Benchmarking of three-dimensional multicomponent lattice Boltzmann equation

We present a challenging validation of phase field multicomponent lattice Boltzmann equation (MCLBE) simulation against the $\text{Re}=0$ Stokes flow regime Taylor-Einstein theory of dilute suspension viscosity. By applying a number of recent advances in the understanding and the elimination of the interfacial microcurrent artefact, extending to a three-dimensional class of stability-enhancing multiple relaxation time collision models (which require no explicit collision matrix, note) and developing new interfacial interpolation schemes, we are able to obtain data that show that MCLBE may be applied in new flow regimes. Our data represent one of the most stringent tests yet attempted on LBE—one which received wisdom would preclude on grounds of overwhelming artefact flow.

Figure 1

Viscous stress field ${\sigma }_{xy}^{\prime }$ in the equatorial plane of a spherical red drop, radius $R=20$, suspended in a blue fluid which is sheared, at $\text{Re}=0$ for two viscosity ratios, $\mathrm{\Lambda }$, with ${\eta }_B=\frac{1}{3}$ (continuous component) and $\frac{R}{L}=\frac{1}{6}$. Case 1, $\mathrm{\Lambda }=\frac{1}{16}$: (a) Taylor's theory, (b) interfacial interpolation based upon a density-weighted harmonic mean of separated fluids' parameter $\tau =\frac{1}{{\lambda }_4}$ [see Eq. (27)], (c) interfacial interpolation based upon a density-weighted harmonic mean of shear viscosity [see Eq. (28)], and (d) interfacial interpolation based upon a density-weighted arithmetic mean of shear viscosity [see Eq. (29)]. Case 2, $\mathrm{\Lambda }=12$: (e) Taylor's theory, (f) interfacial interpolation based upon a density-weighted harmonic mean of separated fluids' parameter $\tau =\frac{1}{{\lambda }_4}$ [see Eq. (27)], (g) interfacial interpolation based upon a density-weighted harmonic mean of shear viscosity [see Eq. (28)], and (h) interfacial interpolation based upon a density-weighted arithmetic mean of shear viscosity [see Eq. (29)].

Figure 2

Effective suspension viscosity, ${\eta }_{\text{eff}}$, as a function of concentration, $c$ (discrete crosses linearly interpolated by dotted lines), for the indicated range of drop/background fluid viscosity ratio, $\mathrm{\Lambda }=\frac{{\eta }_R}{{\eta }_B}$, together with the variation predicted by the Taylor-Einstein result, ${\eta }_{\text{eff}}^{\left(T\right)}={\eta }_B\left[1+\left(\frac{\frac{5}{2}{\eta }_R+{\eta }_B}{{\eta }_R+{\eta }_B}\right)c\right]$ (continuous line of corresponding color). These data were obtained using the interpolation defined in Eq. (27).

Figure 3

Data equivalent to that shown in Fig. 2 using different interfacial interpolation methods. Effective suspension viscosity, ${\eta }_{\text{eff}}$, as a function of concentration, $c$ (discrete crosses linearly interpolated by dotted lines), for the indicated range of drop/background fluid viscosity ratio, $\mathrm{\Lambda }=\frac{{\eta }_R}{{\eta }_B}$, together with the variation predicted by the Taylor-Einstein result, ${\eta }_{\text{eff}}^{\left(T\right)}={\eta }_B\left[1+\left(\frac{\frac{5}{2}{\eta }_R+{\eta }_B}{{\eta }_R+{\eta }_B}\right)c\right]$ (continuous line of corresponding color). These data were obtained using an interpolation based upon (a) the harmonic mean of the separated fluids' shear viscosity defined in Eq. (28) and (b) the arithmetic mean of shear viscosity defined in Eq. (29).

Figure 4

Relative error of interfacial interpolation method, $\epsilon$, for a range of $\mathrm{\Lambda }$, expressed in percentage. For all data in Fig. 2, $\epsilon \equiv \frac{{\eta }_{\text{eff}}^{\left(T\right)}-{\eta }_{\text{eff}}}{{\eta }_{\text{eff}}^{\left(T\right)}}\times 100\%$ was computed for each of the three interfacial interpolation methods considered, as identified in the key.