Lattice Boltzmann simulations of nonequilibrium fluctuations in a nonideal binary mixture

In recent years the lattice Boltzmann (LB) methodology has been fruitfully extended to include the effects of thermal fluctuations. So far, all studied cases pertain to equilibrium fluctuations, i.e., fluctuations with respect to an equilibrium background state. In this paper we take a step further and present results of fluctuating LB simulations of a binary mixture confined between two parallel walls in the presence of a constant concentration gradient in the wall-to-wall direction. This is a paradigmatic setup for the study of nonequilibrium (NE) fluctuations, i.e., fluctuations with respect to a nonequilibrium state. We analyze the dependence of the structure factors for the hydrodynamical fields on the wave vector $\mathit{q}$ in both the directions parallel and perpendicular to the walls, highlighting the long-range (${\sim \left|\mathit{q}\right|}^{-4}$) nature of correlations in the NE framework. Results at the small scales (high wave numbers) quantitatively agree with the predictions of fluctuating hydrodynamics without fitting parameters. At larger scales (low wave numbers), however, results show finite-size effects induced by confinement and call for further studies aimed at controlling boundary conditions in the fluctuating LB framework as well as compressibility effects. Moreover, in the presence of a nonideal equation of state of the mixture, we also observe that the (spatially homogeneous) average pressure changes, due to a genuinely new contribution triggered by NE fluctuations. These NE pressure effects are studied at changing the system size and the concentration gradient. Taken all together, we argue that the results of this article are useful and instrumental to boost the applicability of the fluctuating LB methodology in the framework of NE fluctuations, possibly in conjunction with experiments.

Figure 1

Setup for the numerical simulations. The wall-to-wall distance is $L$ and we take the convention that $z=0$ indicates the center of the channel. A linear concentration background profile $c_0\left(z\right)=1/2+z\nabla c_0$ is imposed, corresponding to a constant concentration gradient $\nabla c_0$ in the vertical direction. The vectors ${\mathit{v}}_{1-8}$, together with ${\mathit{v}}_0\equiv \mathit{0}$, act as lattice links in the D2Q9 LB simulations.

Figure 2

Spectra of fluctuations of hydrodynamical fields around an equilibrium background ($\nabla c_0=0$). Top panel: Velocity in the streamflow ($x$) direction; theoretical prediction in Eq. (22). Central panel: Velocity in the wall-to-wall ($z$) direction; theoretical prediction in Eq. (22). Bottom panel: Concentration fluctuations; theoretical prediction in Eq. (23). All the theoretical predictions refer to unbounded fluids. In all cases, the viscosity and diffusivity have been fixed to their reference values (cf. Sec. 4).

Figure 3

Spectra of fluctuations of hydrodynamical fields around a NE background ($\nabla c_0\ne 0$). Top panel: Velocity in the streamflow ($x$) direction; equilibrium theoretical prediction in Eq. (22). Central panel: Velocity in the wall-to-wall ($z$) direction; equilibrium theoretical prediction in Eq. (22). Bottom panel: Concentration fluctuations; equilibrium theoretical prediction in Eq. (23). In all cases, the simulation parameters have been fixed to their reference values (cf. Sec. 4).

Figure 4

Real space NE correlations of velocity and concentration fluctuations. They are obtained by subtracting to the measured correlations the equilibrium values reported in Eqs. (24) and (25). The simulation parameters have been fixed to their reference values (cf. Sec. 4).

Figure 5

Nonequilibrium structure factor contribution [see Eqs. (27) and (28)] as a function of the dimensionless wave number $\tilde{q}$ [see Eq. (26)]; the theoretical prediction, corresponding to ${\tilde{S}}_{{}^{{}_{\text{NE}}}}\left(\tilde{q}\right)={\tilde{q}}^{-4}$, is obtained from Eqs. (1, 2, 3) for unbounded systems. Top panel: NE structure factor contribution for different concentration gradients $\nabla c_0$. Central panel: NE structure factor contribution for different wall-to-wall separations $L$. Bottom panel: NE structure factor contribution for different kinematic viscosities $\nu$. The simulation parameters not given in the legends are detailed in Sec. 4.

Figure 6

We report the ratio between the nonequilibrium structure factors computed by implementing the noise according to Eqs. (14) and (15) and using the constant reference value for the mass densities (hom) and local space-dependent (loc) density. Different transport coefficient are considered. The simulation parameters not given in the legends are detailed in Sec. 4.

Figure 7

Nonequilibrium structure factor contribution as a function of the dimensionless wavenumber $\tilde{q}$ [see Eq. (26)]. Different analytical formulas are checked, dependently on the boundary conditions Eq. (29). The choice of the simulation parameters is detailed in Sec. 4.

Figure 8

Average total pressure. Top panel: Average total pressure for an ideal binary mixture ($G=0$). Bottom panel: Average total pressure for a nonideal binary mixture ($G>0$). The choice of the simulation parameters is detailed in Sec. 4.

Figure 9

Deviation of the average total mass density from it noiseless value in both the ideal ($G=0$) and nonideal ($G>0$) cases. The choice of the simulation parameters is detailed in Sec. 4.

Figure 10

Scaling laws for the NE contribution to the spatial average pressure computed according to Eq. (31). Top panel: NE pressure contribution as a function of the concentration gradient $\nabla c_0$. Central panel: NE pressure contribution as a function of the wall-to-wall separation $L$. Bottom panel: NE pressure contribution as a function of the thermal energy $k_{{}^{{}_{\text{B}}}}T$. The choice of the simulation parameters not given in the legends is detailed in Sec. 4.