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Schematic of D2Q9 lattice vectors for 2D Lattice Boltzmann. Unlike CFD methods that solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice.
Normal coordinates decouple the classical vibrational motions of the molecule and thus give an easy route to obtaining vibrational amplitudes of the atoms as a function of time. In Wilson's GF method it is assumed that the molecular kinetic energy consists only of harmonic vibrations of the atoms, i.e., overall rotational and translational ...
The solution is the weighted average of six increments, where each increment is the product of the size of the interval, , and an estimated slope specified by function f on the right-hand side of the differential equation.
Here + is the RK4 approximation of (+), and the next value (+) is determined by the present value plus the weighted average of four increments, where each increment is the product of the size of the interval, h, and an estimated slope specified by function f on the right-hand side of the differential equation.
The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element) centered at ...
The mean free time for a molecule in a fluid is the average time between collisions. The mean free path of the molecule is the product of the average speed and the mean free time. [1] These concepts are used in the kinetic theory of gases to compute transport coefficients such as the viscosity. [2] In a gas the mean free path may be much larger ...
The purpose of the method of averaging is to tell us the qualitative behavior of the vector field when we average it over a period of time. It guarantees that the solution y ( t ) {\displaystyle y(t)} approximates x ( t ) {\displaystyle x(t)} for times t = O ( 1 / ε ) . {\displaystyle t={\mathcal {O}}(1/\varepsilon ).}
The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probabilities. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.