Lattice Boltzmann method LBM

LBM is another kind of micro-scale method that is suitable for polymer solution d5mamics. The origin of LBM can be traced back to the lattice-gas cellular automata (LGCA), in which the similar kinetic equation is shared (Lim et al, 2002)  

A typical lattice gas automaton consists of a regular lattice with particles residing on the nodes. A set of Boolean variable fi x,i) i = 1. f ) describing the particle occupation is defined, where k is the number of directions of the particle velocities at each node c,- is the particle velocity, the last term in the equation represents the collision operator in accordance with arbitrary coUision 

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During the past few decades, various theoretical models have been developed to explain the physical properties and to find key parameters for the prediction of the system behaviors. Recent technological trends focus toward integration of subsystem models in various scales, which entails examining the nanophysical properties, subsystem size, and scale-specified numerical analysis methods on system level performance. Multi-scale modeling components including quantum mechanical (i.e., density functional theory (DFT) and ab initio simulation), atom-istic/molecular (i.e., Monte Carlo (MC) and molecular dynamics (MD)), mesoscopic (i.e., dissipative particle dynamics (DPD) and lattice Boltzmann method (LBM)), and macroscopic (i.e., LBM, computational... 

The lattice Boltzmann method (LBM) is a relatively new simulation technique for complex fluid systems and has attracted great interests from researchers in computational physics and engineering. Unlike traditional computation fluid dynamics (CFD) methods to numerically solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy), LBM models the fluid as fictitious particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. Due to its particulate nature and local dynamics, LBM has several advantages over conventional CFD methods, especially in dealing with complex boundaries, incorporation of microscopic interactions, and parallel computation [1, 2]. 

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... 

The lattice Boltzmann method (LBM) has been a very active mesoscopic numerical tool for fluid flow simulations since the late 1980s [4]. Recently, it has been developed to complex flows, including acoustic-fluid interaction, elec-trokinetic flows in complex geometries, red blood cells or bubble deformations in shear flows, and... 

Noncontinuous approach can be deterministic or stochastic. In deterministic approaches, such as the molecular dynamics (MD) method and the lattice Boltzmann method (LBM), the particle or molecule s trajectory, velocity, and intermolecular collision are calculated or simulated in a deterministic manner. In the stochastic approaches, such as the direct simulation Monte Carlo (DSMC) method, randomness is introduced into the solution variables. 

Lattice Boltzmann Method (LBM) Molecular Dynamics Simulation Method... 

The simplified Boltzmann equation can be solved using the Lattice Boltzmann Method (LBM) for the distributed function on a regular lattice. LBM considers each lattice structure as a volume element that consists of a collection of particles in the fluid. This simplified Boltzmann equation approximates the collision term, Q f, / ), in Eq. 37 using a relaxation time, t, providing a linear correlation. The most well-known form of the LBM is the BGKLBM, where the relaxation time is a constant. 

The Lattice Boltzmann Method (LBM), including the method Cellular Automaton (AC), present a powerful alternative to standard apvproaches known like "of up toward down" and "of down toward up". The first approximation study a continuous description of macroscopic phenomenon given for a partial differential equation (an example of this, is the Navier-Stokes equation used for flow of incompressible fluids) some numerical techniques like finite difference and the finite element, they are used for the transformation of continuous description to discreet it permits solve numerically equations in the compniter. 

The Lattice Boltzmann Method (LBM), its simple form consist of discreet net (lattice), each place (node) is represented by unique distribution equation, which is defined by particle s velocity and is limited a discrete group of allowed velocities. During each discrete time step of the simulation, particles move, or hop, to the nearest lattice site along their direction of motion, where they "collide" with other particles that arrive at the same site. The outcome of the collision is determined by solving the kinetic (Boltzmann) equation for the new particle-distribution function at that site and the particle distribution function is updated (Chen Doolen, 1998 Wilke, 2003). Specifically, particle distribution function in each site f[(x,t), it is defined like a probability of find a particle with direction velocity. Each value of the index i specifies one of the allowed directions of motion (Chen et al., 1994 ThAurey, 2003). 

A relatively new approach to the resolution of the fluid dynamics problem is the lattice Boltzmann method (LBM). LBM models the fluid flow as a movement of imaginary particles interacting with the obstacles (fibres)... 

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