Chromodynamic multirelaxation-time lattice Boltzmann scheme for fluids with density difference.

We develop, after Dellar [Phys. Rev. E. 65, 036309 (2002)10.1103/PhysRevE.65.036309; J. Comput. Phys. 190, 351 (2003)10.1016/S0021-9991(03)00279-1], a multiple-relaxation-time (MRT), chromodynamic, multicomponent lattice Boltzmann equation (MCLBE) scheme for simulation of isothermal, immiscible fluid flow with a density contrast. It is based on Lishchuk's method [Brackbill, Kothe, and Zemach, J. Comp. Phys. 100, 335 (1992)10.1016/0021-9991(92)90240-Y; Lishchuk, Care, and Halliday, Phys. Rev. E. 67, 036701, (2003)10.1103/PhysRevE.76.036701] and the segregation of d'Ortona et al. [Phys. Rev. E. 51, 3718, (1995)10.1103/PhysRevE.51.3718]. We focus on fundamental model verifiability but do relate some of our data to that from previous approaches, due to Ba et al. [Phys. Rev. E 94, 023310 (2016)10.1103/PhysRevE.94.023310] and earlier Liu et al. [Phys. Rev. E 85, 046309 (2012)10.1103/PhysRevE.85.046309], who pioneered large density difference chromodynamic MCLBE and showed the practical benefits of an MRT collision model. Specifically, we test the extent to which chromodynamic MCLBE MRT schemes comply with the kinematic condition of mutual impenetrability and the continuous traction condition by developing analytical benchmarking flows. We conclude that our data, taken with those of Ba et al., verify the utility of MRT chromodynamic MCLBE.

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

The utility of an enhanced chromodynamic, color gradient or phase-field multicomponent lattice Boltzmann (MCLB) equation for immiscible fluids with a density difference was demonstrated by Wen et al. [Phys. Rev. E 100, 023301 (2019)2470-004510.1103/PhysRevE.100.023301] and Ba et al. [Phys. Rev. E 94, 023310 (2016)2470-004510.1103/PhysRevE.94.023310], who advanced earlier work by Liu et al. [Phys. Rev. E 85, 046309 (2012)PLEEE81539-375510.1103/PhysRevE.85.046309] by removing certain error terms in the momentum equations. But while these models' collision scheme has been carefully enhanced by degrees, there is, currently, no quantitative consideration in the macroscopic dynamics of the segregation scheme which is common to all. Here, by analysis of the kinetic-scale segregation rule (previously neglected when considering the continuum behavior) we derive, bound, and test the emergent kinematics of the continuum fluids' interface for this class of MCLB, concurrently demonstrating the circular relationship with-and competition between-the models' dynamics and kinematics. The analytical and numerical results we present in Sec. V confirm that, at the kinetic scale, for a range of density contrast, color is a material invariant. That is, within numerical error, the emergent interface structure is isotropic (i.e., without orientation dependence) and Galilean-invariant (i.e., without dependence on direction of motion). Numerical data further suggest that reported restrictions on the achievable density contrast in rapid flow, using chromodynamic MCLB, originate in the effect on the model's kinematics of the terms deriving from our term F_{1i} in the evolution equation, which correct its dynamics for large density differences. Taken with Ba's applications and validations, this result significantly enhances the theoretical foundation of this MCLB variant, bringing it somewhat belatedly further into line with the schemes of Inamuro et al. [J. Comput. Phys. 198, 628 (2004)JCTPAH0021-999110.1016/j.jcp.2004.01.019] and the free-energy scheme [see, e.g., Phys. Rev. E. 76, 045702(R) (2007)10.1103/PhysRevE.76.045702, and references therein] which, in contradistinction to the present scheme and perhaps wisely, postulate appropriate kinematics a priori.

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.

Benchmarking of three-dimensional multicomponent lattice Boltzmann equation.

We present a challenging validation of phase field multicomponent lattice Boltzmann equation (MCLBE) simulation against the 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.

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.

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.

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.

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.

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.

Effective surface viscosities of a particle-laden fluid interface.

The Einstein formula for the effective shear viscosity of low Reynolds number suspension flows is generalized to the case of flat, low-concentration, particle-laden interfaces separating two immiscible fluids. The effective surface shear and dilational viscosities of this system is found to be eta{s}=5/3(eta{1}+eta{2})R phi and zeta{s}=5(eta_{1}+eta_{2})R phi , correspondingly, where eta{1} and eta{2} are the shear viscosities of two bulk fluids and phi is the surface concentration of spherical particles of radius R . The formula is found to be in excellent agreement with data obtained using multicomponent lattice Boltzmann equation simulation.

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.

Kinematic condition for multicomponent lattice Boltzmann simulation.

We present a simple, transferable, efficient, and effective algorithmic enhancement designed to improve the accuracy of all multicomponent lattice Boltzmann methods when applied to the simulation in the continuum approximation of fluid mechanics. By applying a collision parameter (kinematic viscosity) perturbation to reduce velocity gradients in the interfacial region, a kinematic condition is effectively enforced. Matters relating to a variation in the collision parameter are briefly discussed.

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.

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.