Lattice velocity

In the LB technique, the fluid to be simulated consists of a large set of fictitious particles. Essentially, the LB technique boils down to tracking a collection of these fictitious particles residing on a regular lattice. A typical lattice that is commonly used for the effective simulation of the NS equations (Somers, 1993) is a 3-D projection of a 4-D face-centred hypercube. This projected lattice has 18 velocity directions. Every time step, the particles move synchronously along these directions to neighboring lattice sites where they collide. The collisions at the lattice sites have to conserve mass and momentum and obey the so-called collision operator comprising a set of collision rules. The characteristic features of the LB technique are the distribution of particle densities over the various directions, the lattice velocities, and the collision rules. 

For the external, observer, jA +/B = 0. From this condition and the Gibbs-Duhem relation, the local lattice velocity becomes... 

Lattice Boltzmann Method (LBM), Fig. 1 The discrete lattice velocities of a D2Q9 lattice structure Co = (0, 0) corresponds to the rest portion of the particles/o... 

Other parameters, including the lattice sound speed Cs and weight factor t are lattice structure dependent. For example, for a typical D2Q9 (two dimensions and nine lattice velocities see Fig. 1) lattice structure, we have to = 4/9, = 1/9,... 

The local lattice velocity Ul which arises as a result of the production and annihilation of vacancies is given by the following expression as shown in section 5.5.3 (see eq. (5-35)) ... 

Other parameters, including the lattice sound speed Cs and weight factor fj, are lattice structure dependent. For example, for a typical D2Q9 (two dimensions and nine lattice velocities see Fig. 1) lattice structure, we have tQ = 4/9, ii 4 = 1/9, f5 8 = 1/36, and = A /3Afi, where Ax is the spatial distance between two nearest lattice nodes. Through the Chapman-Enskog expansion, one can recover the macroscopic continuity and momentum (Navier-Stokes) equations from the above-defined LBM dynamics ... 

The LBM is similar to the LGA in that one performs simulations for populations of computational particles on a lattice. It differs from the LGA in that one computes the time evolution of particle distribution functions. These particle distribution functions are a discretized version of the particle distribution function that is used in Boltzmann s kinetic theory of dilute gases. There are, however, several important differences. First, the Boltzmann distribution function is a function of three continuous spatial coordinates, three continuous velocity components, and time. In the LBM, the velocity space is truncated to a finite number of directions. One popular lattice uses 15 lattice velocities, including the rest state. The dimensionless velocity vectors are shown in Fig. 66. The length of the lattice vectors is chosen so that, in one time step, the population of particles having that velocity will propagate to the nearest lattice point along the direction of the lattice vector. If one denotes the distribution function for direction i by fi x,t), the fluid density, p, and fluid velocity, u, are given by... 

In (1), the lattice velocity vector is denoted by e,. The distribution functions satisfy a linear algebraic evolution equation that involves a collision term. The most popular version of the LBE is called the BGK formulation and it is based on the simplified model of the Boltzmann kinetic equation that was devised by Bhamagar et al. (1954). In the BGK formulation, the LBE takes the following form ... 

The particle at the node may be moving toward any one of the nine directions as indicated in the figure including zero direction (stationary). At time interval At, the particle moves with velocity e (a = 0,1,2,3,..., 8) to the neighboring nodes and collision happens. Let Ax be the length of the square lattice, and the ratio of = c is called lattice velocity. The ratio c can be letting c = 1 or c = /3RT = /3cs where R is gas constant, T is the absolute temperature, and is cus-... 

Intercalation-induced stresses have been modeled extensively in the Hterature. A one-dimensional model was proposed to estimate stress generation in the lithium insertion process in the spherical particles of a carbon anode [24] and an LiMn204 cathode [23]. In this model, displacement inside a particle is related to species flux by lattice velocity, and total concentration of species is related to the trace of the stress tensor by compressibihty. Species conservation equations and elasticity equations are also included. A two-dimensional porous electrode model was also proposed to predict electrochemicaUy induced stresses [30]. Following the model approach of diffusion-induced stress in metal oxidation and semiconductor doping [31-33], a model based on thermal stress analogy was proposed to simulate intercalation-induced stresses inside three-dimensional eUipsoidal particles [1]. This model was later extended to include the electrochemical kinetics at electrode particle surfaces [2]. This thermal stress analogy model was later adapted to include the effect of surface stress [34]. 

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