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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
ID:(8462, 0)
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:
| $\rho(\vec{x},t) = m\displaystyle\int f(\vec{x},\vec{v},t)d\vec{v}$ |
ID:(8458, 0)
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:
| $\vec{u}(\vec{x},t) = \displaystyle\frac{m}{\rho}\int \vec{v}f(\vec{x},\vec{v},t)d\vec{v}$ |
ID:(8459, 0)
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:
| $T(\vec{x},t) = \displaystyle\frac{m}{3R\rho}\displaystyle\int (\vec{v}\cdot\vec{v})f(\vec{x},\vec{v},t)d\vec{v}$ |
ID:(8460, 0)
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:
| $\sigma_{ij} = m\displaystyle\int (v_i-u_i)(v_j-u_j)f(\vec{x},\vec{v},t)d\vec{v}$ |
ID:(8461, 0)