Multiple-component lattice Boltzmann equation for fluid-filled vesicles in flow

Algorithm to implement unsteady jump boundary conditions within the lattice Boltzmann method

An algorithm is proposed to implement unsteady jump boundary conditions in the lattice Boltzmann method (LBM). This is useful for dealing with problems of mass transfer across membranes that exhibit resistance and discontinuity in concentration. The algorithm is simple to implement into an existing LBM-based code that computes diffusion and advection of a solute. Analytical solutions are recovered in the limiting case of a planar membrane. When combined with the immersed boundary method, the algorithm can handle moving deformable boundaries that adopt arbitrary geometries. Simulations of controlled solute release from stationary rigid and moving deformable particles are given as a proof of concept.

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

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.

Local membrane length conservation in two-dimensional vesicle simulation using a multicomponent lattice Boltzmann equation method

We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013)10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition.

Fluid vesicles in flow

We review the dynamical behavior of giant fluid vesicles in various types of external hydrodynamic flow. The interplay between stresses arising from membrane elasticity, hydrodynamic flows, and the ever present thermal fluctuations leads to a rich phenomenology. In linear flows with both rotational and elongational components, the properties of the tank-treading and tumbling motions are now well described by theoretical and numerical models. At the transition between these two regimes, strong shape deformations and amplification of thermal fluctuations generate a new regime called trembling. In this regime, the vesicle orientation oscillates quasi-periodically around the flow direction while asymmetric deformations occur. For strong enough flows, small-wavelength deformations like wrinkles are observed, similar to what happens in a suddenly reversed elongational flow. In steady elongational flow, vesicles with large excess areas deform into dumbbells at large flow rates and pearling occurs for even stronger flows. In capillary flows with parabolic flow profile, single vesicles migrate towards the center of the channel, where they adopt symmetric shapes, for two reasons. First, walls exert a hydrodynamic lift force which pushes them away. Second, shear stresses are minimal at the tip of the flow. However, symmetry is broken for vesicles with large excess areas, which flow off-center and deform asymmetrically. In suspensions, hydrodynamic interactions between vesicles add up to these two effects, making it challenging to deduce rheological properties from the dynamics of individual vesicles. Further investigations of vesicles and similar objects and their suspensions in steady or time-dependent flow will shed light on phenomena such as blood flow.

Multicomponent lattice Boltzmann equation method with a discontinuous hydrodynamic interface

In the multicomponent lattice Boltzmann equation simulation method (MCLB), applied to the continuum regime of fluid flow, the finite width of the fluid-fluid interface introduces unphysical scales. We present a practical, robust, computationally efficient, and easy to implement solution to this problem which needs only low order interpolation to be stable and accurate and is applicable to any MCLB variant which uses a continuous phase field to distinguish between immiscible fluids with arrested coalescence. Our method extends the ideas of Kim and Pitsch, [Phys. Fluids 19, 108101 (2007)] and uses no external force distribution whatsoever to generate continuum interfacial physics, i.e., the Laplace law and no traction conditions on interfacial stresses. As such, it is amenable to the simplest form of Chapman-Enskog analysis used for lattice Boltzmann models. We assess our method and proceed to compare key results obtained with it against other equivalent data, obtained using the established continuum regime MCLB technique based upon the work of Lishchuk, Care, and Halliday, [Phys. Rev. E 67, 036701 (2003)] and Halliday, Hollis, and Care, [Phys. Rev. E 76, 026708 (2007)], quantifying performance in terms of the minimum feasible capillary available to simulation using that technique.