Resonance of droplets in constricted capillary tubes: Critical factors and nonlinearity

Resonance of droplets in constricted capillary tubes responding to external vibratory excitation can be theoretically characterized by the abstract coefficients such as resonant frequency and rate of damping. A physically sound model, however, is needed to relate these abstract coefficients to the fluid properties of two-phase fluid, geometric properties of tube, and features of excited waves. A hydrodynamic model, based on the moving-boundary control volume concept and the transient oscillatory velocity profile, was developed to characterize the resonance of droplets. This model was validated against computational fluid dynamics simulation results expressed in both time and frequency domains. Dominant factors including Ohnesorge number, viscosity ratio, density ratio, and aspect ratio which control resonance were systematically investigated for their influences on the resonance of droplets. The nonlinearity in light of the initial position of front droplet meniscus and the amplitude of oscillation was distinguished and validated by the theoretical model.

Figure 1

A single flow tube, showing variables and parameters. The fluid flows from left to right.

Figure 2

Comparisons of the displacement of the nonwetting phase with the theoretical model and the CFD simulation: (a) the frequency was $f=5\mathrm{Hz}$; (b) the frequency was $f=0\mathrm{Hz}$; and (c) the frequency was $f=12\mathrm{Hz}$. The solid line denotes the result of the CFD simulation, and the dashed line denotes theoretical result.

Figure 3

Comparison of (a) amplitude and (b) central position of the output displacement at steady state between the theoretical and CFD results. The blue line with circles shows the results of the CFD simulation, and the circles represent CFD data; the black line shows the theoretical results. The horizontal dashed line in (b) corresponds to the initially stationary center at $\varepsilon =-0.6299\mathrm{mm}$. The amplitude of the acceleration is $A=0.1\mathrm{m}/{\mathrm{s}}^2$. The theoretical results were acquired from 100 points in the interval between 1 and 20 Hz. The ultimate position at steady state is the sum of amplitude in (a) and the center position in (b). If this position is positive, then overshoot occurs.

Figure 4

(a) The ratio $r$ with respect to dimensionless exerted frequency $\tilde{f}$ as a comparison reference. The peak ratio $r_0$ and the associated resonant frequency ${\tilde{f}}_0$ are marked by the circle at the coordinate (2.25, 0.0645) in the plot. (b) Dependence of the maximum ratio $r_0$ and the resonant frequency ${\tilde{f}}_0$ on the amplitude, Ã, of the input acceleration. The à was selected as 1.25 in following analysis, and it is marked as the upper triangles on the graph.

Figure 5

Change of amplitude ratio $r$ with respect to the dimensionless excited frequency $\tilde{f}$ under the effect of (a) Oh number, (b) viscosity ratio, (c) density ratio, and (d) aspect ratio.

Figure 6

(a) The geometric function $f\left(\mathrm{\Lambda },\tilde{L},{\varepsilon }^{*}\right)$ with respect to ${\varepsilon }^{*}$; (b) the $r$ with respect to $\tilde{f}$ at three dimensionless initial positions of the front of the meniscus, which are marked correspondingly as circles in (a); (c) variation of $k$ with respect to relative displacement; (d) the $r$ with respect to $\tilde{f}$ at the different dimensionless amplitudes of acceleration.

Figure 7

The comparison of resonance characteristics of nonwetting droplet in constricted tube under lubrication approximation and extended lubrication approximation: (a) the peak ratio with respect to aspect ratio Λ and slope of curved surface $\frac{\mathrm{\Lambda }-1}{\tilde{L}}$, (b) the associated resonant frequency with respect to aspect ratio Λ and slope of curved surface $\frac{\mathrm{\Lambda }-1}{\tilde{L}}$. The solid and dash line denote lubrication approximation and extended lubrication approximation, respectively. The black, red, and blue color denote $\mathrm{\Lambda }=2$, 4, 6, respectively.

Figure 8

The effect of wettability on the resonance curve in based case.

Figure 9

Schematic of nonwetting droplet trapped in the constricted tube, as part of Fig. 1. The geometric parameters are annotated. The black dots mark the locations of corresponding pressure which are listed at bottom.