The objectives are to: (1) Understand the influence in low gravity of flow on interface shape. For example, document and control the influence of axial flow on the Plateau-Rayleigh instability of a liquid bridge; and (2) Extend the ground-based density-matching technique of low gravity simulation to situations with flow; that is, develop Plateau chamber experiments for which flow can be controlled. Containerless containment of liquid by surface tension has broad importance in low gravity. For space vehicles, the behavior of liquid/gas interfaces is crucial to successful liquid management systems. In microgravity science, free interfaces are exploited in various applications. Examples include float-zone crystal growth, phase separation near the critical point of liquid mixtures (spinoidal decomposition) and quenching of miscibility gap molten metal alloys. In some cases, it is desired to stabilize the capillary instability while in others it is desired to induce capillary breakup. In all cases, understanding the stability of interface shape in the presence of liquid motion is central

Steen, Paul H.

English

NASA Technical Reports Server

N95- 14538 
Influence of Flow on Interface Shape Stability in Low Gravity 
Paul H. Steen (PI) 
School of Chemical Engineering 
Cornell University 
Ithaca, NY 14853 
Science objectives 
1)Understand the influence in low gravity of flow on interface shape. For example, document and control 
the influence of axial flow on the Plateau-Rsyleigh instability of a liquid bridge. 
2)Extend the ground-based density-matching technique of low kravity simulation to situations with flow; 
that is, develop Plateau chamber experiments for which flow can be controlled. 
Relevance of science and potential applications 
Containerless containment of liquids by surface tension has broad importance in low gravity. For space 
vehicles, the behavior of liquidlgas interfaces is crucial to successful liquid management systems. In 
microgravity science, free interfaces are exploited in various applications. Examples include float-zone 
crystal growth, phase separation near the critical point of liquid mixtures ( spinoidal decomposition) and 
quenching of miscibility gap molten metal alloys. In some cases, it is desired to stabilize the capillary 
instability while in others it is desired to induce capillary breakup. In all cases, understanding the 
stability of interface shape in the presence of liquid motion is central. 
Research approach 
Both analytical/numerical and experimental approaches are employed. 
Stability analyses include linear and nonlinear techniques. The linear stability approach has been used 
to analyze the shape stability of a cylindrical interface containing axial shear flows, both isothermally- 
and thermocapillary-driven[l, 21. Computational feasibility currently limits this approach to base states 
that are separable flows, effectively, the axial-infinite interfaces. It is now well-known that infinite cylin- 
drical interfaces can be stabiliredE3, 4, 51. For finite interfaces an alternative approach is needed. In 
the limit of no motion, minima of the free-energy functional are obtained using the calculus of varia- 
tions supplemented by numerical branch-tracing/b]. For weak motion (creeping flow), we extended this 
approach below using a modified functional. Near the singularity represented by the Plateau-Rayleigh 
limit, bifurcation theory using Liapunov-Schmidt reduction is a natural tool for the solution of the ap- 
propriate nonlinear Euler-Lagrange equation. All these analytical/numerical tools lend themselves to 
understanding the physics of stability in terms of simple competition mechanisms. 
As for the experimental approach, a dynamic Plateau chamber has been built and is used to study 
liquid bridges held captive by rod-ends and embedded in a controlled surrounding flow. Theory has guided 
the experiments to a particular window in parameter space. Such guidance is crucial since interesting 
stabilization effects occur over narrow parameter ranges for this problem. 
3.07 
https://ntrs.nasa.gov/search.jsp?R=19950008124 2020-06-16T10:24:12+00:00Z
Science results 
The stability of liquid bridges of finite extent near the Plateau-Rayleigh length is the main subject of 
this report. The bridge is sheared d y  by embedding it in a pipe flow. Finite amplitude deviations 
from cylindrical shape are accounted for by a nonlinear theory. The predictions so far are limited to 
small deviations from no flow, no gravity, and the Plateau-Rayleigh limit. They explain the experimental 
observation of a slight stabilization to lengths beyond the Plateau-Rayleigh limit. 
The schematic of the experimental configuration is shown in figure 1. The bridge liquid attaches to 
each rod end with a cicular contact-line of radius T,. A smaller diameter rod (radius ~ i )  connects the 
two end rods. This annular geometry influences the pressure at the interface when the bridge liquid ia 
in motion. In the absence of motion, the interface shape and its stability are controlled by dimensionless 
bridge length L, volume 7, and the gravity to surface tension strength B (Bond number). The strength 
of gravity is controlled by the density imbalance Ap (bridge - surrounding liquid). In the absence of 
motion the connecting rod plays no role. By controlling the flow rate of the external liquid, between the 
outer tube wall (radius ~ t )  and the interface, traction at the interface transmits motion to bridge liquid. 
In summary, with motion, the shape and stability of the inteiface depend on a dimensionless flow rate 
C (capillary number), two radius ratios and the viscosity ratio in addition to the three parameters that 
determine shapes and stability under static conditions. 
Upward flow is driven by a pressure gradient that opposes the hydrostatic gradient of a heavy bridge. 
This is the interesting case. In one protocol, B and C are fixed and the bridge length is quasistatically 
increased maintaining a cylindrical volume (7 = 1). Breakup is recorded at length 4,. The experiment 
is repeated for a different C. The symbols in figure 2 show the results. The opposing effects of gravity 
and flow are evident: a maximun length occurs near B = C. Flow effectively cancels density imbalance. 
The solid curve is the destabilization by gravity (theory) from the Plateau-Rayleigh limit (L, = 2 r )  in 
the absence of motion (C = 0). Unexplained is the apparent stabilisation to length greatel. than 2 r .  
A Werent experimental protocol fixes the bridge length for given B. A volume is chosen and the 
flow rate C is increased/decreased until breakup occurs. Figure 3 show these results. The data are 
connected by a solid line representing interpolation (dotted indicates extrapolation). Note that as the 
bridge length approaches 27r the effects of flow on breakup become n o h e a r  and, indeed for L = 6.099, 
there is apparently an 'island' of stability for cylindrical volumes: bridges exist in the presence of flow 
that would otherwise breakup. Details of these experiments are found 4 7 ,  8, 91. 
The nonlinear analysis is based on a model that accounts for the external flow by imposing a shear 
stress r on the interface of surface tension Q. The dissipation of this single phase configuration with 
deformable interface is minimixed. The normal stress balance is a modified Young-Laplace equation and 
also corresponds to the nonlinear Euler-Lagrange equation of a modified functional. Solutions of the 
nonlinear equation with their stabilities are obtained by Liapunov-Schmidt reduction using bifurcation 
theory. Perturbation parameters measure deviations from no flow, no gravity and the Plateau-Rayleigh 
limit. The following dimensionless groups arise (cylindrical volume is assumed): 
is treated as the primary bifurcation parameter. 7 = (E, B) are the perturbation unfolding parameters. 
In the equlibrium state (7 = 0), the Young-Laplace equation is recovered which has a linear operator 
with null space 4 spanned by sin(k7rx). The bifurcation diagram shows a subcritical pitchfork structure. 
The nontrivial branch near the singular point takes the form: 
€4 + W(.). (5) 
108 
W ( Z )  is in the complement of the null space and is o(E). The bifurcation equation is g = 0 and the 
universal unfolding is 
(6) 
A€ 3r3 
9 = -- - - + (Cp - B)  + Cp?, 2 32 
C = Cp(A) 
Details of the analysis and the form of functions p(A) and q(A) are given elsewhere[lO]. The classical 
subcritical pitchfork is recovered for no flow and no gravity (C = B = 0). For gravity ( B  # 0) and 
no flow (C = 0) the perturbation results oflll] are recovered. The presence of creeping flow without 
gravity is always destabilising (B = 0, C # 0). Thus, the physics at work here is Merent from that in 
the infinite cylinder where inertial effects are responsible for stabilisig pressures. There, finite Reynolds 
number flows can can stabilize without gravity[l]. Our main result is the perturbed bifurcation structure 
with flow and gravity for B = C. This case shows somewhat surprisingly that even though each of two 
effects is separately destabhing, together they can stabhe. The maximum stability limit is plotted in 
figure 4 ( A  = 0.5) as a function of imposed shear and gravity.imbalance. We see that the destabilizing 
effect of gravity can be overcome by the pressure field induced by the shear flow. In the operating regime 
where the stability limit is extended beyond 2r, the liquid column does not have a perfect cylindrical 
shape. The predictions of figure 4 are to be compared to the data of figure 2. 
Theory (additional results from the past year - periferal to the above disscusion) 
1) Stability (instability) of a static bridge equilibrium ( B = 0 ) is immediate once the family of equilibria 
to which it belongs is identified; direct calculation of the second variation is circumvented[6]. 
2) All known families of static bridge equilibria ( B = 0 ) are ultimately connected and thereby inherit 
their states of instability ( number of unstable modes) ultimately from the stability of the sphere[b]. 
Experiment(additiona.l results from the past year - periferd to the above disscusion) 
3) A comparison of pairs of relatively immiscible liquids suitable for use in a dynamic Plateau chamber 
with density balance within loe4 g l m 3  has been presented. Pure water and the isomeric system of 2-, 
3- and Il-fluorotoluene is one preferred pair. Pure water and the homologous system of chlorocyclohexane 
and chlorocyclopentene is another[12]. 
4) Further observation and analysis of the collapse of the soapfilm bridge have been performed. The 
soapfilm collapse is viewed as a prototype collapse. 
Research plan 
Theory 
The nonlinear analysis presented above is limited by the single phase assumption of the model. Moreover, 
even with the model, there is limited validity in amplitude e, length deviation X and C and B. We shall 
attempt to remove these limitations. Another goal is to make precise the relationship between o w  , 
approach and that of energy stability theory (with linking parameters). 
Experiment 
Further tests of stabhation guided by the new theoretical results are planned. Exploratory experiments 
peill probe the influence of a time-dependent driving flow on the stability limit of bridges, and the 
coalesence of droplets in the presence of flow. 
109 
Conclusion 
We have shown in theory and in experiment that shear can stabilise a finite liquid column beyond 
the Plateau-Rayleigh limit, among other results. Surpriseingly, a slight density imbalance is required. 
Two imperfections (density and flow) conspire to stabilise even though each on its own destabilizes. 
Apparently, only lengths slightly longer than the Plateau-Rayleigh length are achievable. 
References 
[l] M.J. Russo and Steen P.H., "Shear Stabilktion of the Capillary Breakup of a Cylindrical Interface," 
[2] H.A. Dijkstra and Steen P.H., " Thermocapillary Stabilisation of the Capillary Breakup of an 
Phys. Fluids A, 1 (1989) 1926-1937. 
Annular Film of Liquid," J. Fluid Me&., 22$ (1991) 205-228. 
[3] T. W. F. Russel and Charles M.E. "The Effect of the less Viscous liquid in the Laminar Flow of 
Two immisicible Liquids," Can. J. Chem. Engng., 39 (1959) 18-24. 
[4] A.L. Frenkel, Babachin A.J., Levich B.G., Shlang T., and Sivashinsky G.I., " Annular Flows Can 
Keep Unstable Films from Breakup: Nonlinear Saturation of Capillary Instability," J. Colloid In- 
terface Sci., 115 (1987) 225-133. 
[5] H.H Hu, Lundgren T.S. and Joseph D.D., "Stability of Coreannular Flow with a Small Viscosity 
Ratio," Phys Fluid A, 2 (1990) 1945-1954. 
cation to the Liquid Bridge," Proc. Roy. SOC. A, accepted. 
[6] B. J. Lowry and Steen P. H.,"Capillary Surfaces: Stability from Families of Equilibria with Appli- 
[7] B.J. Lowry, Ph.D. Dissertation, Cornell University, Ithaca NY, 1994. 
[8] B.J. Lowry and Steen P.H.,"Stabilisation of an Axisymmetric Liquid Bridge by Viscous Flow," Int. 
J. Multiphase Flow, 20(2) (1994) . - 
[9] B. J. Lowry and Steen P. H.,"Flow-influenced Stabilisation of Liquid Columns in a Dynamic Plateau 
[lo] Y. J. Chen and Steen P. H.,"Influence of Flow on Interhe Shape Stability," Proceedings of the 14th 
Chamber," AIAA Paper 93-0255 (1993). 
IMACS World Congress, Atlanta, GA, July 1994. 
Ell] J.M.Vega and J.M.Pearles,"Almost Cylindrical Isorotating Liquid Bridges for Small Bond Numbers," 
ESA SP-191 (1983) 247. 
[12] B. J. Lowry and Steen P. H.,"On the Density Matching of Liquid using Plateau's Method," AIAA 
Paper 940832 (1994) 
110 
I 
Figure 1 Definition sketch of the espexhent. 
6.40 
2% 
6.20 
=c 
6.00 
-0.010 -0.005 0.000 0.005 0.010 
B-C 
2 - dotailof L O B - 0  peakfor 0 - 1.00f0.02 md 
B I 0.0029 with comparison to rWie LOB) E w e  (80% 
4-Fr, 2m2-m, ‘t = 2*130r* ) 
2.4 
2.2 
2.0 
iiT 
1.8 
1.6 
1.4 
1.2 
6.3- 
6.28 
6.26 
6.24 
0.002 
B = 0.0 1 - 1 
- 
- 
6.38 
6.36 
6.34 
6.32 
6.4 1 1 1 1 
0.008 
1 
i 0.c 
0.004 
C 
6 
Figure 4 Critical length L, of bridge versus C 
A = 0.5 
112 


Influence of flow on interface shape stability in low gravity

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