Faraday instability of binary miscible/immiscible fluids with phase field approach

The objective in the present paper is to study binary fluids with phase field modeling coupled with Navier-Stokes equations. An extended free energy is proposed to account for the continuous path from immiscible to miscible states. We consider fluid pairs that are immiscible for temperatures below the critical one (consolute temperature) and miscible above it. Our extended phase field equation permits us to move from the immiscible state (governed by the Cahn-Hilliard equation) to the miscible state (defined by the species diffusion equation). The scaling of interface tension and interface width with the distance to the critical point is highlighted. The whole system is mechanically excited showing Faraday instability of a flat interface. A linear stability analysis is performed for the stable case (interface waves) as well as for the unstable Faraday one. For the latter, a Floquet analysis shows the well-known Arnold's tongues as a function of the consolute temperature and depth layer. Moreover, two-dimensional finite difference simulations have been performed allowing us to model nonlinear flow patterns both in miscible and immiscible phases. Linear theory and nonlinear simulations show interesting results such as the diminishing of the wavelength of Faraday waves or a shift of the critical vibration amplitude when the consolute temperature is approached.

Figure 1

Left: sketch of the bulk potential free energy $f=-\frac{1}{2}{r}_{0}r{\phi }^2+\frac{1}{4}{r}_0\theta \left(r\right)r^q{\phi }^4$ in the miscible phase $r<0$ (dashed), and in the immiscible phase $r>0$ (solid). Right: phase field in the bulk as a function of $r$ for different values of $q$ (${\phi }_m=\pm r^{\frac{1-q}{2}}$ corresponds to the minimum values of the phase field for the two fluid components, for $r>0$).

Figure 2

Sketch of the geometry of the two-fluid layer.

Figure 3

Function $r\left(\vartheta \right)$ for $a=10$ and $\mathrm{Sc}=10$.

Figure 4

(a) Densities of the two phases (solid lines) adjusted to experimental data (circles) by chosing values for $a,\mathrm{Sc}$, and $q$ [32]; (b) surface tension as a function of $T$ for the same parameters and $p=1/7$.

Figure 5

Frequency of the least stable modes (largest real part) over $k$ for $r=1$. Parameters: $\mathrm{At}=0.26$, $\sigma =0.001\mathrm{N}/\mathrm{m}$, ${\rho }_0={10}^3\mathrm{kg}/{\mathrm{m}}^3$, $\nu ={10}^{-6}{\mathrm{m}}^2/\mathrm{s}$; (a): $d=1\mathrm{cm}$, (b): $d=1\mathrm{mm}$. The dashed lines show the exact results for an inviscid liquid, Eq. (26). The insets show the eigenvalues with the largest real part for the viscous case computed by our analysis. In (b), a real-valued eigenvalue crosses a complex one at $k\approx 500/\mathrm{m}$.

Figure 6

Stability diagram of the Faraday instability for ${\omega }_0=50\mathrm{rad}/\mathrm{s}$ and with the same parameters as Fig. 5; solid line $r=1$, dashed line $r=0.5$, and dashed-dotted line $r=0.1$.

Figure 7

Minimum values of acceleration for which the Faraday instability emerges with $k=k_{min}$ (insets). Top: as a function of the depth $d$ for fixed frequency ${\omega }_0=100\mathrm{rad}/\mathrm{s}$; bottom: as a function of ${\omega }_0$ for fixed depth $d=1\mathrm{cm}$. Same parameters as in Fig. 5.

Figure 8

First tongues for ${\omega }_0=100\mathrm{rad}/\mathrm{s}$ and several layer depths. All curves are for $r=1$ except for the dashed-dotted one which is for $r=0.1$.

Figure 9

Time series for $r=1$; length is in cm. Different colors correspond to the values of the phase field. The FD mesh has the size 1024 × 400, corresponding to an aspect ratio of 2.56. See the movie in the Supplemental Material [38] for the time evolution of the phase field for the unforced system from a random-dot initial distribution of the phase field and below the critical temperature (immiscible state at $r=1$).

Figure 10

Run for the initial condition from Fig. 9 last frame ($t=10\mathrm{s}$), but mirrored on the $y=1/2$ plane. Now the heavier fluid is placed on top of the lighter one and the system is gravitationally unstable (Rayleigh-Taylor instability). All parameters are the same as in Fig. 9. See the movie in the Supplemental Material [38] for time evolution of the phase field in Rayleigh-Taylor configuration (heavier fluid placed above the lighter one) far from the critical temperature (at $r=1$).

Figure 11

Time series for $r=0.1$, closer to the critical temperature. Due to the critical slowing down, it takes by a factor $1/r=10$ longer until the sharp interface occurs. See the movie in the Supplemental Material [38] for time evolution of the phase field for the unforced system from a random-dot initial distribution of the phase field, closer to but below the critical temperature at $r=0.1$.

Figure 12

Time series for $r=-0.1$, above the critical temperature. The fluids are now miscible. In the long-time limit the mixture becomes homogeneous and $\phi \to 0$ as expected. See the movie in the Supplemental Material at [38] for time evolution of the phase field for the unforced system from a random-dot initial distribution of the phase field, above the critical temperature (miscible state) at $r=-0.1$.

Figure 13

Initial condition and snapshot at later time for a vibrating layer, ${\omega }_0=100\mathrm{rad}/\mathrm{s}$, $a=0.35$ above Faraday threshold and with immiscible fluids $r=1$.

Figure 14

Phase field at $x=1.28\mathrm{cm}$, $y=0.5\mathrm{cm}$ (middle of the layer) over time for the parameters of Fig. 13.

Figure 15

Phase field contour plots for different values of acceleration parameter $a$ which has been increased in steps after the times $t=3.3,4.0,5.3,5.6,6.0\mathrm{s}$, $r=1$. See the movie in the Supplemental Material at [38] for time evolution of the phase field with different values of the acceleration $a=0.5,1,2,3,4,5,6$ at time $t=3.3,4.0,5.3,5.6,6.0\mathrm{s}$.

Figure 16

Phase field at $x=1.28\mathrm{cm}$, $y=0.5\mathrm{cm}$ (middle of the layer) over time for the case of increasing the amplitude of acceleration $a$ as shown in Fig. 15.

Figure 17

Time evolution of the phase field for Faraday instability for $a=0.5$. The fluid is immiscible ($r=1$) up to $t=5.72\mathrm{s}$; then $r$ is switched to $r=-0.1$. See the movie in the Supplemental Material at [38] for time evolution of the phase field for Faraday instability, and time evolution from an immiscible state ($r=1$) to a miscible state ($r=-0.1$).

Figure 18

Phase field at $x=1.28\mathrm{cm}$, $y=0.5\mathrm{cm}$ (middle of the layer) over time for the series of Fig. 17.

Figure 19

Time evolution of the phase field for $a=0.5$ with $r$ changing from $r=1$ to $r=0.1$ after time $t=5.72\mathrm{s}$.

Figure 20

Kinetic energy (dimensionless) over time for the sequence shown in Fig. 19.