Chapman–Enskog theory

Definition

Consider the Boltzmann equation in the scaling $$\partial_t f + v \cdot \nabla_x f = \frac{1}{\varepsilon} Q(f, f),$$ where $\varepsilon > 0$ is the Knudsen number and $Q$ is the collision operator. Let $\rho$, $u$ and $T$ be the density, bulk velocity and temperature of $f$, as in the kinetic theory of gases. The Chapman–Enskog expansion seeks solutions of the form $$f = f^{(0)} + \varepsilon f^{(1)} + \varepsilon^2 f^{(2)} + \cdots$$ subject to the following conditions.

1. Each $f^{(j)}$ depends on $(t, x)$ only through $\rho$, $u$, $T$ and their spatial derivatives at $(t, x)$.
2. $f^{(0)}$ is the local Maxwell–Boltzmann distribution with the fields $\rho$, $u$, $T$, and $f^{(j)}$ for $j \ge 1$ has zero mass, momentum and energy moments.
3. The fields obey conservation laws whose fluxes are expanded in the same way, $\partial_t = \partial_t^{(0)} + \varepsilon \partial_t^{(1)} + \cdots$, and each order is solved in turn.

Chapman–Enskog theory is the derivation of fluid equations from this expansion by truncation at a given order.