Stability of the fluid interface in a Hele-Shaw cell subjected to horizontal vibrations

The stability of the horizontal interface of two immiscible viscous fluids in a Hele-Shaw cell subject to gravity and horizontal vibrations is studied. The problem is reduced to the generalized Hill equation, which is solved analytically by the multiple scale method and numerically. The long-wave instability, the resonance (parametric resonance) excitation of waves at finite frequencies of vibrations (for the first three resonances), and the limit of high-frequency vibrations are studied analytically under the assumption of small amplitudes of vibrations and small viscosity. For finite amplitudes of vibrations, finite wave numbers, and finite viscosity, the study is performed numerically. The influence of the specific natural control parameters and physical parameters of the system on its instability threshold is discussed. The results provide extension to other results [J. Bouchgl, S. Aniss, and M. Souhar, Phys. Rev. E 88, 023027 (2013)], where the authors considered a similar problem but took into account viscosity in the basic state and did not consider it in the equations for perturbations.

Figure 1

Dependence of $q$ on the wave number $k$ in the long-wave limit for $H=1, \overline{H}=1, \mathrm{We}=5, \overline{\mu }=1, \overline{\rho }=0.8$: 0, $F=0$; 1, $F=0.5$; 2, $F=1$; 3, $F=1.5$.

Figure 2

Dependence of $q$ on the wave number $k$ in the long-wave limit for $H=2, \overline{H}=1, \mathrm{We}=5, \overline{\mu }=1, \overline{\rho }=0.8$: 0, $F=0$; 1, $F=0.5$; 2, $F=1$; 3, $F=1.5$.

Figure 3

Dependence of $q$ on the wave number $k$ for the long-wave instability and the first and second resonances at $H=1, \overline{H}=1, \mathrm{We}=5, \overline{\mu }=1, \overline{\rho }=0.8$: 0, $F=0$; 1, $F=0.5$; 2, $F=1$; 3, $F=1.5$.

Figure 4

Dependence of $q$ on the wave number $k$ for the long-wave limit and the first and second resonances at $H=2, \overline{H}=1, \mathrm{We}=5, \overline{\mu }=1, \overline{\rho }=0.8$: 0, $F=0$; 1, $F=0.5$; 2, $F=1$; 3, $F=1.5$.

Figure 5

Dependence of $q$ on the wave number $k$ for the long-wave instability and the first, second, and third resonances at $H=1, \overline{H}=1, \mathrm{We}=5, \overline{\mu }=1, \overline{\rho }=0.8$: 0, $F=0$; 1, $F=0.15$; 2, $F=0.3$; 3, $F=0.45$.

Figure 6

Dependence of $q_0$ on the viscosity contrast $\overline{\mu }$ for $\overline{H}=1, H=2, \mathrm{We}=5, F=0.4$: 1, $\overline{\rho }=0.1$; 2, $\overline{\rho }=0.4$; 3, $\overline{\rho }=0.6$; 4, $\overline{\rho }=0.8$.

Figure 7

Dependence of $q_0$ on the viscosity contrast $\overline{\mu }$ for $\overline{H}=1, H=2, \mathrm{We}=5, \overline{\rho }=0.1$: 1, $F=0.4$; 2, $F=1$; 3, $F=2$; 4, $F=3$.

Figure 8

Neutral curve at the following values of the control parameters: $\overline{\rho }=0.8, \overline{\mu }=0.5, H=2, \overline{H}=1, \mathrm{We}=5$, and $F=0.4$.

Figure 9

Dependence of $k_{m0}, k_{m1}, q_{m0}$, and $q_{m1}$ on the parameter $We$ at fixed values of other parameters: $\overline{\rho }=0.8, \overline{\mu }=0.5, H=2, \overline{H}=1$, and $F=0.4$.

Figure 10

Dependence of $k_{m0}, k_{m1}, q_{m0}$, and $q_{m1}$ on the parameter $\overline{\rho }$ at fixed values of other parameters: $\overline{\mu }=0.5, H=2, \overline{H}=1, \mathrm{We}=5$, and $F=0.4$.

Figure 11

Dependence of $k_{m0}, k_{m1}, q_{m0}$, and $q_{m1}$ on the parameter $\overline{\mu }$ at fixed values of other parameters: $\overline{\rho }=0.8, \overline{H}=1, H=2, \mathrm{We}=5$, and $F=0.4$.

Figure 12

Dependence of $k_{m0}, k_{m1}, q_{m0}$, and $q_{m1}$ on the parameter $\overline{\mu }$ at fixed values of other parameters: $\overline{\rho }=0.1, \overline{H}=1, H=2, \mathrm{We}=5$, and $F=0.4$.

Figure 13

Dependence of $k_{m0}, k_{m1}, q_{m0}$, and $q_{m1}$ on the parameter $F$ at the following fixed values of other parameters: $\overline{\rho }=0.4, \overline{\mu }=0.5, H=2, \overline{H}=1$, and $\mathrm{We}=5$.

Figure 14

Comparison of neutral curves for different values of the parameter $H$. Other parameters are the same: $\overline{\rho }=0.8, \overline{\mu }=0.5, \overline{H}=1, \mathrm{We}=5$, and $F=0.4$. Solid line, $H=1.4$, dashed line, $H=2$.

Figure 15

Comparison of the dependence of $q_0$ on the parameter $\mathrm{We}$ obtained numerically (dashed line) and analytically (solid line) for $\overline{\rho }=0.8, \overline{\mu }=0.5, H=2, \overline{H}=1$, and $F=0.4$.

Figure 16

Comparison of analytical and numerical dependencies of $q_0$ on the parameter $\overline{\mu }$ for $\overline{\rho }=0.1, H=2, \overline{H}=1, \mathrm{We}=5$, and $F=0.4$. Solid line, numerical results; dashed line, analytical results.

Figure 17

Comparison of analytical and numerical results on the dependence of critical vibration amplitude $q$ on the wave number $k$ for $\rho =0.8, \mu =0.5, H=1.4, \overline{H}=1, \mathrm{We}=5$, and $F=0.1$: solid lines, analytical results; dashed lines, numerical results.

Figure 18

Comparison of analytical and numerical results on the dependence of critical vibration amplitude $q$ on the wave number $k$ for $\rho =0.4, \mu =0.5, H=2, \overline{H}=1$, and $\mathrm{We}=5$: curves 1a and 1b, $F=0.1$; curves 2a and 2b, $F=1$; solid lines 1a and 2a, analytical results; dashed lines 1b and 2b, numerical results.