
100
200
300
A1=33 px, B1=18 px
400
500
600
100
0
0
50
300
400
500
600
0
100
200
300
400
500
600
400
500
600
A2=-9.4 px, B2=-2.8 px
400
500
600
100
0
0
100
200
300
100
50
50
A3=0.0055 px, B3=-0.046 px
0
200
50
A2=-0.015 px, B2=-0.16 px
0
100
100
0
100
200
300
A3=1.7 px, B3=-2.4 px
400
500
600
0
0
100
200
300
10
Stability margin: 100x s
5
b
[pixels]
a
0
-5
-160
-140
-120
-100
-80
time t [s]
-60
-40
-20
0
Evolution of the first sine term (a) and cosine term (b)
10
5
[pixels]
a
b
0
-5
-50
-45
-40
-35
-30
-25
-20
time t [s]
-15
-10
-5
0
Necking dynamics
2
r(z)
1
0
rmin(t)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
-50
1
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
-45
-40
-35
-30
-25
-20
-15
time t [s]
-10
-5
0
1
0.5
z rmin(t)
0.5
Vfraction(t)
0
-50
1
0.5
0
-50
Stability margin
z
r  z , t   req  z   a (t ) sin
Λ
d 2 a(t )
da(t )  v
3

3
m

mC



a
(
t
)

a
(
t
)
0


2
dt
dt
4
2

5
4
amax,cyl
[mm]
3
amax
2
1
a
0
85
86
87
88
89
90
L [mm]
91
92
93
94
95
Linear inviscid stable and unstable response

if
v



ma      a  0 
2

if


m
v

     0, T0  2
  v / 2
2

m
v

     0, T2  ln 2
 v/ 2
2

30
25
20
[s]
15
T0
10
5
0
85
T2
86
87
88
89
90
L [mm]
91
92
93
94
Diverging amplitude (amphora-type deformation)
30
2
d𝑎
d𝑎
»𝑚 2 + 𝑚𝐶 +
d𝑡
d𝑡
𝑣
2
3
4
− 𝜆 𝑎 − 𝑎3 = 0,
with
𝑎 𝑡0 = 𝑎0
𝑎 𝑡0 = 𝑎0
25
20
a [mm]

t 
a  a0 exp  ln  2  
T2 

15
10
5
0
-10
-9
-8
-7
-6
-5
-4
time t [s]
-3
-2
-1
0
Conclusions
• Non-linear dynamic simulation of the breaking process has yield a perfect
matching with experimental results, which linear theories did not achieve.
• Many small details in the experimental results are still unexplained (e.g. the
lack of decay in the small free oscillations; g-jitter?).
• Automated image analysis has progressed a lot, but small problems remain (full
image analysis helps a lot, but details of the discs are not visible).
• Digital imaging nowadays would solve many of the old video problems.
• Actual liquid column in space appear always oscillating (microgravity):
– Around an equilibrium shape that is unexpected (a residual load shows up)
– With a small but non-decaying amplitude (0.3 mm peak-to-peak)
– With a frequency very close to the first natural frequency (axial, and lateral).
• Useful experimental time in space is always very scarce (e.g. a couple of
minutes in half an hour, here).
• Unique experiments may have some unknown boundary conditions (repetition
is a must, but these experiments have not been reproduced yet).