Shear viscosity of a two-dimensional emulsion of drops using a multiple-relaxation-time-step lattice Boltzmann method

Abstract

An extended Benzi-Dellar lattice Boltzmann equation scheme [R. Benzi, S. Succi, and M. Vergassola, Europhys. Lett. 13, 727 (1990)EULEEJ0295-507510.1209/0295-5075/13/8/010; R. Benzi, S. Succi, and M. Vergassola, Phys. Rep. 222, 145 (1992)PRPLCM0370-157310.1016/0370-1573(92)90090-M; P. J. Dellar, Phys. Rev. E 65, 036309 (2002)1063-651X10.1103/PhysRevE.65.036309] is developed and applied to the problem of confirming, at low Re and drop fluid concentration, c, the variation of effective shear viscosity, η_{eff}=η_{1}[1+f(η_{1},η_{2})c], with respect to c for a sheared, two-dimensional, initially crystalline emulsion [here η_{1} (η_{2}) is the fluid (drop fluid) shear viscosity]. Data obtained with our enhanced multicomponent lattice Boltzmann method, using average shear stress and hydrodynamic dissipation, agree well once appropriate corrections to Landau's volume average shear stress [L. Landau and E. M. Lifshitz, Fluid Mechanics, 6th ed. (Pergamon, London, 1966)] are applied. Simulation results also confirm the expected form for f(η_{i},η_{2}), and they provide a reasonable estimate of its parameters. Most significantly, perhaps, the generality of our data supports the validity of Taylor's disputed simplification [G. I. Taylor, Proc. R. Soc. London, Ser. A 138, 133 (1932)1364-502110.1098/rspa.1932.0175] to reduce the effect of one hydrodynamic boundary condition (on the continuity of the normal contraction of stress) to an assumption that interfacial tension is sufficiently strong to maintain a spherical drop shape.

Kinematics of chromodynamic multicomponent lattice Boltzmann simulation with a large density contrast

“…Ba et al derived an equivalent scheme based upon a multiple-relaxation-time LB variant, after Lallemand and Luo [24]. One could also derive schemes based upon inverse multiple-relaxation-time LB variants [25,26]. However, all would contain equivalent evolution equation source terms, i.e., those independent of F β in Eq.…”

Benchmarking of three-dimensional multicomponent lattice Boltzmann equation

“…For D3Q19, we extend the set developed for D2Q9 [9], using Gram-Schmidt orthogonalization to which in Table I(a). Four degenerate eigenvectors necessarily project hydrodynamic modes ρ ≡ i f i and ρu ≡ i f i c i [9] with λ i = 0, six project components of stress P αβ , four "ghosts" are chosen to project N , J (following Refs. [2][3][4]) and five new eigenvectors are denoted E 1 , E 2 , E 3 , X x , and X y .…”

“…(a) Collision matrix left-row eigenvector notation and properties, with eigenvalues and the corresponding equilibria and sources used in Eq. (9). Here, ρu α , P αβ , and (0) αβ represent the α and αβ components of momentum, viscous stress tensor, and momentum flux tensor, respectively.…”

“…The past two decades have seen steady growth in interest in multirelaxation time (MRT) lattice Boltzmann (LB) schemes, which offer enhanced simulation stability [1][2][3][4][5][6][7][8], etc. We extend to D3Q19 a recent D2Q9 variant [9], in which the usual collision matrix is only implicit, being represented by a carefully chosen, modal eigenbasis, which is subject to forced, scalar relaxation. As well as the usual advantages, the new method has transparent analytic properties: its orthogonal modes are defined as polynomials in the lattice basis, as are the elements of the transformation matrix between the distribution function and the mode space.…”

“…where α and β represent the x, y, or z directions, λ p denotes the pth eigenvalue of A ij , and [9]. For eigenvalues, their corresponding left-row eigenvectors, h (p) , p ∈ [0,18], and the modes they project, see Table I(a).…”