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

We document the derivation and implementation of extensions to a two-dimensional, multicomponent lattice Boltzmann equation model, with Laplace law interfacial tension. The extended model behaves in such a way that the boundary between its immiscible drop and embedding fluid components can be shown to describe a vesicle of constant volume bounded by a membrane with conserved length, specified interface compressibility, bending rigidity, preferred curvature, and interfacial tension. We describe how to apply this result to several, independent vesicles. The extended scheme is completely Eulerian, and it represents a two-way coupled vesicle membrane and flow within a single framework. Unlike previous methods, our approach dispenses entirely with the need explicitly to track the membrane, or boundary, and makes no use whatsoever of computationally expensive and intricate interface tracking and remeshing. Validation data are presented, which demonstrate the utility of the method in the simulation of the flow of high volume fraction suspensions of deformable objects.

In silico multi-scale model of transport and dynamic seeding in a bone tissue engineering perfusion bioreactor.

Computer simulations can potentially be used to design, predict, and inform properties for tissue engineering perfusion bioreactors. In this work, we investigate the flow properties that result from a particular poly-L-lactide porous scaffold and a particular choice of perfusion bioreactor vessel design used in bone tissue engineering. We also propose a model to investigate the dynamic seeding properties such as the homogeneity (or lack of) of the cellular distribution within the scaffold of the perfusion bioreactor: a pre-requisite for the subsequent successful uniform growth of a viable bone tissue engineered construct. Flows inside geometrically complex scaffolds have been investigated previously and results shown at these pore scales. Here, it is our aim to show accurately that through the use of modern high performance computers that the bioreactor device scale that encloses a scaffold can affect the flows and stresses within the pores throughout the scaffold which has implications for bioreactor design, control, and use. Central to this work is that the boundary conditions are derived from micro computed tomography scans of both a device chamber and scaffold in order to avoid generalizations and uncertainties. Dynamic seeding methods have also been shown to provide certain advantages over static seeding methods. We propose here a novel coupled model for dynamic seeding accounting for flow, species mass transport and cell advection-diffusion-attachment tuned for bone tissue engineering. The model highlights the timescale differences between different species suggesting that traditional homogeneous porous flow models of transport must be applied with caution to perfusion bioreactors. Our in silico data illustrate the extent to which these experiments have the potential to contribute to future design and development of large-scale bioreactors.

Multi-scale interaction of particulate flow and the artery wall.

We discuss, from the perspective of basic science, the physical and biological processes which underlie atherosclerotic (plaque) initiation at the vascular endothelium, identifying the widely separated spatial and temporal scales which participate. We draw on current, related models of vessel wall evolution, paying particular attention to the role of particulate flow (blood is not a continuum fluid), and proceed to propose, then validate all the key components in a multiply-coupled, multi-scale modeling strategy (in qualitative terms only, note). Eventually, this strategy should lead to a quantitative, patient-specific understanding of the coupling between particulate flow and the endothelial state.

Zenithal bistable device: Comparison of modeling and experiment.

A comparative modeling and experimental study of the zenithal bistable liquid crystal device is presented. A dynamic Landau de Gennes theory of nematic liquid crystals is solved numerically to model the electric field induced latching of the device and the results are compared with experimental measurements and theoretical approximations. The study gives a clear insight into the latching mechanism dynamics and enables the dependence of the device latching on both material parameters and surface shape to be determined. Analytical approximation highlights a route to optimize material selection in terms of latching voltages and the numerical model, which includes an accurate surface representation, recovers the complex surface shape effects. Predictions of device performance are presented as a function of both surface anchoring strength and surface shape and grating pitch. A measurement of the homeotropic anchoring energy has been undertaken by comparing the voltage response as a function of cell gap; we find the homeotropic anchoring energies can be varied in the range 0.5 to 4 ( 10^{-4} J m^{-2} ).

Mesh-free modeling of liquid crystals using modified smoothed particle hydrodynamics.

We present a generalization of the modified smooth particle hydrodynamics simulation technique capable of simulating static and dynamic liquid crystalline behavior. This generalization is then implemented in the context of the Qian-Sheng description of nematodynamics. To test the method, we first use it to simulate switching in both a Fréedericksz setup and a chiral hybrid aligned nematic cell. In both cases, the results obtained give excellent agreement with previously published results. We then apply the technique in a three-dimensional simulation of the switching dynamics of the post aligned bistable nematic device.

Lattice Boltzmann equation method for multiple immiscible continuum fluids.

This paper generalizes the two-component algorithm of Sec. , extending it, in Sec. , to describe N>2 mutually immiscible fluids in the isothermal continuum regime. Each fluid has an independent interfacial tension. While retaining all its computational advantages, we remove entirely the empiricism associated with contact behavior in our previous multiple immiscible fluid models [M. M. Dupin, Phys. Rev. E 73, 055701(R) (2006); Med. Eng. Phys. 28, 13 (2006)] while solidifying the physical foundations. Moreover, the model relies upon a fluid-fluid segregation which is simpler, computationally faster, more free of artifacts (i.e., the interfacial microcurrent), and upon an interface-inducing force distribution which is analytic. The method is completely symmetric between any numbers of immiscible fluids and stable over a wide range of directly input interfacial tension. We present data on the steady-state properties of multiple interface model, which are in good agreement with theory [R. E. Johnson and S. S. Sadhal, Annu. Rev. Fluid Mech. 17, 289 (1985)], specifically on the shapes of multidrop systems. Section is an analysis of the kinetic and continuum-scale descriptions of the underlying two-component lattice Boltzmann model for immiscible fluids, extendable to more than two immiscible fluids. This extension requires (i) the use of a more local kinetic equation perturbation which is (ii) free from a reliance on measured interfacial curvature. It should be noted that viewed simply as a two-component method, the continuum algorithm is inferior to our previous methods, reported by Lishchuk [Phys. Rev. E 67, 036701 (2003)] and Halliday [Phys. Rev. E 76, 026708 (2007)]. Greater stability and parameter range is achieved in multiple drop simulations by using the forced multi-relaxation-time lattice Boltzmann method developed, along with (for completeness) a forced exactly incompressible Bhatnagar-Gross-Krook lattice Boltzmann model, in the Appendix. These appended schemes closely follow those developed by Guo [Phys. Rev. E 65, 046308 (2002)] for the single-relaxation-time scheme.

Validation of multicomponent lattice Boltzmann equation simulations using theoretical calculations of immiscible drop shape.

Quantitative comparison between the measured deformation of a neutrally buoyant drop, obtained with an appropriately conceived three-dimensional, multicomponent lattice Boltzmann equation simulation methods for continuum multicomponent hydrodynamics [Phys. Rev. E 76, 026708 (2007); 76, 026709 (2007)], are shown to be in agreement with the theoretical predictions of Taylor and Acrivos [J. Fluid. Mech. 18(3), 466 (1964)].

Modelling the interaction of haemodynamics and the artery wall: current status and future prospects.

Clinical research has historically focused on the two main strategies of in vivo and in vitro experimentation. The concept of applying scientific theory to direct clinical applications is relatively recent. In this paper we focus on the interaction of wall shear stress with the endothelium and discuss how 'state of the art' computer modelling techniques can provide valuable data to aid understanding. Such data may be used to inform experiment and further, may help identify the key features of this complex system. Current emphasis is on coupling haemodynamics with models of biological phenomena to test hypotheses or predict the likely outcome of a disease or an intervention. New technologies to enable the integration of models of different types, levels of complexity and scales, are being developed. As will be discussed, the ultimate goal is the translation of this technology to the clinical arena.

Multicomponent lattice Boltzmann method for fluids with a density contrast.

We present and verify a multicomponent lattice Boltzmann simulation scheme for two immiscible and incompressible fluids with a large density contrast. Our method is constructed from a continuum approximation description of a single inhomogeneous, and essentially incompressible, fluid. The equations that arise from this analysis are mapped onto an established multicomponent lattice Boltzmann method. The approach avoids the computational expense of a numerical solution of the fluid pressure field in a separate step. We present results obtained with our model which validate the initial assumptions and verify correct static and dynamic operation of the model up to a fluid density contrast ratio of more than 500. The paper concludes with an example that illustrates the potential utility of the approach by modeling a gas bubble rising under gravity and breaking through a free surface.

Lattice Boltzmann algorithm for continuum multicomponent flow.

We present a multicomponent lattice Boltzmann simulation for continuum fluid mechanics, paying particular attention to the component segregation part of the underlying algorithm. In the principal result of this paper, the dynamics of a component index, or phase field, is obtained for a segregation method after U. D'Ortona [Phys. Rev. E 51, 3718 (1995)], due to Latva-Kokko and Rothman [Phys. Rev. E 71 056702 (2005)]. The said dynamics accord with a simulation designed to address multicomponent flow in the continuum approximation and underwrite improved simulation performance in two main ways: (i) by reducing the interfacial microcurrent activity considerably and (ii) by facilitating simulational access to regimes of flow with a low capillary number and drop Reynolds number [I. Halliday, R. Law, C. M. Care, and A. Hollis, Phys. Rev. E 73, 056708 (2006)]. The component segregation method studied, used in conjunction with Lishchuk's method [S. V. Lishchuk, C. M. Care, and I. Halliday, Phys. Rev. E 67, 036701 (2003)], produces an interface, which is distributed in terms of its component index; however, the hydrodynamic boundary conditions which emerge are shown to support the notion of a sharp, unstructured, continuum interface.

Control of the nematic-isotropic phase transition by an electric field.

We use a relatively simple continuum model to investigate the effects of dielectric inhomogeneity within confined liquid-crystal cells. Specifically, we consider, in planar, cylindrical, and spherical geometries, the stability of a nematic-isotropic interface subject to an applied voltage when the nematic liquid crystal has a positive dielectric anisotropy. Depending on the magnitude of this voltage, the temperature, and the geometry of the cell, the nematic region may shrink until the material is completely isotropic within the cell, grow until the nematic phase fills the cell, or, in certain geometries, coexist with the isotropic phase. For planar geometry, no coexistence is found, but we are able to give analytical expressions for the critical voltage for an electric-field-induced phase transition as well as the critical wetting layer thickness for arbitrary applied voltage. In cells with cylindrical and spherical geometries, however, locally stable nematic-isotropic coexistence is predicted, the thickness of the nematic region being controllable by alteration of the applied voltage.

Shear viscosity of bulk suspensions at low Reynolds number with the three-dimensional lattice Boltzmann method.

We report three-dimensional parallel Lagrangian particle simulations using the lattice Boltzmann method, conducted at a low Reynolds number. Using modified Lees-Edwards boundary conditions and directly calculated viscous dissipation, we show that it is possible to recover excellent agreement with the Einstein viscosity formula in the low concentration limit and to predict viscosity corrections for larger concentrations.

Improved simulation of drop dynamics in a shear flow at low Reynolds and capillary number.

The simulation of multicomponent fluids at low Reynolds number and low capillary number is of interest in a variety of applications such as the modeling of venule scale blood flow and microfluidics; however, such simulations are computationally demanding. An improved multicomponent lattice Boltzmann scheme, designed to represent interfaces in the continuum approximation, is presented and shown (i) significantly to reduce common algorithmic artifacts and (ii) to recover full Galilean invariance. The method is used to model drop dynamics in shear flow in two dimensions where it recovers correct results over a range of Reynolds and capillary number greater than that which may be addressed with previous methods.

A multi-component lattice Boltzmann scheme: towards the mesoscale simulation of blood flow.

While blood at the macroscopic scale is frequently treated as a continuum by techniques such as computational fluid dynamics, its mesoscale behaviour is not so well investigated or understood. At this scale, the deformability of each cell within the plasma is important and cannot be ignored. However there is currently a lack of efficient computational techniques able to simulate a large number of deformable particles such as blood cells. This paper addresses this problem and demonstrates the applicability of the authors' recent multi-component lattice Boltzmann method for the simulation of a large number of mutually immiscible liquid species [Dupin MM, Halliday I, Care CM. Multi-component lattice boltzmann equation for mesoscale blood flow. J Phys A: Math Gen 2003;36:8517-34]. In here, biological cells are treated as immiscible, deformable, and relatively viscous drops (compared to the surrounding fluid). The validation of the model is based on the work of Goldsmith on the flow of solid particles, deformable particles and red blood cells [Goldsmith HL, Marlow JC. Flow behavior of erythrocytes. II. Particle motions in concentrated suspensions of ghost cells. J Colloid Interf Sci 1979;71:383-407]. We demonstrate, in particular, that the model recovers Goldsmith's observations on the flow properties of red blood cells and also the experimental observations of Frank on the flow of solid beads [Frank M, Anderson D, Weeks ER, Morris JF. Particle migration in pressure-driven flow of a brownian suspension. J Fluid Mech 2003;493:363-78]. The current article is the first validation of our new lattice Boltzmann model for a large number of deformable particles in this context and demonstrates that the method provides a new, and effective, approach for the modeling of mesoscale blood flow.

Lattice Boltzmann scheme for modeling liquid-crystal dynamics: Zenithal bistable device in the presence of defect motion.

A lattice Boltzmann scheme is presented which recovers the dynamics of nematic and chiral liquid crystals; the method essentially gives solutions to the Qian-Sheng [Phys. Rev. E 58, 7475 (1998)] equations for the evolution of the velocity and tensor order-parameter fields. The resulting algorithm is able to include five independent Leslie viscosities, a Landau-deGennes free energy which introduces three or more elastic constants, a temperature dependent order parameter, surface anchoring and viscosity coefficients, flexoelectric and order electricity, and chirality. When combined with a solver for the Maxwell equations associated with the electric field, the algorithm is able to provide a full "device solver" for a liquid crystal display. Coupled lattice Boltzmann schemes are used to capture the evolution of the fast momentum and slow director motions in a computationally efficient way. The method is shown to give results in close agreement with analytical results for a number of validating examples. The use of the method is illustrated through the simulation of the motion of defects in a zenithal bistable liquid crystal device.

Comment on "Fluctuations of elastic interfaces in fluids: theory, lattice-Boltzmann model, and simulation".

The formulas for the force exerted by the interface upon the fluids, given by Stelitano and Rothman [Phys. Rev. E 62, 6667 (2000)] are corrected.

Shape of an isotropic droplet in a nematic liquid crystal: the role of surfactant.

We investigate theoretically, and numerically, the shape of a droplet of an isotropic fluid immersed in a nematic liquid crystal in the presence of an interfacial layer of surfactant; the droplet size is assumed to be small compared to the extrapolation length of the nematic and homeotropic alignment is favored by the anchoring energy at the nematic-isotropic interface. In a certain range of droplet sizes, the droplets are found to be lens shaped with the rotation axis aligned along the imposed director field and the aspect ratio dependent upon the ratio of anchoring strength and surface tension coefficients. For anchoring strengths large compared to the surface tension, the curvature of the edge of lens is controlled by the bending rigidity of surfactant.

A lattice Boltzmann model of flow blunting.

We review our recent multi-component lattice Boltzmann equation method for the simulation of a large number of mutually immiscible liquid species and then apply it to the simulation of dense volume fraction suspensions of deformable particles in internal geometry. In particular, we illustrate the scope of our method by applying it to the simulation of pipe flows containing a high volume fraction of monodisperse suspended, deformable particles. The particles are modelled as immiscible, relatively viscous liquid drops. We modify the 'solidity' of the particles by modifying their viscosity and surface tension and demonstrate the effect of the solidity upon the blunting of the velocity profile.

A many-component lattice Boltzmann equation simulation for transport of deformable particles.

We review the analysis of single and N-component lattice Boltzmann methods for fluid flow simulation. Results are presented for the emergent pressure field of a single phase incompressible liquid flowing over a backward-facing step, at moderate Reynolds Number, which is compared with the experimental data of Denham & Patrick (1974 Trans. IChE 52, 361-367). We then access the potential of the N-component method for transport of high volume fraction suspensions of deformable particles in pressure-driven flow. The latter are modelled as incompressible, closely packed liquid drops. We demonstrate the technique by investigating the particles' transverse migration in a uniform shear ('lift'), and profile blunting and chaining.

Lattice Boltzmann algorithm for surface tension with greatly reduced microcurrents.

We present an algorithm for inserting an interface between the immiscible phases of a multicomponent lattice Boltzmann fluid which is based solely upon the appropriate continuum physics: stress boundary conditions and continuity of velocity. Results are presented for the algorithm when applied to static, neutrally buoyant drops. It is shown that the present algorithm gives a significant reduction in the spurious velocities which are reported for previous schemes and a concomitant improvement in the isotropy of the interface.