We simulate by lattice Boltzmann the steady shearing of a binary fluid mixture undergoing phase separation with full hydrodynamics in two dimensions. Contrary to some theoretical scenarios, a dynamical steady state is attained with finite domain lengths Lx,y in the directions (x, y) of velocity and velocity gradient. Apparent scaling exponents are estimated as Lx ∼γ −2/3 and Ly ∼γ −3/4 . We discuss the relative roles of diffusivity and hydrodynamics in attaining steady state.PACS numbers: 47.11.+jSystems that are not in thermal equilibrium play a central role in modern statistical physics, and arise in areas ranging from soap manufacture to subcellular biology [1]. Such systems include two important classes: those that are evolving towards Boltzmann equilibrium (e.g., by phase separation following a temperature quench), and those that are 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 it is not known [2,3] whether coarsening continues indefinitely, as it does without shear, or whether a steady state 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.) Experimentally, saturating length scales are reportedly reached after a period of anisotropic domain growth [2,4]. However, the extreme elongation of domains along the flow direction means that, even in experiments, finite size effects could play an essential role in such saturation [5]. 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 [5]. In real fluids, however, the velocity fluctuates strongly in nonlinear response to the advected concentration field, and hydrodynamic scaling arguments, balancing either interfacial and viscous or interfacial and inertial forces, predict saturation (e.g., L ∼γ −1 or L ∼γ −2/3 ) [3,6,7]. Given these experimental and theoretical differences of opinion, computer simulations of sheared binary fluids, with full hydrodynamics, are of major interest.The aforementioned scaling arguments cannot really distinguish one Cartesian direction from another, but even in theories that can do so, a two dimensional (2D) representation, suppressing z, is expected to capture the main physics [5]. (Without shear, subtle non-scaling effects arise in 2D from the formation of disconnected droplets [8], but shear seems to suppress these [9].) Performing simulations in 2D is therefore a fair compromise, especially given the extreme computational demands of the full 3D problem [3,10]. But, apart from [9,11]...

Abstract. We present a progress report on our work on lattice Boltzmann methods for colloidal suspensions. We focus on the treatment of colloidal particles in binary solvents and on the inclusion of thermal noise. For a benchmark problem of colloids sedimenting and becoming trapped by capillary forces at a horizontal interface between two fluids, we discuss the criteria for parameter selection, and address the inevitable compromise between computational resources and simulation accuracy.

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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...