Self-similarity and coarsening rate of a convecting bicontinuous phase separating mixture: Effect of the viscosity contrast

We present a computational study of the hydrodynamic coarsening in three dimensions of a critical mixture using the Cahn-Hilliard/Navier-Stokes model. The topology of the resulting intricate bicontinuous microstructure is analyzed through the principal curvatures to prove self-similar morphological evolution. We find that the self-similarity exists for both systems: isoviscous and with variable viscosity. However, the two systems have a distinct topological character. Moreover, an effective viscosity that accurately predicts the coarsening rate is proposed.

Figure 1

Perspective view of the isosurface $c=0.5$ at different times of a simulation. The volume fraction is $\phi =0.5$, the viscosity contrast between the phases is 128, and the viscosity is $\nu =2$.

Figure 2

(a) Growth velocity as a function of viscosity for a volume fraction of 0.5. In the inset, a zoom on the linear regime is shown. (b) $v/v_0=dl/dt/v_0$ as a function of $l$ for $\rho =1$, 0.25, and 0.0625 and $\nu =8$, 16, and 32, shown as solid, long, and short dashes, respectively. Here, $v_0$ is the value computed in the case $\rho =0.0625$, $\nu =32$ (rescaled for $\rho =1$ and 0.25).

Figure 3

(a) Growth rate as a function of $\sqrt{{\nu }_l{\nu }_h}$ for different values of ${\nu }_h$ ranging from 0.0625 to 32 and of ${\nu }_h/{\nu }_l$ equal to 2, 4, 16, and 32. The points corresponding to ${\nu }_h/{\nu }_l=1$ are the purple $+$. The line is a guide to the eye. (b) Same data with a zoom on the vicinity of the origin. (c) and (d) Plot of the coarsening velocity vs (c) ${\nu }_{\mathrm{eff}}=({\nu }_h+{\nu }_l)/2$ and (d) $1/{\nu }_{\mathrm{eff}}=(1/{\nu }_h+1/{\nu }_l)/2$.

Figure 4

(a), (b) Contour plots of the PDF of the principal curvatures rescaled by the characteristic length for two values of the viscosity contrast: 1 (128) taken at two times corresponding to (solid) $l\approx 23$ (30) and to (dashed) $l\approx 88$ (84) (dashed). The isolevels are 0.0001, 0.0002, 0.0003, 0.0004, and 0.0005. (c) PDF of the Gaussian curvature for three different values of the viscosity contrast (solid: 1; long dashed: 16; and short dashed: 128). (d) Plot of the normalized structure functions taken at different times corresponding to the value of $l$ indicated on the graph. The very good collapse of the curves for the lowest values of $l$ confirms the self-similar nature of the coarsening process. The low wave-number departure from the collapse for $l=63$ and $l=82$ can be attributed to discretization effects [12].

Figure 5

(a) Plot of the rescaled genius as a function of $l$ for $VC=1$ (+), 16 ($\times$), and 128 ($\square$). (b) Plot of the rescaled average mean curvature as a function of $l$ for $VC=1$ (+), 16 ($\times$), and 128 ($\square$).