Turbulence and Coarsening in Active and Passive Binary Mixtures

Phase separation between two fluids in two dimensions is investigated by means of direct numerical simulations of coupled Navier-Stokes and Cahn-Hilliard equations. We study the phase ordering process in the presence of an external stirring acting on the velocity field. For both active and passive mixtures we find that, for a sufficiently strong stirring, coarsening is arrested in a stationary dynamical state characterized by a continuous rupture and formation of finite domains. Coarsening arrest is shown to be independent of the chaotic or regular nature of the flow.

Figure 1

(a) $L(t)$ vs $t$ obtained by DNS of (2, 3) with $\xi =0.015$, $\nu ={10}^{-3}$, without external forcing. (b) Kinetic energy $\mathcal{K}=⟨v^2⟩/2$ (bottom) and enstrophy $\Omega =⟨{\omega }^2⟩/2$ (top) vs $t$ in the same run.

Figure 2

Snapshots of $\theta$ at time $t=4000$ at varying the forcing intensity $F$ with ${\ell }_f=26\xi$ (left) and ${\ell }_f=84\xi$ (right). Black/white codes $\theta =\pm 1$. Other parameters are as in Fig. 1

Figure 3

$L$ vs $t$ at varying $F$, from top $F=0$ (thick curve) and $F=0.05,0.10,0.15,\dots ,0.30$. Data refer to DNS with ${\ell }_f=84\xi$ (the case with ${\ell }_f=26\xi$ is qualitatively similar). The straight line displays the scaling $t^{2/3}$. Inset: $L^{*}$ vs $u_{\mathrm{rms}}/L^{*}$; the straight line has slope $-0.29$ the point size is of the order of the statistical error.

Figure 4

Results of DNS in the passive case. (a) $L$ vs $t$ with (from top) $\beta =0.25,0.5,1,2,4$. Forcing parameters are $k_f=70$ and $F=3\times {10}^{-5}$. The straight line has slope $1/3$. (b) Same as (a), in linear scale, for a frozen velocity field (see text), from top $\beta =1,2,4,10$. Inset: $L^{*}$ vs $\beta$; the straight line corresponds to $L^{*}\sim {\beta }^{-0.28}$; the point size is of the order of the statistical error. DNS were performed with hyperdissipation $-{\nu }_2{\Delta }^2\mathit{v}$ with ${\nu }_2={10}^{-7}$, and friction coefficient $\alpha =0.1$. The parameters of Eq. (2) are $\xi =0.015$, $\Gamma =0.02$.

Figure 5

(Left) Snapshots of $\theta$ at $t=6\times {10}^4$ for the cellular flow with different $U$ and $K=8$; here $\xi =0.018$, $\Gamma =0.1$. (Right) $L$ vs $t$, from top to bottom $U=0,0.125,0.25,0.5,1.0,2.0,4.0$. Inset: the same in linear scale.