This paper deals with the dynamic response of long cylindrical viscous liquid bridges subjected to an oscillatory microgravity field whose frequency varies linearly with time. The problem has been solved by using a one-dimensional model for the dynamics, derived from Cosserat theory for continuum, in which the axial velocity is considered to be constant over each cross-section of the liquid column. The dynamic response of the liquid bridge has been obtained by applying the Laplace transform to the problem formulation. The results obtained show that a variable -frequency excitation could give rise to erroneous measurements of the resonance frequencies of viscous liquid bridges

Meseguer Ruiz, José

Perales Perales, José Manuel

English

Archivo Digital UPM

A linear analysis of g-jitter effects on viscous cylindrical liquid bridges,

A.: A theoretical approach to impulsive motion of viscous liquid bridges,

Axial forcing of an inviscid finite length fluid cylinder,

Axisymmetric long liquid bridges in a timedependent microgravity field,

Axisymmetric long liquid bridges stability and resonances,

Frequency response of axisymmetric liquid bridges to an oscillatory microgravity field,

One-dimensional linear analysis of the liquid injection or removal in a liquid bridge,

Sensitivity of liquid bridges subject to axial residual acceleration, Phys. Fluids A 2,

The breaking of axisymmetric slender liquid bridges,

The influence of the outer bath on the dynamics of axisymmetric liquid bridges,

Theoretical and experimental study of the vibration of axisymmetric liquid bridges, Phys. Fluids A, in press

Non-steady Phenomena in the Vibration of Viscous Cylindrical Long Liquid Bridges

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

J. Meseguer and J. M. Per ales 
Non Steady Phenomena in the Vibration of 
Viscous, Cylindrical, Long Liquid Bridges1 
This paper deals with the dynamic response of long cylindri-
cal viscous liquid bridges subjected to an oscillatory micro-
gravity field whose frequency varies linearly with time. The 
problem has been solved by using a one-dimensional model 
for the dynamics, derived from Cosserat theory for contin-
uum, in which the axial velocity is considered to be constant 
over each cross-section of the liquid column. The dynamic 
response of the liquid bridge has been obtained by applying 
the Laplace transform to the problem formulation. The re-
sults obtained show that a variable -frequency excitation could 
give rise to erroneous measurements of the resonance fre-
quencies of viscous liquid bridges. 
1 Introduction 
The fluid configuration analyzed in this paper consists of a 
column of liquid spanning between two coaxial solid disks 
of the same radius, R<}, placed a distance L apart, as 
sketched in fig. 1. The liquid bridge is assumed to be 
isothermal and the properties of both the liquid (density, Q, 
and viscosity, v) and the interface (surface tension, a) are 
assumed to be uniform and constant, and the effects of the 
gas surrounding the liquid bridge negligible. Additional 
characteristics of the fluid configuration here analyzed are 
that the interface must remain anchored to the edges of the 
disks and that the volume of liquid equals that of a cylinder 
of radius R„ and length L. 
In the last decade a large number of papers dealing with 
the dynamics of liquid bridges have been published, most of 
them being devoted to the analysis of resonance phenom-
ena of such fluid configurations. Early studies [1,2] were 
related to the analysis of the frequencies of resonance 
(eigenfrequencies) whereas in the last years the attention 
has been focused on the analysis of the response of the 
liquid bridge when subjected to harmonic perturbations 
[3-8], and only a few attempts have been made to analyze 
non-harmonic perturbations [9-11]. In one of these papers 
[10] the dynamic response of long cylindrical liquid bridges 
when subjected to a small change in the value of the axial 
acceleration acting on the liquid bridge was calculated by 
V 
Fig. I. Geometry and coordinate 
system for the liquid bridge prob-
lem 
using a one-dimensional Cosserat model, widely used in 
capillary jet problems as well as in liquid bridge problems. 
Such a solution of the dynamics of the liquid bridge is 
used here to analyze several aspects of the liquid bridge 
dynamics which are of importance from the experimental 
point of view, namely the onset of the vibration of the 
liquid bridge and the forced vibration of the liquid bridge 
when the frequency of the perturbation varies with time. 
In the following all physical quantities are made di-
mensionless using the characteristic length R„ and the char-
acteristic time, tc = ((?/?'/cr)"2. The set of dimensionless 
parameters defining the fluid configuration are the slender-
ness, A = LI(2R„), the capillary number C = V(QI<TR„)112, 
the Bond number B — QgRlJa, g being the axial accelera-
tion, and the dimensionless volume of liquid, V = 2nA, 
which has been made dimensionless with Rl. 
2 Theoretical Background 
Under the assumptions done, the set of nondimensional 
differential equations and boundary conditions for the 
axisymmetric, non rotating, viscous flow, according to the 
Cosserat model are [10]: 
S, + Q:=0, (1) 
DS-l\s D:~^(QIS); 
-SP;-l-C[S2(QIS)::l: 3C[5(S/5)=] (2) 
69 
where 
D = Q, + (Q
2IS); (3) 
P = 4(25 + 5 2 - 55 I 2)(45 + S2:)-y2 + B{t)z. (4) 
In these expressions 5 = F2 and Q = SIV, F(z, t) being the 
dimensionless equation of the liquid gas interface and 
W{z, t) the dimensionless axial velocity which, in this 
model, is uniform in each plane parallel to the disks. 
Boundary conditions state that the interface must remain 
anchored to the disk edges, S{±A,t) — \, and that the 
axial velocity is zero at each one of the disks, Q( ± A, t) = 0, 
whereas the initial conditions are that the liquid bridge is at 
rest at / = 0, Q(z, 0) = 0, its shape being that of a cylinder, 
S{z, 0) = 1, which implies 5 ( 0 = 0 for / < 0. 
In [10] it was assumed that the Bond number changes at 
t = 0 from the initial value B = 0 to a new one B = e, e « 1, 
that is B{t) = eJf(t), ^ ( 0 being the Heaviside function 
(yf = 0 for t < 0 and J f = 1 for t > 0). In view of such 
perturbation the variables were expanded as 5 = 1 + 
«„(z, t) + O(e), Q = eq(z, t) + O(e). The introduction of 
these expressions in the problem formulation, neglecting 
second order terms, and the elimination of the variable 
s„(z, t) from the momentum equation by using the continu-
ity equation allows to formulate the problem in terms only 
of the variable q(z, t): 
<1„ ~ g q„zz + 2 <?---- + 2 ^ " + g Cq,"~-t ~ 3 Cq,~ = -yf> (5) 
boundary conditions being q(±A,t)=0, q.(±A, t) =0, 
and initial conditions q{z, 0) = 0, q,(z, 0) = 0. 
The application of the Laplace transform to the last 
equation allows to calculate the liquid bridge response in 
the Laplace domain as well as in the time domain by using 
the Inversion Theorem, in such a way that the dependence 
of the liquid bridge interface on the dimensionless time is 
given by 
s„(z,t)=2\z-- A 
sin A sin z 
D (- h ) 
(6) 
where 
D(z, h) = 02 sinh 02A cosh 0,z - 0, sinh 0XA cosh 02z, (7) 
Dizji) =0 ,0 2 ( s inh0 2 / l sinh 0,z - sinh 0,/l sinh 02z), (8) 
and 
d£)(/l, /;) 
ii = ii„ 
D„(A,h„) d/i (9) 
In these expressions h„ are the roots of D(A, Ii) = 0, which 
in general are complex, //„ = y„ + iwn\ 0, and 02 are related 
to /; and C through the characteristic equation 
+ -hC)04 + [ 1 --h2~6hC)02 + 2h2- •0. (10) 
Once the liquid bridge response to a perturbation consisting 
of a sudden change in the value of Bond number (B = s Jf) 
is calculated, s„(z, t), the response s{z, t) to any variation of 
Bond number B(t) = i:b(i) can be calculated by applying the 
Duhamel's theorem 
s{z, t) = b(t)su(z, t) + /;(T).V,„(/ - r) dr (11) 
0 
and taking into account that ,v„(z, 0) = 0 finally results 
/;(r)cxp [/;„(/-!)] dr. (12) * . o - i D',;''° 
• l>lD„(AJi„) } 
3. Results 
The considered perturbation has been b(l) = sin at2, that is, 
the axial microgravity varies sinusoidally with a frequency 
that increases linearly with the time (note that the instanta-
neous frequency is defined as co(f) = 2a/). Such a kind of 
perturbation could be considered as an idealization of the 
one available in the Fluid Physics Module, one of the 
multiuser experimental facilities provided by the European 
Space Agency for fluid experimentation aboard space plat-
forms like Spacelab (this equipment has been flown in past 
Spacelab missions and it will be flown in the next Spacelab-
D2 mission). In the Fluid Physics Module one of the disks 
supporting the liquid bridge can be vibrated at constant 
amplitude with frequencies that change linearly with time, 
the minimum rate being 1/90 Hzs~ ' . Of course this kind of 
perturbation is not exactly equal to a time variation of the 
Bond number, the main difference being that, as stated 
elsewhere [6] an oscillatory Bond number only excites non-
symmetric oscillation modes (in respect'to the middle plane 
parallel to the supporting disks) whereas the vibration of 
one of the disks excites both non-symmetric and symmetric 
oscillation modes. However this difference is not relevant if 
the interest is mainly focused on the' measurement of the 
frequency of resonance corresponding to the first oscillation 
mode, so that in the following the analysis is kept within 
this boundary. 
Since there are a large number of parameters involved in 
the theoretical model, it is convenient to fix the values of 
some of them in order to limit the boundaries of the study. 
Therefore, taking into account that silicone oils are nor-
mally used as working liquid for liquid bridge experi-
mentation either on Earth or in space laboratories, we 
have selected the physical properties of these liquids 
(Q » 103 kg m~3, <T« 0.02 N m~', and viscosities ranging 
from 10~ 6 m 2 s _ i to 10~5m2s~') to estimate the values of 
the parameters involved. Then, assuming that the radius of 
the disks is R„ = 0.015 m, the value of the characteristic 
time becomes t,. x 0.4 s whereas the parameter of viscosity 
ranges from 10~3 to 10"2. On the other hand, taking into 
account the above mentioned value of the frequency ramp 
in the Fluid Physics Module, one gets a x 0.005. 
The introduction of the selected law for the time varia-
tion of Bond number (b(t) = sin at2) in expression (12) 
allows to calculated the time variation of the liquid bridge 
interface. Unfortunately, for the selected perturbation the 
integrals appearing in (12) must be computed numerically, 
and some care is needed during such calculations in order 
to avoid aliasing errors. To measure the response of the 
liquid bridge to the imposed perturbation the adopted crite-
rion has been to calculate the variation with time of the 
diameter of the interface at the section situated at a quarter 
of the total length, s(A/2, t); note that this section is close 
to the one where the deformation of the liquid bridge 
interface becomes maximum. As already stated, the atten-
tion has been mainly focused on the resonance correspond-
ing to the first oscillation mode. 
The response of a liquid bridge with A =2 and C = 0.05 
to a perturbation characterized by the value a = 0.005 is 
shown in fig. 2. In this plot, as well as in the following 
figures, the abscissae are the dimensionless pulsation, co = 
2at instead of the dimensionless time. As it can be observed 
the amplitude of the deformation of the liquid bridge inter-
face increases as the frequency grows until a relative maxi-
mum appears (which roughly corresponds to the first 
resonance), then the amplitude decreases and it will increase 
again close to the next relative maximum (that correspond-
ing to the third resonance, not shown in fig. 2, according to 
the type of perturbation here considered) and so on. 
For frequencies below the maximum of the interface 
deformation, the liquid bridge response is a wave of 
monotonically increasing amplitude and frequency, but the 
behaviour is rather different for frequencies larger than the 
resonance. The reason for this difference is that once the 
resonance is excited the liquid bridge starts to oscillate with 
its natural frequency (although, because of viscosity, the 
amplitude of such oscillation decreases with time) and at 
the same time the liquid bridge is subjected to a forced 
oscillation whose frequency increases as the time grows 
(such frequencies are greater than the one corresponding to 
the resonance). 
\p_ooi 
C=0.05 
\ 0.005 
0.001 0005 C=0.10 
0.* 0 6 0.8 i.o 1.2 
Fig. 3. Variation with dimensionless pulsation, co (or dimensionless 
time, t = ml2a) of the amplitude of the deformation of the liquid . 
bridge interface at the section z = A\2, A = max \s(A\2, l)\. The 
results correspond to liquid bridges with slenderness A=2 and 
different values of the viscosity parameter C subjected to a perturba-
tion whose frequency varies linearly with time with a sweep a. 
Numbers on the curves indicate the value of a 
s(A/2,t) , 1 
\l 
j 
u In 
i 
V 
<&^^ 
/ 5 ^ 0 
/J^^ 
0.06 
Fig. 2. Variation with dimensionless pulsation, w (or dimensionless 
time, t = co 12a) of the deformation of the liquid bridge interface at 
the section z = Aj2, s(Aj2, t). The results correspond to a liquid 
bridge with slenderness A = 2 and viscosity parameter C = 0.05 
subjected to a perturbation whose frequency varies linearly with time 
with a sweep a = 0.005 
Fig. 4. Variation with the frequency sweep a and the parameter of 
viscosity C of the dimensionless pulsation of resonance corresponding 
to the first oscillation mode co,. The results correspond to a liquid 
bridge with slenderness A =2 
71 
Another characteristic of the liquid bridge response to be 
pointed out is that a criterion based on the maximum 
interface deformation for the measurement of resonances 
could lead to unacceptable errors in an experiment with a 
procedure similar to the one here analyzed. In effect, if this 
criterion were adopted to analyze the results shown in fig. 2, 
the measured dimensionless resonance pulsation corre-
sponding to the first oscillation mode would be o>, «0.90 
instead of the correct value to, = 0.73 which results from a 
harmonic excitation analysis [6]. This shift on the apparent 
frequency of resonance depends on the viscosity of the 
liquid as well as on the frequency sweep a, as shown in fig. 
3, where the liquid bridge responses of the same fluid 
configuration (/I = 2) with different viscosities and different 
frequency ramps have been represented. The curves shown 
in fig. 3 are the locii of the maxima of the absolute value of 
,y(/l/2, t), so that they represent the variation with time of 
the amplitude of the deformation of the radius at the 
section z = /1/2 (according to the definition S = 1 + es = 
F2 = (l + r / ) 2 = 1 +2e.f+0(s2), so that, within this ap-
proximation s(z, t) = 2f(z, t), f(z, t) being the expanded de-
formation of the radius of the interface at section z). Note 
that, in order to compare the effect of the frequency ramp 
of the excitation, in the abscissae frequency instead of time 
has been represented, that means that, since / = w/2a, in 
each of the plots each one of the curves has a different time 
scale (depending on the value of a) as illustrated in the 
insert. 
Finally, the influence of both a and C on the pulsation of 
resonance, defined as the pulsation for which the interface 
deformation is maximum, is shown in fig. 4. Note that the 
differences between the measured frequencies and the real 
frequency of resonance increases as the frequency ramp 
increases, and that the contrary occurs with the viscosity. 
Acknowledgement 
This work has been supported by the Spanish Ministerio de 
Educacion y Ciencia (S.G.C.I) under grants HA-19B and HA-010. 
References 
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6. Perales, J. M., Meseguer, J.: Theoretical and experimental 
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Non-steady Phenomena in the Vibration of Viscous Cylindrical Long Liquid Bridges

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