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Método de Celdas de Boltzmann (LBM)
Storyboard 
El método de celdas de Boltzmann o lattice Boltzmann Model emplea un sistema de ecuaciones basados en la teoría de transporte de Boltzmann para calcular la velocidad de un fluido.
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Density
Equation
If the parameters are calculated by averaging over the speed using
| \chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t) |
the mass density estimation is obtained by:
| \rho(\vec{x},t) = m\displaystyle\int f(\vec{x},\vec{v},t)d\vec{v} |
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Speed of the Flow
Equation
If the parameters are calculated by averaging over the speed using
| \chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t) |
the velocity of the flow is calculated by integrating the velocity distribution function on all velocities by weighing the velocities:
| \vec{u}(\vec{x},t) = \displaystyle\frac{m}{\rho}\int \vec{v}f(\vec{x},\vec{v},t)d\vec{v} |
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Temperature
Equation
If the parameters are calculated by averaging over the speed using
| \chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t) |
and the equipartition theorem is considered, the temperature can be estimated by integrating the kinetic energy weighted by the velocity distribution divided by the gas constant:
| T(\vec{x},t) = \displaystyle\frac{m}{3R\rho}\displaystyle\int (\vec{v}\cdot\vec{v})f(\vec{x},\vec{v},t)d\vec{v} |
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Tension tensor
Equation
If the parameters are calculated by averaging over the speed using
| \chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t) |
the flow tensor is calculated by integrating the velocity distribution function on all velocities by weighing the velocity differences:
| \sigma_{ij} = m\displaystyle\int (v_i-u_i)(v_j-u_j)f(\vec{x},\vec{v},t)d\vec{v} |
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Discretization function
Equation
In the case of the discretization in the LBM models we work not with functions of the speed if not with discrete components. In this way the
| f_i(\vec{x},t)=w_if(\vec{x},\vec{v}_i,t) |
where
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Densidad del Gas
Equation
la densidad en un punto \vex{x} y tiempo se calcula simplemente sumando todas las distribuciones de particulas f_i en dicho punto y tiempo:
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Densidad de momento del gas
Equation
Con la descritización
| f_i(\vec{x},t)=w_if(\vec{x},\vec{v}_i,t) |
la ecuación
| \vec{u}(\vec{x},t) = \displaystyle\frac{m}{\rho}\int \vec{v}f(\vec{x},\vec{v},t)d\vec{v} |
pasa a ser
| \rho(\vec{x},t)\vec{u}(\vec{x},t)=m\sum_i\vec{e}_if_i(\vec{x},t) |
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Temperatura del Gas
Equation
la densidad en un punto \vec{x} y tiempo se calcula simplemente sumando todas las distribuciones de particulas f_i en dicho punto y tiempo:
| T(\vec{x},t)=\displaystyle\frac{m}{3R\rho}\sum_i(\vec{e}_i\cdot\vec{e}_i)f_i(\vec{x},t) |
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Boltzmann equation
Equation
The Boltzmann function describes the transport of a particle system described by the velocity distribution function:
| \displaystyle\frac{\partial f}{\partial t}+v_i\displaystyle\frac{\partial f}{\partial x_i}=C(f) |
Where the term
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Teoría de Grad de los 13 momentos
Equation
La distribución de velocidades se puede representar como un polinomio ortogoan de Hermite
| f^N(\vec{x},\vec{v},t)=\omega(\vec{v})\displaystyle\sum_{n=0}^N\displaystyle\frac{1}{n!}a^{(n)}(\vec{x},t)\mathcal{H}(\vec{v}) |
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Coeficiente de Orden 0
Equation
El coeficiente de orden cero es
| a^{(0)}=\displaystyle\int f(\vec{x},\vec{v},t)d\vec{v}=\rho |
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Coeficiente de Orden 1
Equation
El coeficiente de orden cero es
| a^{(1)}=\displaystyle\int \vec{v}f(\vec{x},\vec{v},t)d\vec{v}=\rho\vec{u} |
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Coeficientes
Equation
Los coeficientes son:
| a^{(n)}(\vec{x},\vec{v},t)=\displaystyle\int f(\vec{x},\vec{v},t)\mathcal{H}^{(n)}(\vec{v})d\vec{v}=\displaystyle\sum_if_i(\vec{x},t)\mathcal{H}^{(n)}(\vec{v}) |
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