Nonequilibrium Steady States in Sheared Binary Fluids

Abstract

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]...

Binary fluids under steady shear in three dimensions

“…(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.…”

“…Even given the excellent parallel scaling of LB on multiprocessor machines, each one of these 12 datasets required more computational resource than the entirety of Ref. [6]. The production runs reported here were performed using 1024 processors of the IBM Blue Gene/L machine at the University of Edinburgh.…”

“…Although we neglect thermal fluctuations in our fluid, as appropriate for dynamics near a zero-temperature fixed point [17], a fluctuating local velocity field still arises via nonlinear interaction between the order parameter field and flow field. To help control errors, we adhered as far as possible to previously used parameter values and protocols [6,12]. However, the sheared 3D case showed significant stability problems compared with either the 3D unsheared case or 2D sheared case.…”

“…In [12], finite size effects (in the absence of shear) were considered quantitatively under control when the correlation length L was less than 1/4 of the system size Λ. In [6] this criterion was applied to time-averaged correlation lengths L x,y in the 2D sheared system. However, the actual system size dependence of L x,y,z in both 2D [6] and 3D (this work) suggests that under shear this criterion is unnecessarily strict, at least if the purpose is to eliminate the qualitatively artefactual states that arise directly from finite size effects.…”

Lattice Boltzmann Simulations of Microemulsions and Binary Immiscible Fluids Under Shear

“…Our findings are in agreement with Ginzburg-Landau and Langevin calculations [7,8] as well as two-dimensional lattice-Boltzmann simulations of binary immiscible fluid mixtures as presented in [13,34,38]. However, to the best of our knowledge, there are no detailed theoretical studies of the dependence of domain growth properties on the surfactant concentration.…”