Conservation Laws - Equations of Hydrodynamics

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

Suppose that k — k(x, v) is some quantity associated with a hard-sphere such that in any collision between spheres that takes place at position x, we have that 

General Conservation Theorem If k x, v) is a conserved quantity and C[f] is the collision term in the Boltzman equation (i.e. C f] is equal to the RHS of either equation 9.32 or equation 9.40), then 

Proof We begin by explicitly setting C[f] equal to the RHS of equation 9.40 in equation 9.48  

Since k is a conserved quantity, equation 9.47 holds, and f d v k C[/] is identically equal to zero. -QED- I 

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