^{1}and J. M. Montanero

^{1}

### Abstract

We analyzed experimentally the damping of both axial and lateral free oscillations of small amplitude in axisymmetric liquid bridges. We excited the first oscillation mode in nearly inviscid and in moderately viscous liquid bridges, and measured the parameters which characterize that mode. The axial spatial dependence of those parameters was determined, and the influence of the equilibrium shape on the oscillation frequency and damping rate was analyzed by considering liquid bridges with very different volumes. The experimental results were compared with the solution of the Navier–Stokes equations in the limit of zero viscous Capillary number and of two one-dimensional models. These theoretical approaches predicted accurately the axial spatial dependence of the parameters characterizing the oscillation mode. Comparison with the experimental data showed remarkable agreement for the oscillation frequency, while significant discrepancies were found for the damping rate.

This research was supported by the Ministerio de Educación y Ciencia (Spain) through Grant No. DPI2007-63559. Partial support from the Junta de Extremadura through Grant No. GRU07003 is also acknowledged.

I. INTRODUCTION

II. FORMULATION OF THE PROBLEM

III. METHODS

A. Experimental setup

B. Experimental procedure

C. Image processing and fitting methods

1. Equilibrium images

2. Dynamic images

3. Fitting method

IV. RESULTS

A. Temporal dependence

B. Axial dependence

C. Influence of the liquid bridge equilibrium shape

V. CONCLUSIONS

### Key Topics

- Viscosity
- 37.0
- Free oscillations
- 25.0
- Free surface
- 16.0
- Surface tension
- 15.0
- Liquid bridge flows
- 9.0

## Figures

Liquid bridge configuration.

Liquid bridge configuration.

Experimental apparatus: upper needle (A), bottom disk (B), liquid bridge cell (C), vibrating platform (D), electrodynamic shaker (E), camera (F), optical lenses (G), micrometer screws (H, I, and J), optical fiber (K), frosted diffuser (L), and optical table (M).

Experimental apparatus: upper needle (A), bottom disk (B), liquid bridge cell (C), vibrating platform (D), electrodynamic shaker (E), camera (F), optical lenses (G), micrometer screws (H, I, and J), optical fiber (K), frosted diffuser (L), and optical table (M).

Surface tension values measured from the images acquired before (open symbols) and after (filled symbols) the damping process of lateral oscillations of hexadecane (circles) and 35-cSt silicone oil (triangles). The symbols are the values averaged over the sets of ten images, and the error bars are the corresponding standard deviations. The dotted line indicates the literature value for hexadecane.

Surface tension values measured from the images acquired before (open symbols) and after (filled symbols) the damping process of lateral oscillations of hexadecane (circles) and 35-cSt silicone oil (triangles). The symbols are the values averaged over the sets of ten images, and the error bars are the corresponding standard deviations. The dotted line indicates the literature value for hexadecane.

(a) Two images of a hexadecane liquid bridge oscillating laterally. (b) Temporal evolution of the free surface position (symbols) at the height indicated in the images, and fit (1) (solid line) for the damping period. The parameters characterizing the experiment were , , , , and .

(a) Two images of a hexadecane liquid bridge oscillating laterally. (b) Temporal evolution of the free surface position (symbols) at the height indicated in the images, and fit (1) (solid line) for the damping period. The parameters characterizing the experiment were , , , , and .

Oscillation frequency and damping rate measured for consecutive periods of time of 83.3 ms as a function of the maximum free surface deformation measured in the corresponding period of time. The symbols are the averages over the intervals of the axis considered, and the error bars are the corresponding standard deviations. The open and filled symbols correspond to the axial and lateral oscillations of hexadecane, respectively. The parameters characterizing the experiments with axial and lateral oscillations were { , , , , and } and { , , , , and }, respectively.

Oscillation frequency and damping rate measured for consecutive periods of time of 83.3 ms as a function of the maximum free surface deformation measured in the corresponding period of time. The symbols are the averages over the intervals of the axis considered, and the error bars are the corresponding standard deviations. The open and filled symbols correspond to the axial and lateral oscillations of hexadecane, respectively. The parameters characterizing the experiments with axial and lateral oscillations were { , , , , and } and { , , , , and }, respectively.

Dependence of the oscillation parameters on for an experiment with axial oscillations of hexadecane. The symbols are the experimental data. The solid line in (a) is the solution to the Young–Laplace equation. The solid lines in (b)–(e) correspond to the Cosserat model. The dashed lines in (b) and (d) are the NS results. The dotted line in (b) is a function proportional to . The parameters characterizing the experiment were , , , , and .

Dependence of the oscillation parameters on for an experiment with axial oscillations of hexadecane. The symbols are the experimental data. The solid line in (a) is the solution to the Young–Laplace equation. The solid lines in (b)–(e) correspond to the Cosserat model. The dashed lines in (b) and (d) are the NS results. The dotted line in (b) is a function proportional to . The parameters characterizing the experiment were , , , , and .

Oscillation frequency and damping rate for an experiment with axial oscillations of 35-cSt silicone oil. The parameters characterizing the experiment were , , , , and .

Oscillation frequency and damping rate for an experiment with axial oscillations of 35-cSt silicone oil. The parameters characterizing the experiment were , , , , and .

Dependence of the oscillation parameters on for an experiment with lateral oscillations of hexadecane. The symbols are the experimental data. The solid line in (a) is the solution to the Young–Laplace equation. The solid lines in (b)–(e) correspond to the 1D slice model proposed in Ref. ^{ 16 } . The dashed lines in (b) and (d) are the NS results. The dotted line in (b) is a function proportional to . The parameters characterizing the experiment were , , , , and .

Dependence of the oscillation parameters on for an experiment with lateral oscillations of hexadecane. The symbols are the experimental data. The solid line in (a) is the solution to the Young–Laplace equation. The solid lines in (b)–(e) correspond to the 1D slice model proposed in Ref. ^{ 16 } . The dashed lines in (b) and (d) are the NS results. The dotted line in (b) is a function proportional to . The parameters characterizing the experiment were , , , , and .

Dependence of the oscillation parameters on for an experiment with lateral oscillations of 35-cSt silicone oil. The symbols are the experimental data. The solid line in (a) is the solution to the Young–Laplace equation. The solid lines in (b)–(e) correspond to the 1D results. The parameters characterizing the experiment were , , , , and .

Dependence of the oscillation parameters on for an experiment with lateral oscillations of 35-cSt silicone oil. The symbols are the experimental data. The solid line in (a) is the solution to the Young–Laplace equation. The solid lines in (b)–(e) correspond to the 1D results. The parameters characterizing the experiment were , , , , and .

Dependence of the oscillation frequency and damping rate on the liquid bridge volume for the axial oscillations of hexadecane. The left and right axes show the values of and and their normalized values and , respectively, where and are the oscillation frequency and damping rate obtained for the largest volume. The symbols are the experimental data calculated as the average over the liquid bridge free surface, while the error bars are the corresponding standard deviations. The solid and dashed lines correspond to the 1D and NS results, respectively. The parameters characterizing the experiment were , , , and .

Dependence of the oscillation frequency and damping rate on the liquid bridge volume for the axial oscillations of hexadecane. The left and right axes show the values of and and their normalized values and , respectively, where and are the oscillation frequency and damping rate obtained for the largest volume. The symbols are the experimental data calculated as the average over the liquid bridge free surface, while the error bars are the corresponding standard deviations. The solid and dashed lines correspond to the 1D and NS results, respectively. The parameters characterizing the experiment were , , , and .

The same as in Fig. 10 but for lateral oscillations. The parameters characterizing the experiment were , , , and .

The same as in Fig. 10 but for lateral oscillations. The parameters characterizing the experiment were , , , and .

The same as in Fig. 10 but for lateral oscillations of 35-cSt silicone oil. The parameters characterizing the experiment were , , , and .

The same as in Fig. 10 but for lateral oscillations of 35-cSt silicone oil. The parameters characterizing the experiment were , , , and .

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