Past exam of the kinetic theory course of the University of Oxford / 2025 / 1 / c

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- Solution c

Solution

Use the BGK model from b. Multiply it by

v_{i}

and integrate over velocity space. Matching the collision invariants makes the collision contribution vanish. For the external force term, integration by parts gives

- \partial_{j} U \int v_{i} \partial_{v_{j}} f d^{3} v = ρ \partial_{i} U .

Define the momentum flux and force density by

Π_{ij} = \int v_{i} v_{j} f d^{3} v, F_{i} = - ρ \partial_{i} U .

The resulting momentum conservation equation is

\partial_{t} (ρ u_{i}) + \partial_{j} Π_{ij} = F_{i}, Π = ρ uu + P, F = - ρ \nabla U .

The pressure tensor is the second central velocity moment,

P_{ij} = \int (v_{i} - u_{i}) (v_{j} - u_{j}) f d^{3} v .

The zeroth velocity moment also supplies the continuity equation

\partial_{t} ρ + \partial_{j} (ρ u_{j}) = 0

, which will be needed in d.

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