Interaction pressure tensor on high-order lattice Boltzmann models for nonideal fluids

In this work we address the application of pseudopotentials directly on high-order lattice Boltzmann models. We derive a general expression for the pressure tensor on high-order lattices considering all nonideal interactions, including intra- and intermolecular interactions, following the discrete lattice theory introduced by X. Shan [Phys. Rev. E 77, 066702 (2008)]. From the derived expression, a generalized continuum approximation, truncated at fourth-order isotropy, is obtained that is readily applicable to high-order lattices. With this, we demonstrate that high-order lattice models with pseudopotentials can satisfy thermodynamic consistency. The derived generalized expression and continuum approximation are validated for the case of a flat interface and compared against the standard definition available from the literature. The generalized expression is also shown to accurately reproduce the Laplace experiment for a variety of high-order lattice structures. This work sets the preliminary steps towards the application of high-order lattice models for simulating nonideal fluid mixtures.

Figure 1

Two-dimensional high-order lattice model constructed from projection of D1Q7 [37], i.e., a total of $7^2=49$ lattice velocities. We denote this lattice structure by Q49ZOTT, based on its D1Q7 construction on zero-one-two-three (ZOTT) velocity, analogous to the zero-one-three (ZOT) construction used for D2Q25ZOT in [37]. Refer to Appendix pp1 for details.

Figure 2

Illustration of the force vectors along the vectors ${\mathit{\xi }}_{\alpha }$ that belong to symmetry group $\overline{z}$ and pass through (bold solid arrows) the infinitesimal horizontal $d{\mathrm{A}}^{\left(x\right)}$ and vertical $d{\mathrm{A}}^{\left(y\right)}$ unit area elements (bold red line) centered at $\mathit{x}$. In reference to the Q49ZOTT lattice in Fig. 1, force vectors along (a) ${\mathit{\xi }}_{25}$ ($\overline{z}=9$) and (b) ${\mathit{\xi }}_{45}$ ($\overline{z}=18$) are shown. In both (a) and (b), there is a total of six ($2{\mathcal{E}}^z$) force vectors contributing equally to their respective interaction pressure tensor, i.e., ${\mathcal{P}}_{ij}^{\text{int},\overline{z}=9}$ and ${\mathcal{P}}_{ij}^{\text{int},\overline{z}=18}$. The six force vectors consist of four different classes of vectors; one (${\mathit{F}}_1$) starts and another ends (accounted for along the opposite ${\mathit{\xi }}_a$) at $\mathit{x}$ and two (${\mathit{F}}_2$ and ${\mathit{F}}_3$) pass through $\mathit{x}$. Labeled force vectors are those to be explicitly considered along each respective ${\mathit{\xi }}_{\alpha }$. The unlabeled bold dotted arrows are not considered along vectors ${\mathit{\xi }}_{\alpha }$ but are instead accounted for along the opposite ${\mathit{\xi }}_a$ (hence the direction of arrows) as a result of symmetry and summation around the lattice, for example, in this case (a) ${\mathit{\xi }}_{27}$ and (b) ${\mathit{\xi }}_{47}$. These are automatically considered in Eq. (15).

Figure 3

Illustration of the force vectors along the vectors ${\mathit{\xi }}_{\alpha }$ (of the Q49ZOTT lattice in Fig. 1) that belong to symmetry group $\widehat{z}$ and pass through (bold solid arrows) the vertical infinitesimal $d{\mathrm{A}}^{\left(y\right)}$ unit area element (red bold solid line). The unlabeled bold dotted arrows which pass through the red bold dotted line ($d{\mathrm{A}}^{\left(y\right)}$) are accounted for along ${\mathit{\xi }}_a$. In both (a) and (b), it is clear that considering contributions to $d{\mathrm{A}}^{\left(y\right)}$ in turn correctly accounts for those passing through the horizontal infinitesimal $d{\mathrm{A}}^{\left(x\right)}$ unit area element (red bold solid line) centered at $\mathit{x}$. (a) For vector ${\mathit{\xi }}_{30}$ there are six ($2{\mathcal{E}}^z$) force vectors with equal contribution to ${\mathcal{P}}_{ij}^{\text{int},\widehat{z}=10}$. (b) For vector ${\mathit{\xi }}_{38}$ there are also six force vectors, however, they do not contribute equally to ${\mathcal{P}}_{ij}^{\text{int},\widehat{z}=13}$. The contributions to $\widehat{z}=10$ are automatically considered in Eq. (16), whereas those in $\widehat{z}=13$ can still be obtained with Eq. (16) but require a unique condition ($C_{\beta }$) to correct the different contributions of ${\mathit{F}}_2$ and ${\mathit{F}}_3$ (see the text).

Figure 4

Theoretical prediction of equation of state $P_0$ (dashed and dotted curved lines) for a range of $\mathcal{G}/G_{\text{crit}}$ up to $G_{\text{max}}=1.225G_{\text{crit}}$ (red dot-dashed line), solved for (a) the Q9 lattice model ($\epsilon =0$) with $\psi \left(\rho \right)=e^{-1/\rho }$ and (b) the Q49ZOTT lattice model with Eqs. (30) and (35). The thermodynamic coexistence curve (bold solid green line) drawn within the shaded region is obtained from the Maxwell construction (34). Numerical simulations were conducted for $\mathcal{G}/G_{\text{crit}}=\{1.05,1.075,1.1,1.125,1.15,1.175,1.2,G_{\text{max}}\}$ where the bulk properties of each phase ($•$), namely, densities and corresponding static pressure, are obtained from Eqs. (7a), (9), and (17).

Figure 5

Numerical results at equilibrium for the flat-interface test using the Q9 lattice model. The pseudopotential interactions (11) are deployed (a) directly on the lattice structure (LS) and (b) with the E8 model. The interaction component of $P_{xx}^{\text{tot}}$ obtained numerically $P_{xx}$ ($\times$) from Eq. (17) and its continuum approximation $P_{xx}^{\text{CA}}$ ($\circ$) obtained from Eqs. (29) and (30) are compared against the standard solution $P_{xx}^{*\text{CA}}$ (solid line) from Eq. (19). (c) Constancy $\mathrm{\Delta }{P}_{xx}\%$ of the $P_{xx}$ obtained numerically (17), tested for the Q9 LS, E8, E10, and E12 along spatial dimension $x$. For each lattice, the constancy $\mathrm{\Delta }{P}_{xx}\%$ is calculated as the percentage difference of $P_{xx}$ along $x$ against the reference $P_{xx}\left(x\right)$ measured at center $x=100$.

Figure 6

Numerical results at equilibrium for the flat-interface test for a variety of lattice structures. (a) Pseudopotential interactions (11) are deployed directly on the Q25ZOT lattice structure. The interaction component of $P_{xx}^{\text{tot}}$ obtained numerically $P_{xx}$ ($\times$) from Eq. (17) and its continuum approximation $P_{xx}^{\text{CA}}$ ($\circ$) from Eqs. (29) and (30) are compared against the standard solution $P_{xx}^{*\text{CA}}$ (solid line) from Eq. (19). (b) Constancy $\mathrm{\Delta }{P}_{xx}\%$ of the $P_{xx}$ obtained numerically with Eq. (17) along spatial dimension $x$, tested for Q25ZOT on the LS and with E12, in addition to Q17ZOT and Q49ZOTT. Here, for Q49ZOTT, the sound speed $c_s$ was modified to match that of the ZOT variants.

Figure 7

Single static circular droplet results for Q49ZOTT* with modified $c_s$. (a) Mismatch between normal and tangential components of the total pressure tensor (9), (10), and (17). (b) Cross-sectional profiles along $x$ at $y=n_y/2$, comparing the interaction component of $P_{xx}^{\text{tot}}$ obtained numerically $P_{xx}$ ($\times$) from Eq. (17), its continuum approximation $P_{xx}^{\text{CA}}$ ($\circ$) from Eqs. (29) and (30), and the standard solution $P_{xx}^{*\text{CA}}$ (solid line) from Eq. (19).

Figure 8

Laplace experiment for the single static circular droplet test including intra- and inter-interactions for 11 different lattice models. The numerical results (symbols) are obtained by setting the initial $R_0$ [in Eq. (D2)] and then at equilibrium, by postprocessing, measuring the actual $R$ and calculating the difference between the equation of state, computed by Eqs. (9) and (17), of the two bulk phases, i.e., $\mathrm{\Delta }P=P_0^{\varphi }-P_0^{\phi }$. The theoretical approximations (lines) are obtained by computing $\sigma$ from Eqs. (37) and (38), treated as a one-dimensional problem, using the density profiles from equilibrated numerical results (for each respective lattice) for the case $R\approx R_0=25$. The Laplace $\mathrm{\Delta }P=\sigma /R$ is then solved for the range $R=12,\cdots ,25$ directly using this approximated $\sigma$. For visual purposes, all 11 different lattices are presented over three plots (a-c) and theoretical approximations between different lattices are differentiated using solid, dashed, and dotted lines. Note in (b) the results for Q49ZOTT* ($⧫$), both the theoretical approximation and the numerical, are the same as those presented in (c).