- #1
Monty Hall
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I would have asked in math, but I was hoping the context of lattice Boltzmann may make my question clearer. Given f is the number density of particles, v velocity, and u equilibrium velocity.
In a book(http://www.ndsu.edu/fileadmin/physics.ndsu.edu/Wagner/LBbook.pdf equation 3.14), he derives mass conservation/continuity equation in terms of distribution function f:
NOTE: He's using Einstein index notation.
[tex]
\begin{align}
\partial_{t}\int f d v +
\partial_\alpha\int f v_\alpha d v +
F \int \partial_v f d v & =
\frac{1}{\tau}\int (f^0 - f)d v \\
\partial_t n + \partial_\alpha(n u_\alpha) & = 0
\end{align}
[/tex]
where I'm using equilbrium distribution f0 in place of f:
[tex]
f^0(v)=\frac{n}{(2\pi\theta)^{3/2}}\exp{-\frac{(v-u)^2}{2\theta}}
[/tex]
I'm reading along and get the first 2 terms of the lhs and rhs to be n, n u_alpha, and 0 respectively, but the third term on the lhs looks like it evaluates to zero. My question is, it looks like he's integrating a derivative of f wrt to a vector. I'm assuming this is a directional derivative and looked it up in wikipedia. However, when I throw it into mathematica (using the equilibrium distrubtion f0) and triple integrate, I don't get zero. Anybody shed some light on how he gets this? (If you can show me the integral in mathematica - so I can see that it works - that would be a bonus)
In a book(http://www.ndsu.edu/fileadmin/physics.ndsu.edu/Wagner/LBbook.pdf equation 3.14), he derives mass conservation/continuity equation in terms of distribution function f:
NOTE: He's using Einstein index notation.
[tex]
\begin{align}
\partial_{t}\int f d v +
\partial_\alpha\int f v_\alpha d v +
F \int \partial_v f d v & =
\frac{1}{\tau}\int (f^0 - f)d v \\
\partial_t n + \partial_\alpha(n u_\alpha) & = 0
\end{align}
[/tex]
where I'm using equilbrium distribution f0 in place of f:
[tex]
f^0(v)=\frac{n}{(2\pi\theta)^{3/2}}\exp{-\frac{(v-u)^2}{2\theta}}
[/tex]
I'm reading along and get the first 2 terms of the lhs and rhs to be n, n u_alpha, and 0 respectively, but the third term on the lhs looks like it evaluates to zero. My question is, it looks like he's integrating a derivative of f wrt to a vector. I'm assuming this is a directional derivative and looked it up in wikipedia. However, when I throw it into mathematica (using the equilibrium distrubtion f0) and triple integrate, I don't get zero. Anybody shed some light on how he gets this? (If you can show me the integral in mathematica - so I can see that it works - that would be a bonus)
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