Flow-parametric regulation of shear-driven phase separation in two and three dimensions

The Cahn-Hilliard equation with an externally prescribed chaotic shear flow is studied in two and three dimensions. The main goal is to compare and contrast the phase separation in two and three dimensions, using high-resolution numerical simulation as the basis for the study. The model flow is parametrized by its amplitudes (thereby admitting the possibility of anisotropy), length scales, and multiple time scales, and the outcome of the phase separation is investigated as a function of these parameters as well as the dimensionality. In this way, a parameter regime is identified wherein the phase separation and the associated coarsening phenomenon are not only arrested but in fact the concentration variance decays, thereby opening up the possibility of describing the dynamics of the concentration field using the theories of advection diffusion. This parameter regime corresponds to long flow correlation times, large flow amplitudes and small diffusivities. The onset of this hyperdiffusive regime is interpreted by introducing Batchelor length scales. A key result is that in the hyperdiffusive regime, the distribution of concentration (in particular, the frequency of extreme values of concentration) depends strongly on the dimensionality. Anisotropic scenarios are also investigated: for scenarios wherein the variance saturates (corresponding to coarsening arrest), the direction in which the domains align depends on the flow correlation time. Thus, for correlation times comparable to the inverse of the mean shear rate, the domains align in the direction of maximum flow amplitude, while for short correlation times, the domains initially align in the opposite direction. However, at very late times (after the passage of thousands of correlation times), the fate of the domains is the same regardless of correlation time, namely alignment in the direction of maximum flow amplitude. A theoretical model to explain these features is proposed. These features and the theoretical model carry over to the three-dimensional case, albeit that an extra degree of freedom pertains, such that the dynamics of the domain alignment in three dimensions warrant a more detailed consideration, also presented herein.

Figure 1

(Circles) Average value of the maximal fully converged infinite-time Lyapunov exponent $\mathrm{\Lambda }(N,k_0,\tau )$ of the flow (8) and (9), with $\tau =0.01$ and $k_0=2\pi$ (box size $L=1$). Circles: 3D case; Crosses: 2D case.

Figure 2

Flow-pattern map showing the results of the different simulations. Filled symbols indicate situations where decay of the concentration variance occurs. Part of the boundary between the regime of variance decay and coarsening arrest has been filled in. Circles: FDCH3D-long; crosses: FDCH3D-short; squares: FDCH2D-long; stars: FDCH2D-short; diamonds: 2DLA.

Figure 3

Snapshot results for FDCH3D-long. Shown are isovolume plots based on $\varphi =(C+1)/2$. Top row: $\mathcal{D}={10}^{-5}$, with $t=1,4,8$ reading from left to right; middle row: $\mathcal{D}={10}^{-3}$, with $t=1,4,8$ reading from left to right; bottom row: $\mathcal{D}={10}^{-1}$, with $t=1,2,4$ reading from left to right.

Figure 4

Results from the simulation FDCH3D-long: (a) the concentration variance; (b) the length scale $\ell \propto 1/(1-{\sigma }^2)$, shown as a function of time for various values of the inverse Péclet number $\mathcal{D}$. The saturation of $\ell$ at late times reveals itself more convincingly in a plot wherein a linear scale is used on both axes. This can be seen (albeit indirectly) by comparing (a) and (b): the variance clearly saturates for $\mathcal{D}={10}^{-3},{10}^{-2},{10}^{-1}$ in (a) (linear scales) and hence it can be concluded that $1/(1-{\sigma }^2)$ saturates for the same values of $\mathcal{D}$ in (b).

Figure 5

The time-average of ${(1-{\sigma }^2)}^{-1}$ corresponding to coarsening arrest.

Figure 6

The time average of the Batchelor scales $k_2$ (a) and $k_4$ (b). Plot key: (circles) 2DLA; (squares) FDCH2D-long; (diamonds) FDCH2D-short; (stars) FDCH3D-long. The case FDCH3D-short is not shown because convergence to a statistically steady state was not reached by $t=10$.

Figure 7

PDF of concentration for decaying cases: (a) a particular long-correlation-time realization of 2DLA, with ${\tau }_{\mathrm{corr}}=0.5$ and $\mathcal{D}=2\times {10}^{-3}$, (b) FDCH2D-long, $\mathcal{D}={10}^{-5}$, $t\in [4,10]$; (c) FDCH3D-long, $\mathcal{D}={10}^{-5}$, $t\in [4,8]$. The inset in each panel shows the kurtosis defect (flatness) of the PDF as a function of time.

Figure 8

Results for the simulation 2DAniso1 $(N=100)$. (a) Batchelor length scales as a function of time (the inset is the same plot, only shown over a much longer time interval). (b), (c) Snapshots of concentration: (b) just before saturation of ${\ell }_x$, with $t=0.5$ (c) after saturation of ${\ell }_x$, with $t=5$.

Figure 9

Results for the simulation 2DAniso2 $(N=1)$.

Figure 10

Results for the simulation 2DAniso2 $(N=1)$, with $t=5,10,15,20,25$ [reading from (a) to (e)].

Figure 11

Typical length scales ${\ell }_x=2\pi /k_x$, ${\ell }_y=2\pi /k_y$ and ${\ell }_z=2\pi /k_z$ corresponding to the simulations 3DAniso1 and 3DAniso2 respectively. (a) $N=100$; (b) $N=1$.

Figure 12

Comparison between linear theory and direct numerical simulation. Model parameters: $D=1$ and $\gamma =5\times {10}^{-4}$. Simulation parameters: $\mathrm{\Delta }x=1/304$, $\mathrm{\Delta }t={10}^{-6}$.

Figure 13

Comparison between weakly linear theory and direct numerical simulation. Model parameters: $D=1$, $L=1$, and $\gamma =1/8{\pi }^2$. Simulation parameters: $\mathrm{\Delta }x=1/304$, $\mathrm{\Delta }t={10}^{-8}$. The simulation results are shown in thick lines: solid line: $|A_1|$, dotted line: $|A_3|$, dashed line: $|A_5|$. The predictions from weakly nonlinear theory are shown in thin lines with symbols. Squares and circles: predictions based on Eq. (A4) for $|A_3|$ and $|A_5|$ respectively. Diamonds: prediction based on Eq. (A5) for $|A_1|$. The main figure is presented again in the inset using a log-log scale to show the initial layer of the dynamics before the onset of slaving.

Figure 14

Results for a simulation on a ${505}^3$ grid with $\mathrm{\Delta }t={10}^{-5}$. Time dependence of $k_1$, with fitted power-law exponent $0.324$ close to the theoretical Lifshitz-Slyozov exponent $1/3$.