There are many different variants can be used to create holistic model, for example, DEM-FEM-CA based model. Most of the combinations are significantly inhomogeneous or heterogeneous multi-scale models that would require complicated interfaces between the various components. It almost eliminates the possibility to complete full scale calculations within a single numerical model without running independently the individual modules and combining them afterwards. As a result, difficulties and limitations for development of such multiscale heterogeneous holistic models arise and, at the same time, arises the question of feasibility of such approaches. However, development of the holistic model based entirely on one or two homogeneous methods allows for modelling very complex physical phenomena accompanying the manufacturing process and for elimination of the complicated interface. The approaches based on CA and LatticeBoltzmannmethods (CA-LBM) are among the methods that satisfy the above mentioned criteria.
Rooted from the gas kinetics, the latticeBoltzmann method (LBM) is a popular and powerful tool in model- ing the NS hydrodynamics and beyond. Historically, the SRT-LBM is firstly developed as an alternative solver for the NS equations. Since it uses a very limited but highly optimized number of discrete velocities, e.g. the D2Q9 scheme for 2D problems, the SRT-LBM can be viewed as a special type of DVM to solve the BGK equation . To capture the non-equilibrium effects beyond the NS hydro- dynamics, a rigorous procedure for obtaining high-order approximations to the BGK equation is proposed  and tested in various canonical problems [23–29]. The basic idea is that, since non-equilibrium effects are re- lated to high-order moments of the VDF, a higher-order quadrature must be used in the simulation of rarefied gas flows. By expanding the VDF into high-order Gauss- Hermite polynomials or Gauss quadrature in the spher- ical coordinate system, it is found that, with a not very large number of discrete lattice velocities, e.g. D2Q36 or D2Q64, high-order LBM schemes can accurately describe rarefied gas flows up to the early transition flow regime, i.e. Kn . 0.5.
We have chosen to model two dimensional cancer growth where the biological cells are repre- sented by cellular automata and the oxygen in the environment is modelled as a two-phase fluid using the lattice Boltzman method. Most models in the literature currently restrict themselves to two dimensions as it is more computationally feasible and since cancer does not grow in a sphere but rather an oblate spheroid. Our 2D simulation is easily compared with both existing models and 2D biopsy slices. Here we will present a high level outline of the method, and each section will be covered in more detail below. Pseudocode describing the simulation is provided in Section 3.2.4. The simulation begins with a single healthy cell at the center of a 2-dimensional grid. An event queue keeps track of cellular events, and initially a single mitotic event is placed on the queue for the healthy cell. Each event dequeued from the queue is an- other loop in the model and puts that cell through a life cycle. The cell is checked for whether it still has enough oxygen to survive, is in a location with growth factor, has access to blood, has space to grow, and has su ffi ciently long telomeres. If all of these checks are successful, or if mutations confer these abilities, the cell enters a mitotic event. This creates a daughter cell and potentially introduces mutations into the daughter or parent. Both cells have events scheduled for some point in the future and are added to the event queue, then the next event is popped. Oxygen is consumed by cells when they divide or every 25 time steps of the main simulation if they are not actively dividing.
sumali and Karlin to the finite Knudsen range, introduc- ing a relation between the Knudsen number and the re- laxation time and adopting the respective diffuse boundary condition to include velocity slip. Works into similar directions have been published at about the same time by Tang et al. . Sbragaglia and Succi presented a new formulation of kinetic boundary conditions for flows at finite Knudsen numbers in ; therefore, they pro- posed models based on slip, reflection and accomodation coefficients. Toschi and Succi proposed a stochastic han- dling of finite Knudsen number flows within the context of LatticeBoltzmann simulations in . Virtual wall collisions of the LatticeBoltzmann particles are incorpo- rated into the LatticeBoltzmann model, yielding satis- factory results for flow regimes up to Knudsen numbers ~30. Zhang et al. report a successful qualitative Knudsen minimum prediction in . Their results show good agreement for Knudsen numbers up to ~0.4 and only differ for higher numbers, due to numerical errors in- duced by the increasing value of the BGK-relaxation- time . In order to suppress artificial slip effects near walls, Verhaeghe et al.  propose a MRT-based model, including a particular tuning of the relaxation parameters. Their results show excellent agreement in the slip flow regime, however, they point out deficiencies of the slip flow model for higher Knudsen numbers. A respective extension to the transition flow regime has been devel- oped recently by Li et al. . Another approach to rare- fied gas modeling using LatticeBoltzmannmethods is reported in [22,23] where—based on the Hermite projec- tion method—higher-order LatticeBoltzmann models are constructed and yield promising results for both slip and transition flow regime.
Historically LatticeBoltzmannmethods derived from Lattice Gas Cellular Au tomata. As mentioned in Chapter 2, one of the main problems arising in LGCA was noise in the simulations, which was overcome by using an ensemble average as a primary quantity as opposed to boolean quantities. Other problems with LGCA arise in the form of sensitivity to lattice structure; only ceftain choices of local lat tice structure (i.e. the unit cell) retain sufficient symmetry in terms of their isotropy to . allow successful recovery of Navier-Stokes behaviour . Further problems arise when carrying out simulations in three dimensions, there are no unit cells that have sufficient symmetry to recover the appropriate hydrodynamics. In order to achieve three dimensional simulations with LGCA it is necessary to work in four dimensions to increase the isotropy of the system. This obviously increases the complexity of the simulation and requires large run times or the compilation of large collision rule tables, using large amounts of system memory.
Numerical methods for the 1-D Dirac equation based on operator splitting and on the quantum latticeBoltzmann (QLB) schemes are reviewed. It is shown that these discretizations fall within the class of quantum walks, i.e. discrete maps for complex ﬁelds, whose continuum limit delivers Dirac-like relativistic quantum wave equations. The correspondence between the quantum walk dynamics and these numerical schemes is given explicitly, allowing a connection between quantum computations, numerical analysis and latticeBoltzmannmethods. The QLB method is then extended to the Dirac equation in curved spaces and it is demonstrated that the quantum walk structure is preserved. Finally, it is argued that the existence of this link between the discretized Dirac equation and quantum walks may be employed to simulate relativistic quantum dynamics on quantum computers.
In this paper, we developed new four- and eight-velocity lattice models of the internal energy density distribution function for incompressible flow where the compression work done by the pressure and viscous heat dissipation can be neglected. The rest of the paper is organized as follows. In Secs. 2 and 3, we show the theory of the internal energy density distribution function and the discritization procedure of continuous Boltzmann equation which will lead in developing of our new type of four- and eight-velocity lattice models. In Sec. 4, we employ our models to simulate the porous plate Couette flow and natural convection in a cubic cavity for two and three dimensional problems respectively. The final section concludes this study.
The importance of understanding fluid flow with a change in phase arises from the fact that many industrial processes rely on these phenomena for materials processing or for energy transfer, e.g. petroleum processing, paper-pulping, power plants and boiling water reactor. There are many common examples of multiphase flow not only in industrial processes but also everyday life. Thus the understanding of multiphase flow is essential for both fundamental research and engineering applications. However, due to the complex nature of multiphase flow, theoretical solutions are generally limited to relatively simple cases. Meanwhile, the experimental approaches for multiphase flow are very expensive if not impossible, depending on the scale and/or fluid composition. Therefore, it is reasonable to say that numerical simulations are primarily useful in studying the underlying physics of multiphase flow and providing information about the details of processes that are difficult to obtain by theoretical analysis or by experiments. Recently, simulating multiphase flow with LatticeBoltzmann Method (LBM) has attracted much attention. Microscopically, the phase segregation and surface tension in multiphase flow are because of the interparticle forces/interactions. Due
The use of latticeBoltzmann method has been growing continuously in the last 30 years. Historically, LBM originated based on the lattice gas automata (LGA), which can be seen as a simpliﬁed, ﬁctitious version of Molecular Dynamics model . In these methods the ﬂuid is considered by particles that move on a regular lattice which often must be chosen in a special way. Lattice gas automata suﬀers from several na- tive defects including the lack of Galilean invariance, presence of statistical noise and exponential complexity for three-dimensional lattices . In order to overcome the deﬁciencies of lattice-gas methods and improve the situation, several lattice-Boltzmann models were formulated. The statistical noise of LGA can be removed by replacing the Boolean particle number with density distribution function. The ﬁrst LBM is named as nonlinear lattice-Boltzmann model  in which a mean-value representation of par- ticles eliminated the problem of statistical noise. Then, a linearized enhanced-collision method was developed to overcome the complexity of the lattice-gas collision opera- tor used also in the lattice-Boltzmann models . Furthermore, the inclusion of rest particles and Maxwellian velocity distribution achieved the Galilean invariant macro- scopic behaviour . The appearance of the lattice Bhatnagar-Gross-Krook (BGK) model make the LBM reach maturity and be widely used in industrial and academic works. In the lattice BGK, the collision operator is based on the single-relaxation-time approximation to the local equilibrium distribution . In addition, the numerical stability of LBM, especially in the high-Reynolds number simulations was increased by the multi-relaxation time lattice-Boltzmann model (MRT).
Multiphase latticeBoltzmann simulations have been carried out to study droplet flow in three different microchannel networks. The latticeBoltzmann method was shown to be suitable for simulating microdroplet flows in complex microchannel networks. We found that droplets that are large enough to occupy the full width of a microchannel, upon reaching a channel intersection, will travel through the downstream branch with the highest local velocity. Our results suggest that this may apply to arbitrary geometry. We also revealed that single phase flow rates cannot be used to predict the proportion of droplets travelling through the individual branches of a microchannel network.
The past two decades have seen steady growth in in- terest in multi-relaxation time (MRT) latticeBoltzmann (LB) schemes which offer enhanced simulation stability , , , , , ,  and  etc. We extend to D3Q19 a recent D2Q9 variant , in which the usual collision matrix is only implicit, being represented by a carefully chosen, modal eigen-basis which is subject to forced, scalar relaxation. As well as the usual advan- tages, the new method has transparent analytic proper- ties: its orthogonal modes are defined as polynomials in the lattice basis, as are the elements of the transforma- tion matrix between the distribution function and the mode space. This uniquely allows for the direct recon- struction of a post-collision distribution function, which is effectively parameterized by the eigenvalue spectrum. Our purpose in developing a new model is to stabilize multi-component LBE (MCLBE) so as to attempt the challenge of recovering the Taylor-Einstein theory of sus- pension viscosity , .
To simulate the dynamics of liquid crystals, it is common to solve numerically the con- tinuum equations of Beris-Edwards and Navier-Stokes [3–5]. A hybrid method of latticeBoltzmann  and finite differences  is used. Special care has to be taken in order to simu- late interfaces due to spurious currents that arise in these numerical methods, which can lead to interfacial motion or changes in the director field [6, 8].
The past two decades have seen steady growth in interest in multirelaxation time (MRT) latticeBoltzmann (LB) schemes, which offer enhanced simulation stability [1–8], etc. We extend to D3Q19 a recent D2Q9 variant , 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. This uniquely allows for the direct reconstruction of a post-collision distribution function, which is effectively parameterized by the eigenvalue spectrum. Our purpose in developing a new model is to stabilize multicomponent LBE (MCLBE) so as to attempt the challenge of recovering the Taylor-Einstein theory of suspension viscosity [10,11].
mean free path of a molecule . This is to ensure good statistics of the different quantities within the volume of fluid (particle) and consequently their averages and evolution. When this applies (for most gas problems at normal pressure and for most liquid problems), the fluid can be treated as a continuous medium and two distinct alternatives of fluid specification are possible, differing in their reference to the fluid. The first one, the Lagrangian description, considers a fixed volume of fluid V given at a time reference (t = 0 generally) advecting with the flow. The flow observables are defined as functions of time and of the choice of material element of fluid, and describe the dynamical history of this selected fluid element (contained in the moving volume V). Useful in certain special contexts, it leads however to rather cumbersome analysis and is at a disadvantage in not giving directly spatial gradients in the fluid (). That is why a rather less complicated approach is used: the Eulerian approach. Here the velocities considered are at a fixed point. At each instant of time, the velocity field describes the velocity of different fluid particles (of volume V). Methods describing explicitly each molecule are qualified as Lagrangian while any grid-based approach is therefore qualified as Eulerian. LB is Eulerian.
on diﬀerent theoretical frameworks. While informa- tion transferring from the kinetic model to the contin- uum model is usually a well-deﬁned process, the reverse process is more problematic . It is diﬃcult to re- cover non-equilibrium information lost by the continuum solvers which is required by the kinetic method. Al- though the kinetic model can provide necessary infor- mation for the continuum model, it can be computation- ally expensive . The statistical noise associated with the particle methods may also aﬀect the accuracy and stability of the hybrid solver . To eﬀectively model mixed-Kn ﬂows, we introduce a multiscale lattice Boltz- mann (LB) method to utilize various order LB models. Since this multiscale method is based on a same theoret- ical framework, it has distinguished advantages, which has also been demonstrated recently by the uniﬁed gas- kinetic scheme.
Boltzmann equation for theoretical analysis and numerical method is challenging by direct solution. LBM is a method which is quite different from the traditional computational fluid dynamics (CFD) algorithms. From the physical essence, the kinetic behavior of multiphase flow system is the result of the microscopic interaction among fluid phase. Based on the theory of molecular motion, LBM is especially suitable for describe the com- plex multiphase flow from the underlying . Meanwhile, the CFD method such as volume of fluid (VOF)  level set method (LSM) , how to consider the pressure and interaction between electromagnetic field and ca- pillary effect in the multiphase flow calculation has been a challenge all the time. For the flow of the surface tension, the VOF and LSM are unstable near the interface of numeric. Therefore, as a powerful tool for the nu- merical simulations and investigation of multiphase flows, the LBM has multiple advantages including time and space efficient computations that are straightforward to parallelize, handles complex boundaries without diffi- culty, and directly link between microscope and macroscopic phenomenon.
flow rate are of interest which can determine the bulk motion characteristics. Therefore, Molecular Dynamics (MD), Direct Simulation Monte Carlo method (DSMC) and direct numerical simulation of the Boltzmann equa- tion are too computationally expensive and impractical for applications where the microscopic details are not re- quired. Significant effort has been made to improve and extend the validity of the Navier-Stokes equations beyond Knudsen numbers of 0.1, or to construct complicated constitutive laws involving high order terms of Knud- sen number which leads to Burnett-type equations . The LBE method has the potential to improve this situ- ation because it is efficient comparable to Navier-Stokes solvers and it can recover the Navier-Stokes equations. A preliminary link between the LBE and the Burnett-type equations has also been established . In addition, the continuum, slip and transition flow regimes may exist together in microfluidic devices, e.g. a long microchan- nel. Hybrid algorithms that couple DSMC and Navier- Stokes methods have been tried to model these mixed flow regimes . However, large errors can arise from inappropriate assumptions regarding, for example, the velocity distribution for gas molecules at the matching interface between two solutions . Furthermore, these hybrid algorithms entail intensive computational effort for three-dimensional flow simulations. In principle, the LBE is valid throughout these mixed flow regimes and avoids any coupling problem. Consequently, the LBE may be a better method for gas flows in microdevices, particularly where mixed flow regimes are encountered.
Knudsen number and low speed, these methods become not only expensive but also inaccurate. When the Knudsen number is less than 0.1, the NSF equations with slip boundary conditions can provide results with reasonable accuracy. Therefore, we mainly compare the present thermal LBE solution with the solutions of the NSF equations in order to test whether the present thermal LBE is valid in the slip flow regime (0.001<Kn<0.1). Note, the velocity slip and temperature jump coefficients are only weakly correlated with the molecular model . The effect of the molecular model is implemented through the viscosity-temperature power law as given by Eq. (9). Through the Prandtl number, the influence of the temperature on the thermal diffusivity can also be determined. For consistent comparisons, the Maxwellian molecule model will be used for both thermal LBE and NSF simulations.
Abstract. In this paper, we recall the linear version of the latticeBoltzmann schemes in the frame- work proposed by d’Humi` eres. According to the equivalent equations we introduce a definition for a scheme to be isotropic at some order. This definition is chosen such that the equivalent equations are preserved by orthogonal transformations of the frame. The property of isotropy can be read through a group operation and then implies a sequence of relations on relaxation times and equilibrium states that characterizes a latticeBoltzmann scheme. We propose a method to select the parameters of the scheme according to the desired order of isotropy. Applying it to the D2Q9 scheme yields the classical constraints for the first and second orders and some non classical for the third and fourth orders.
To solve multiphase flow among various models D2Q9 LatticeBoltzmann model is found to be very much effective. In order to apply this model the equation of state of fluid under consideration is to be known. In this EOS the proper detection of metastable region is a very important issue. This metastable region lies between two spinodal points having densities ρ 1 and ρ 2 . Values of ρ 1 and ρ 2 can be obtained by solving the conditions of