Influence of gravity on the frozen wave instability in immiscible liquids

The influence of the gravity level on the frozen wave instability in finite containers of immiscible liquids is investigated numerically to shed light on the transformation from a supercritical pitchfork bifurcation in normal gravity, which produces stable finite-amplitude waves, to an apparently degenerate bifurcation in weightlessness, leading to large columnar patterns that collide with the container walls. The vibroequilibria effect grows in importance as gravity is reduced and has a symmetry-breaking effect on the bifurcation, selecting frozen waves with the heavier fluid displaced upward along the lateral walls. Three possible growth regimes are identified. With finite gravity, there is an initial gravity-capillary regime with the square-root growth predicted by weakly nonlinear theory, associated with small-amplitude waves. This is followed by a linear growth regime that is dominated by gravity and associated with larger finger-like waves. Lastly, in some cases with reduced gravity, we observe a third regime characterized by the close proximity of the frozen wave crests to the upper wall and an interval of nearly saturated wave amplitude. The defining parameters of the bifurcation diagram are measured and presented as a function of Bond number and, when possible, compared with theoretical predictions. In particular, there is excellent agreement between theory and simulations for the weakly nonlinear cubic (branching) coefficient and reasonable qualitative agreement for the threshold. Differences are attributed to finite-size effects, vibroequilibria, and viscosity.

Figure 1

Sketch illustrating the two-layer system of immiscible liquids. The rectangular container of size $L\times 2H$ is subjected to vibrations of frequency $\omega$ and amplitude $A$ along the $x$ axis and parallel to the initial flat interface (dashed line).

Figure 2

Time evolution (averaged over one excitation period) of the interface and mean velocity field in a container of size $L\times 2H=30\times 15\mathrm{mm}$ ($\mathrm{\Gamma }=2$) under normal gravity ($g=g_0, \text{Bo}=81.57$) with different forcing amplitudes: (a) $v=0.380\mathrm{m}/\mathrm{s}$, (b) $0.432\mathrm{m}/\mathrm{s}$, (c) $0.456\mathrm{m}/\mathrm{s}$, and (d) $0.577\mathrm{m}/\mathrm{s}$, which correspond to $\mathcal{F}$ values of 12.65, 14.38, 15.19, and 19.22, respectively. The forcing frequency is fixed at $f=55\mathrm{Hz}$ while the amplitude $A$ is increased (1.1, 1.25, 1.32, and $1.67\mathrm{mm}$). The critical forcing is $v_c=0.429\mathrm{m}/\mathrm{s}$ (${\mathcal{F}}_c=14.25$) as shown in Sec. 3b. The velocity field during the transient portion of the simulation is shown with scaled arrows (scale differs between snapshots). The selected time of the intermediate snapshot in (d) is less than that of (a)–(c) due to the more rapid growth of that pattern. The final pattern in (d) also illustrates how the amplitude $\mathrm{\Delta }h$ is measured from the contour $\varphi =0.5$.

Figure 3

Bifurcation diagram showing the trough-to-crest amplitude $\mathrm{\Delta }h$ of the frozen wave patterns versus the vibrational velocity $v$ for $g=g_0$ in a container of size $L\times 2H=30\times 15\mathrm{mm}$. The numerically obtained points are fitted to (a) the perturbed pitchfork bifurcation of Eq. (17) or (b) the square-root dependence of Eq. (15), both shown as solid curves, and to the linear function (19), shown with a dotted line. These fits correspond to the two regimes identified by Jalikop and Juel [17], with the transition between them marked by a vertical line. The three solid markers correspond, in order of increasing $v$, to the snapshots of Figs. 2–2.

Figure 4

Radius of curvature $r_k$ at the wave crests as a function of the applied velocity $v$ for (a) $g=g_0$ and (b) $g=0.5g_0$. Data points are fitted with a solid curve for visual reference. Asymptotic behavior is seen at large $v$ when $r_k$ is comparable to the capillary length (shown with a dashed line) given by Eq. (21). The two solid markers in (a) correspond to the snapshots of Figs. 2 and 2 while those in (b) correspond to the snapshots of Figs. 6 and 6, shown below.

Figure 5

(a) Bifurcation diagrams showing $\mathrm{\Delta }h\left(v\right)$ with $g=2g_0$ ($\mathrm{Bo}=163.14$, squares) and $g=g_0$ ($\mathrm{Bo}=87.57$, circles) in a container of size $L\times 2H=30\times 15\mathrm{mm}$ ($\mathrm{\Gamma }=2$). The initial capillary-gravity regime of each diagram, with the interval given in Table 2 by $(v^{*}-v_c)$, is shaded. (b) Bifurcation diagrams under normal gravity, $g=g_0$, in containers of size $L\times 2H=30\times 15\mathrm{mm}$ ($\mathrm{\Gamma }=2$, circles) and $L\times 2H=60\times 15\mathrm{mm}$ ($\mathrm{\Gamma }=4$, triangles). The insets show representative patterns, to be compared with Fig. 2, for (a) the hypergravity case and (b) the longer container with $\mathrm{\Gamma }=4$.

Figure 6

Time evolution (averaged over one excitation period) of the interface and mean velocity field in a container of size $L\times 2H=30\times 15\mathrm{mm}$ ($\mathrm{\Gamma }=2$) under reduced gravity $g=0.5g_0$ ($\text{Bo}=40.79$). Increasing vibrational forcing is applied: (a) $v=0.282$, (b) 0.344, (c) 0.364, and (d) $0.410\mathrm{m}/\mathrm{s}$, corresponding to $\mathcal{F}=9.41$, 11.49, 12.14, and 13.65, respectively. The frequency is fixed at $45\mathrm{Hz}$ with the amplitude increased from 1 to $1.45\mathrm{mm}$. The critical forcing is $v_c=0.347\mathrm{m}/\mathrm{s}$ (${\mathcal{F}}_c=11.55$); see Sec. 4b. The velocity field during the transient portion of the simulation is shown with scaled arrows (scale differs between snapshots). The selected times of series (a)–(c) are labeled at the top while, due to more rapid growth, the intermediate snapshot in (d) is taken at 0.7 s instead of 0.9 s.

Figure 7

Bifurcation diagram showing the trough-to-crest distance $\mathrm{\Delta }h$ of the frozen wave patterns versus vibrational velocity $v$ for $g=0.5g_0$ in a container of size $L\times 2H=30\times 15\mathrm{mm}$. The measurements in the gravity-capillary regime are fitted to the branching equation (17) of a perturbed pitchfork (dashed curve) and of the unperturbed ($\epsilon =0$) case (solid curve). The gravity regime is fitted to the linear function (19) (dotted curve). The transitions between different regimes (labeled above) are highlighted by vertical lines, including the new finite-depth regime where the crest approaches the upper boundary. The five points distinguished by open black circles correspond, in order of increasing $v$, to the snapshots of Figs. 6–6 and Figs. 8 and 8. The diagram of Fig. 3 for normal gravity is included for comparison (open gray circles).

Figure 8

Time evolution (averaged over one excitation period) of the interface and mean velocity field illustrating the frozen wave instability for $g=0.5g_0$ in the finite-depth regime in a container of size $L\times 2H=30\times 15\mathrm{mm}$ vibrated at $f=45\mathrm{Hz}$ and amplitudes (a) $A=1.52\mathrm{mm}$ ($v=0.430\mathrm{m}/\mathrm{s}, \mathcal{F}=14.31$), (b) $A=1.55\mathrm{mm}$ ($v=0.438\mathrm{m}/\mathrm{s}, \mathcal{F}=14.59$). The time of the snapshot is labeled between the columns. The velocity field is shown using scaled arrows (scale differs between snapshots). The $\mathrm{\Delta }h$ values obtained from these two simulations are marked with open circles in Fig. 7.

Figure 9

Time evolution (averaged over one excitation period) of the frozen wave instability with $g=0.5g_0$ in a container of size $L\times 2H=30\times 15\mathrm{mm}$ vibrated at $f=45\mathrm{Hz}$ with $A=1.6\mathrm{mm}$ ($v=0.452\mathrm{m}/\mathrm{s},\mathcal{F}=15.06$). A mode transition between the $\mathcal{K}=3$ and $\mathcal{K}=2$ patterns is observed. The snapshot times are labeled and the mean velocity field during the transient period is shown using scaled arrows (scale differs between snapshots). For $t\lesssim 0.5\mathrm{s}$ the evolution is similar to that of Fig. 8.

Figure 10

(a) Evolution of the symmetric vibroequilibria state in microgravity with increasing velocity $v$ in a container of size $L\times 2H=30\times 15\mathrm{mm}$. The difference $\mathrm{\Delta }h$ between the height of the contact points and the midpoint of the interface is shown and fitted (dashed line) to a quadratic function as per the theoretical prediction. The inset illustrates a typical vibroequilibria state and the $\mathrm{\Delta }h$ measurement. (b) Inverse square of the frozen wave growth time, ${\tau }^{-2}$ (open squares), for the smallest wave number $\mathcal{K}=1$ versus $v$. The observed linear dependence allows for an estimation of the instability threshold (black marker). The inset shows the final columnar pattern resulting from the $\mathcal{K}=1$ mode.

Figure 11

Bifurcation diagrams obtained for the $L\times 2H=30\times 15\mathrm{mm}$ container with varying gravity levels (in units of $g_0$): 0.25, 0.5, 0.75, 1, 1.5, 2. The data sets are distinguished with different markers and labeled with the gravity level used. Note that for $g=0.25g_0$ and $0.5g_0$, the finite-depth regime discussed in Sec. 4b is visible at large amplitude.

Figure 12

Dimensionless fitting parameters characterizing the frozen wave bifurcation diagram as a function of the Bond number Bo: (a) critical forcing ${\mathcal{F}}_c$ (open circles for $\widehat{\epsilon }=0$, crosses for $\widehat{\epsilon }\ne 0$) and the point of transition to the gravity regime ${\mathcal{F}}^{*}$ (solid circles), (b) branching coefficient $\widehat{K}$ of the gravity-capillary regime (open circles for $\widehat{\epsilon }=0$, crosses for $\widehat{\epsilon }\ne 0$), (c) slope $\widehat{\mathcal{M}}$ of the gravity regime, and (d) symmetry-breaking parameter $\widehat{\epsilon }$. The predicted value for ${\mathcal{F}}_c$ from Eq. (12) is shown with a solid curve in (a) while the theoretical formula for $\widehat{K}$ from Eq. (16) is shown in (b). The dashed curve in (c) shows a fit to ${\text{Bo}}^{-1}$ while the dotted curve in (d) is intended only as a visual aid. The values of ${\mathcal{F}}^{*}$ and $\widehat{\mathcal{M}}$ change imperceptibly between the $\widehat{\epsilon }=0$ and $\widehat{\epsilon }\ne 0$ cases.