Binary fluids under steady shear in three dimensions

Abstract

We simulate by lattice Boltzmann the steady shearing of a binary fluid mixture with full hydrodynamics in three dimensions. Contrary to some theoretical scenarios, a dynamical steady state is attained with finite correlation lengths in all three spatial directions. Using large simulations we obtain at moderately high Reynolds numbers apparent scaling exponents comparable to those found by us previously in 2D. However, in 3D there may be a crossover to different behavior at low Reynolds number: accessing this regime requires even larger computational resource than used here. PACS numbers: 64.75.+g, 47.11.Qr Systems that are not in thermal equilibrium play a central role in modern statistical physics [1]. They include two important classes: those evolving towards Boltzmann equilibrium (e.g., by phase separation following a temperature quench), and those maintained in nonequilibrium by continuous driving (such as a shear flow). Of fundamental interest, and surprising physical subtlety, are systems combining both features -such as a binary fluid undergoing phase separation in the presence of shear. Here a central issue [2,3] is whether coarsening continues indefinitely, as it does without shear, or whether a nonequilibrium steady state (NESS) is reached, in which the characteristic length scales L x,y,z of the fluid domain structure attain finiteγ-dependent values at late times. (We define the mean velocity as u x =γy so that x, y, z are velocity, velocity gradient and vorticity directions respectively;γ is the shear rate.)Our recent simulations, building on earlier work of others [4,5], have shown that in two dimensions (2D), a NESS is indeed achieved [6]. In 3D, the situation is more subtle. Fourier components of the composition field whose wavevectors lie along the vorticity direction feel no direct effect of the mean advective velocity [2,7]. Therefore it might be possible for coarsening to proceed indefinitely by pumping through tubes of fluid oriented along z [3]. Another crucial difference is that in 2D fluid bicontinuity is possible only by fine tuning to a percolation threshold at 50:50 composition (assuming fluids of equal viscosity) so that the generic situation is one of droplets. (Indeed, for topological reasons, droplets are implicated even at threshold [4].) In contrast, in 3D both fluids remain continuously connected across the sample throughout a broad composition window either side of 50:50.In 3D experiments, saturating length scales are reportedly reached after a period of anisotropic domain growth [2,8]. However, the extreme elongation of domains along the flow direction means that, even in experiments, finite size effects could play a role in such saturation [9]. Theories in which the velocity does not fluctuate, but does advect the diffusive fluctuations of the concentration field, predict instead indefinite coarsening, with length scales L y,z scaling asγ-independent powers of the time t since quench, and (typically) L x ∼γtL y [9]. As emphasized in [6], in real fluids, however, the velocit...

Role of inertia in nonequilibrium steady states of sheared binary fluids

“…Despite previous experimental [4,5,6,7,8,9,10,11,12,13,14], numerical [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] and theoretical [18,19,20,21,29,32,33,34,35,36,37] work, this question remained open until the recent simulation studies of Stansell et al in Refs [38,39]. Using Lattice Boltzmann techniques, they gave convincing evidence for the formation of nonequilibrium steady states, with domains of a finite size set by the inverse shear rate, independent of the system size.…”

“…The same scalings L ∼γ −2/3 and L ⊥ ∼γ −3/4 were suggested for the major and minor principal lengths of the domains. Slightly different scalings, discussed below, were obtained by the same group in a more recent 3D study [39].…”

“…[38,39] have non zero fluid inertia. Indeed, even when attempting to access the limit of zero inertia, a small but non-zero inertia remains a practical requirement of the Lattice Boltzmann technique.…”

“…A remarkable achievement of Refs. [38,39] was the exploration, by judicious parameter steering, of a range of inverse shear rates spanning six decades on a scaling plot. For reasons discussed below this range would, a priori, be expected to encompass two different regimes, separately dominated by viscous and inertial dynamics.…”

Spinodal decomposition of asymmetric binary fluids in a micro-Couette geometry simulated with molecular dynamics

“…25,27,28 The domains cannot grow indefinitely in the shear-gradient direction but can attain a ratedependent nonequilibrium steady state; it is not clear whether the growth in the two perpendicular directions also saturates. 25,29 Shear may also be applied to induce phase separation in dynamically asymmetric mixtures. 30,31 Deviations from the ideal case also occur in the presence of walls and other fixed obstacles, which limit the attainable domain size, interfere with the buildup of a hydrodynamic flow field, and may have a preference for wetting by either constituent of the mixture.…”