The late-stage demixing following spinodal decomposition of a three-dimensional symmetric binary fluid mixture is studied numerically, using a thermodynamically consistent lattice Boltzmann method. We combine results from simulations with different numerical parameters to obtain an unprecedented range of length and time scales when expressed in reduced physical units. (These are the length and time units derived from fluid density, viscosity, and interfacial tension.) Using eight large (2563) runs, the resulting composite graph of reduced domain size l against reduced time t covers 1 [less, similar] l [less, similar] 105, 10 [less, similar] t [less, similar] 108. Our data are consistent with the dynamical scaling hypothesis that l(t) is a universal scaling curve. We give the first detailed statistical analysis of fluid motion, rather than just domain evolution, in simulations of this kind, and introduce scaling plots for several quantities derived from the fluid velocity and velocity gradient fields. Using the conventional definition of Reynolds number for this problem, Reφ = ldl/dt, we attain values approaching 350. At Reφ [greater, similar] 100 (which requires t [greater, similar] 106) we find clear evidence of Furukawa's inertial scaling (l [similar] t2/3), although the crossover from the viscous regime (l [similar] t) is both broad and late (102 [less, similar] t [less, similar] 106). Though it cannot be ruled out, we find no indication that Reφ is self-limiting (l [similar] t1/2) at late times, as recently proposed by Grant & Elder. Detailed study of the velocity fields confirms that, for our most inertial runs, the RMS ratio of nonlinear to viscous terms in the Navier-Stokes equation, R2, is of order 10, with the fluid mixture showing incipient turbulent characteristics. However, we cannot go far enough into the inertial regime to obtain a clear length separation of domain size, Taylor microscale, and Kolmogorov scale, as would be needed to test a recent 'extended' scaling theory of Kendon (in which R2 is self-limiting but Reφ not). Obtaining our results has required careful steering of several numerical control parameters so as to maintain adequate algorithmic stability, efficiency and isotropy, while eliminating unwanted residual diffusion. (We argue that the latter affects some studies in the literature which report l [similar] t2/3 for t [less, similar] 104.) We analyse the various sources of error and find them just within acceptable levels (a few percent each) in most of our datasets. To bring these under significantly better control, or to go much further into the inertial regime, would require much larger computational resources and/or a breakthrough in algorithm design

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Abstract. We report a molecular dynamics computation of the entropic depletion force induced between two large spheres (colloidal particles) immersed in a fluid of small spheres. The effective pair potential obtained by numerical integration of the force is used in a Monte Carlo study of the phase behaviour of the binary mixture. The simulation results are compared with the relevant theoretical predictions that follow from various integral equations for liquid mixtures. The simulations provide evidence for a spinodal instability in a liquid mixture of hard spheres with a size ratio of 0.1.

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We simulate late-stage coarsening of a 3D symmetric binary fluid using a lattice Boltzmann method. With reduced lengths and times, l and t respectively (with scales set by viscosity, density and surface tension) our data sets cover 1 < ∼ l < ∼ 10 5 , 10 < ∼ t < ∼ 10 8 . We achieve Reynolds numbers approaching 350. At Re > ∼ 100 we find clear evidence of Furukawa's inertial scaling (l ∼ t 2/3 ), although the crossover from the viscous regime (l ∼ t) is very broad. Though it cannot be ruled out, we find no indication that Re is self-limiting (l ∼ t 1/2 ) as proposed by M. Grant and K. R. Elder [ Phys. Rev. Lett. 82, 14 (1999) ].PACS numbers: 64.75+g, 07.05. Tp, 82.20.Wt When an incompressible binary fluid mixture is quenched far below its spinodal temperature, it will phase separate into domains of different composition. Here we consider only fully symmetric 50/50 mixtures in three dimensions, for which these domains will, at late times, form a bicontinuous structure, with sharp, well-developed interfaces. The late-time evolution of this structure remains incompletely understood despite theoretical [1-4], experimental [5] and simulation [6-10] work.As emphasized by Siggia [1] and Furukawa [2], the physics of spinodal decomposition involves capillary forces, viscous dissipation, and fluid inertia. Thus, assuming that no other physics enters, the control parameters are interfacial tension σ, fluid mass density ρ, and shear viscosity η. From these can be constructed only one length, L 0 = η 2 /ρσ and one time T 0 = η 3 /ρσ 2 . We define the lengthscale L(T ) of the domain structure at time T via the structure factor S(k) as L = 2π S(k)dk/ kS(k)dk. The exclusion of other physics in late stage growth then leads us to the dynamical scaling hypothesis [1,2]: l = l(t), where we use reduced time and length variables, l ≡ L/L 0 and t ≡ (T − T int )/T 0 . Since dynamical scaling should hold only after interfaces have become sharp, and transport by molecular diffusion ignorable, we have allowed for a nonuniversal offset T int ; thereafter the scaling function l(t) should approach a universal form, the same for all (fully symmetric, deepquenched, incompressible) binary fluid mixtures.It was argued further by Furukawa [2] that, for small enough t, fluid inertia is negligible compared to viscosity, whereas for large enough t the reverse is true. Dimensional analysis then requires the following asymptotes:where, if dynamical scaling holds, amplitudes b, c (and the crossover time t * ) are universal. The Reynolds number, conventionally defined as, Re = ρ/ηLdL/dT = ll, becomes indefinitely large in the inertial regime, Eq. (2).In a recent paper, Grant and Elder have argued [4] that the Reynolds number cannot, in fact, continue to grow indefinitely. If so, Eq. (2) is not truly the large t asymptote, which must instead have l ∼ t α with α ≤ 1 2 . Grant and Elder argue that at large enough Re, turbulent remixing of the interface will limit the coarsening rate [4], so that Re stays bounded. A saturating Re (which they estimate as R...