Role of inertia in nonequilibrium steady states of sheared binary fluids

We study numerically phase separation in a binary fluid subject to an applied shear flow in two dimensions, with full hydrodynamics. To do so, we introduce a mixed finite-differencing and spectral simulation technique, with a transformation to render trivial the implementation of Lees-Edwards sheared periodic boundary conditions. For systems with inertia, we reproduce the nonequilibrium steady states reported in a recent lattice Boltzmann study. The domain coarsening that would occur in zero shear is arrested by the applied shear flow, which restores a finite-domain-size set by the inverse shear rate. For inertialess systems, in contrast, we find no evidence of nonequilibrium steady states free of finite-size effects: Coarsening persists indefinitely until the typical domain size attains the system size, as in zero shear. We present an analytical argument that supports this observation and that furthermore provides a possible explanation for a hitherto puzzling property of the nonequilibrium steady states with inertia.

Figure 1

Flow geometries in the (a) unsheared and (b) sheared cases. The crossed connected by thin dotted lines show representative pairs of points connected by periodic boundary conditions.

Figure 2

(a) Scaled domain size versus scaled time for coarsening in zero shear in the diffusive and viscous hydrodynamic regimes. Segments from left to right correspond to runs DV1u–DV6u. The vertical arrows show the values of scaled reciprocal shear rate used in each corresponding run DV1s–DV6s. Circles show the times of the top two snapshots of Fig. 3 below. (b) Scaled domain size versus scaled time for coarsening in zero shear in the viscous and inertial hydrodynamic regimes. Segments from left to right correspond to runs R028u, R022u, R029u, R020u, R030u, R019u, and R032u. The vertical arrows show the values of scaled reciprocal shear rate used in each corresponding run R028s–R032s. Circles show the times of the bottom two snapshots of Fig. 3 below.

Figure 3

Snapshots of domain morphology during coarsening in zero shear at the times shown by circles in Fig. 2. (a) Diffusive regime run DV1u at $t=1052$. (b) Viscous hydrodynamic regime run DV5u at $t=131$. (c) Viscous hydrodynamic regime run R022 at $t=78.9$. (d) Inertial hydrodynamic regime run R019 at $t=78.9$. Each subfigure shows the entire simulation domain, ${\Lambda }_x\times {\Lambda }_y=1\times 1$ in our units.

Figure 4

Dimensionless scaling plot of lengths vs shear rate. Solid symbols: average of time series of Fig. 5 for $L_{\parallel },L_{\perp }$ for strains $\dot{\gamma }t>100$. Symbols from left to right correspond to R028bs, R022bs, R029bs, R020s, R030s, R019s, and R032s. Errors bars show the standard deviation. Solid lines: power law fits to the solid symbols, suggesting $L_{\parallel }\sim {\dot{\gamma }}^{-0.619}$ and $L_{\perp }\sim {\dot{\gamma }}^{-0.693}$. Upper set of crosses shows the scaled system size ${\Lambda }_y∕L_{\mathrm{VI}}$; lower set shows the scaled interfacial width $l∕L_{\mathrm{VI}}$. Open symbols: data from Ref. 38 for $L_{\parallel },L_{\perp }$, reproduced here for comparison by kind permission of the authors of that study.

Figure 5

(Color online) (a) Larger domain length vs strain $\dot{\gamma }t$. Data sets for decreasing values of long-term temporal average correspond to R020s, R030s, R028b, R029bs, R022bs, R019s, and R032s. (b) Smaller domain length vs strain. Data sets for decreasing values of long-term temporal average correspond to R020s, R030s, R019s, R029bs, R032s, R022bs, and R028b.

Figure 6

Snapshots of the steady-state order parameter for R022b at strain $\dot{\gamma }t=112$ (a) and R032 at strain $\dot{\gamma }t=161$ (b).

Figure 7

(a) Larger domain length $L_{\parallel }$ vs strain $\dot{\gamma }t$. (b) smaller domain length $L_{\perp }$ vs strain $\dot{\gamma }t$. In each case, decreasing values of $L$ at fixed $\dot{\gamma }t=15$ correspond to runs DV5s, DV6s, DV4s, DV3s, DV2s, and DV1s.

Figure 8

Snapshot domain morphologies after many strain units $S=\dot{\gamma }t$ for reciprocal shear rates that, as equivalent times ${\dot{\gamma }}^{-1}$, would be shown by the arrows in Fig. 2a. (a) DV1s, $\dot{\gamma }t=78.9$; (b) DV2s, $\dot{\gamma }t=78.9$; (c) DV3s, $\dot{\gamma }t=78.9$; (d) DV4s, $\dot{\gamma }t=63.2$; (e) DV5s, $\dot{\gamma }t=55.3$; (f) DV6s, $\dot{\gamma }t=47.4$. All these runs have zero inertia, $\rho =0$.