Faraday instability in double-interface fluid layers

We investigate, both by way of theory and by experiments, the mechanically forced Faraday instability in immiscible three-fluid systems that is two-interface fluid layers. The theoretical model suggests that two-interface fluid layers offer underlying physics quite distinct from the typical single-interface system due to the coupling of the fluid interfaces, resulting in alternating double-tongued stability curves. This allows for the possibility of unique codimension points unattainable in traditional two-fluid systems. In addition, the presence of a third fluid in the problem can lead to either enhanced or delayed destabilization of the system at target frequencies. Experimental results qualitatively support the theory, though precise agreement between the theory and the experiment is hindered due to the sidewall damping.

Figure 1

Schematic of the three-fluid mechanically oscillated system. The normal vectors point into the lighter fluid at each interface. The lower interface is taken to be located at $z=0$ and both interfaces are initially flat.

Figure 2

Predicted stability curve and relative interfacial deflections for a three-fluid mechanically forced system. (a) Theoretical predictions for critical amplitudes utilizing the fluid combination summarized in Table 1. Double-tongued stability curves are observed for three-fluid Faraday instability systems, where each section of a tongue is dominated by one of the interfaces. The testing frequency is 9 Hz. (b) Relative interfacial deflection $\varphi$ vs wave number. Here $\varphi =1$ indicates a comparable deflection of the interfaces, $\varphi <1$ indicates higher deflection on the top interface, and $\varphi >1$ indicates higher deflection on the bottom interface. For example, for $k=250{\text{m}}^{-1}$, the subharmonic and harmonic tongues in (a) indicate that the bottom interface has the higher deflection and at $k=500{\text{m}}^{-1}$, the upper interface has the higher deflection for the subharmonic tongue while the reverse is true for the harmonic tongue. Note that the harmonic tongues at $k=250$ and $500{\text{m}}^{-1}$ appear at very large vibrational amplitudes not depicted within the scale of (a).

Figure 3

Effect of viscosity on three-fluid instability thresholds. (a) The black lines are obtained using the fluid properties summarized in Table 1, while the red lines are obtained by increasing the viscosity of the middle fluid by a factor of 20. As predicted, increasing the middle fluid viscosity causes a shift of the threshold to higher amplitude and higher wave number. The testing frequency is 10 Hz. (b) Effect obtained when the top or bottom fluid is much more viscous than the other two. The testing frequency is 9 Hz. The fluid properties are the same as those shown in Table 1 with the exception of ${\nu }_1$, which is set at 120 cSt.

Figure 4

Effect of increasing frequency on the stability of the system. In general, an increase in frequency causes the tongues to shift to higher wave number and lower amplitude. The fluid properties are those shown in Table 1.

Figure 5

Comparison between the three-fluid stability thresholds and the corresponding two-fluid cases for tall middle fluid heights. (a) Three-fluid calculation versus corresponding two-fluid calculations for high middle fluid heights. The properties are those shown in Table 1 and layer heights are 0.5 cm for the bottom, 1 cm for the middle, and 0.5 cm for the top. Solid lines show the three-fluid stability curve prediction and symbols show the corresponding top-middle fluid and middle-bottom fluid two-fluid calculations. The agreement between the tongues generated by the separate two-fluid calculations and the three-fluid calculation is excellent, with the only discrepancies occurring at low wave number. (b) Relative deflection versus wave number for the tall middle fluid system. The relative deflection can be accurately predicted by determining which eigenvalue branch occurs at lower amplitude in (a).

Figure 6

Comparison between the three-fluid stability thresholds and the corresponding two-fluid cases for short middle fluid heights. (a) The properties are those shown in Table 1 and the layer heights are 0.5 cm for the bottom, 0.000 01 cm for the middle, and 0.5 cm for the top. Solid lines show the three-fluid stability curve prediction and symbols show the corresponding top-bottom two-fluid calculation with an assumed interfacial tension of $\gamma ={\gamma }_a={\gamma }_b$. The two different predictions agree only at low wave number and differ dramatically with increasing wave number. (b) Relative deflection versus wave number for the short middle fluid system. The relative deflection approaches unity for the subharmonic tongues, but always favors the interface with the higher $\mathrm{\Delta }\rho$ for the harmonic tongues. (c) Three-fluid small middle height calculation versus the corresponding two-fluid calculation with no density difference bias. The fluid properties are the same as in (a), but with ${\rho }_3=520\text{kg}{\text{m}}^{-3}$. (d) Relative deflection for nonbiased three-fluid system. Here $\varphi$ approaches unity for both harmonic and subharmonic responses in the absence of a $\mathrm{\Delta }\rho$ bias.

Figure 7

Comparison between the stability thresholds for a two-fluid case and a three-fluid case with the same total height where the third fluid is added at the top. The forcing frequency $\omega$ is 2 Hz. The fluid parameters for the two-fluid case are ${\rho }_1=1000\text{kg}{\text{m}}^{-3}, {\nu }_1=12\text{cSt}, h_1=1.27\text{cm}, {\rho }_2=846\text{kg}{\text{m}}^{-3}, {\nu }_2=1.5\text{cSt}, h_2=1.27\text{cm}$, and $\gamma =0.007\text{kg}{\text{m}}^{-2}$. The fluid parameters for the three-fluid case are ${\rho }_1=1000\text{kg}{\text{m}}^{-3}, {\nu }_1=12\text{cSt}, h_1=1.27\text{cm}, {\rho }_2=846\text{kg}{\text{m}}^{-3}, {\nu }_2=1.5\text{cSt}, h_2=0.635\text{cm}, {\rho }_3=600\text{kg}{\text{m}}^{-3}, {\nu }_3=1\text{cSt}, h_3=0.635\text{cm}$, and ${\gamma }_a={\gamma }_b=0.007\text{kg}{\text{m}}^{-2}$. Multiple regions of stabilization and destabilization can be observed. Harmonic solutions are omitted for clarity.

Figure 8

Three-fluid stability curve with one interface containing $\mathrm{\Delta }\rho =0$. (a) Stability curve for the fluids described in Table 1 with ${\rho }_2$ set to 1880 $\text{kg}{\text{m}}^{-3}$, therefore making $\mathrm{\Delta }{\rho }_b$ equal to 0. The testing frequency is 9 Hz. (b) Relative deflections of the instability at onset versus wave number. A wave-number range at which the bottom interface deflects more than the top interface can be seen, even though its $\mathrm{\Delta }\rho$ is zero.

Figure 9

Four possible codimension points obtainable in the three-fluid mechanically forced Faraday instability. (a) Codimension point where relative deflection between the interfaces is equal to 1. At this point, both interfaces will become unstable simultaneously with comparable interfacial deflections. (b) Codimension point between tongues with opposing relative interfacial deflections, resulting in mode expressions that are biased on each interface based upon the relative deflections. (c) Codimension point for a single mode at multiple harmonicities. This type of point is unachievable in two-fluid systems because the harmonic and subharmonic tongues never meet. (d) Example of the type of codimension point achievable in two-fluid systems. One eigenvalue branch exhibits the same critical amplitudes for multiple wave numbers, resulting in a codimension point. This can also be achieved for multiple wave numbers at different harmonicities.

Figure 10

Theoretical waveform representation for competing (0,1) and (2,1) modes. The top interface exhibits a mode which is primarily (0,1), while the bottom interface exhibits a mode which is primarily (2,1).

Figure 11

Theoretical waveform representation for two out-of-phase (3,1) modes. Both interfaces exhibit comparable relative deflection, yet their waveforms appear out of phase with one another.

Figure 12

Relative deflection before taking the absolute value for the three-fluid system described in Fig. 7 versus wave number. The relative deflection is negative at various wave numbers. However, the negative values typically occur when $\left|\varphi \right|\ll 1$ or $\left|\varphi \right|\gg 1$.

Figure 13

Assembled testing cell and experimental setup. The base plate attaches to the mechanical shaker and is secured with four screws. Backlighting allows for the clear visualization of the interface and any deflection using a high-speed camera.

Figure 14

Theoretical amplitude versus frequency plot for the three-fluid system described by Table 2. The black circled point is an example of a three-fluid exclusive codimension point shown in Fig. 9. Distinct modes are enclosed between pairs of dashed lines and labeled, with an s denoting a subharmonic response and an h denoting a harmonic response.

Figure 15

Experimental results for the stability threshold obtained for the three-fluid system described in Table 2 in a 5.08-cm-diam cylindrical geometry. Modes are boxed by wave number, which can be compared to the theoretical predictions shown in Fig. 14. The theoretical ${(0,1)}_h$ mode is also shown to account for its experimental appearance. Modal transitions and qualitative behavior are accurately predicted by the model, though quantitative agreement suffers due to the nonideal sidewall conditions.

Figure 16

Mode examples with large, moderate, and small relative interfacial deflection ratios. (a) The (2,1) mode with $\varphi \gg 1$. The deflection of the bottom interface is much smaller than that of the top interface. The testing frequency and amplitude are 4.07 Hz and 8.6 mm, respectively. (b) The (1,1) mode with $\varphi \approx 1$. Both interfaces have comparable in-phase deflections. The testing frequency and amplitude are 5.18 Hz and 4 mm, respectively. (c) The (2,1) mode with $\varphi \ll 1$. The deflection of the bottom interface is much higher than that of the top interface. The testing frequency and amplitude are 6.93 Hz and 3 mm, respectively.

Figure 17

Experimental results for the stability threshold obtained for the three-fluid system described in Table 2 in a 12.7-cm-diam cylindrical geometry. Agreement between theory and experiment is obtained in some frequency bands whose stability thresholds are primarily governed by the top interface.

Figure 18

Corresponding two-fluid comparisons for the system shown in Fig. 17. (a) Two-fluid comparison for silicone oil (1.905 cm) and water (1.27 cm). (b) Two-fluid comparison for water (1.27 cm) and FC-70 (1.905 cm). All of the labeled theoretical modes are subharmonic with the exception of the (4,1)${}_h$ mode in (b). Agreement between theory and experiment is obtained for both systems, but to a greater extent in (a), where the stress-free sidewall assumption is better upheld.