36 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD
while collisions between different species must conserve the total momentum
of both species together so
∫
dvmivCie(fi) +∫
dvmevCei(fe) = 0. (2.15)(c) Conservation of energy –Collisions between particles of the same species can-
not change the total energy of that species so
∫
dvmσv^2 Cσσ(fσ) = 0 (2.16)while collisions between different species must conserve the total energy of
both species together so
∫
dvmiv^2 Cie(fi) +∫
dvmev^2 Cei(fe) = 0. (2.17)2.4 Two-fluid equations
Instead of just taking moments of the distribution functionfitself, moments will now
be taken of the entire Vlasov equation to obtain a set of partial differential equations re-
lating the mean quantitiesn(x),u(x),etc. We begin by integrating the Vlasov equation,
Eq.(2.12), over velocity for each species. This first and simplest step in the procedure is
often called taking the “zeroth” moment, since we are multiplying by unity which for con-
sistency with later “moment-taking”, can be considered as multiplying theentire Vlasov
equation byvraised to the power zero. Multiplying the Vlasov equation by unity and then
integrating over velocity gives
∫[
∂fσ
∂t+
∂
∂x·(vfσ) +∂
∂v·(afσ)]
dv=∑
α∫
Cσα(fσ)dv. (2.18)The velocity integral commutes with both the time and space derivativeson the left hand
side becausex,v,andtare independent variables, while the third term on the left hand side
is the volume integral of a divergence in velocity space. Gauss’s theorem[i.e.,
∫
voldx∇·
Q=
∫
sfcds·Q] givesfσevaluated on a surface atv=∞.However, becausefσ→^0
asv→∞, this surface integral in velocity space vanishes. Using Eqs.(2.10), (2.11), and
(2.13), we see that Eq.(2.18) becomes the speciescontinuity equation
∂nσ
∂t+∇·(nσuσ) = 0. (2.19)Now let us multiply Eq.(2.12) byvand integrate over velocity to take the “first moment”,
∫
v[
∂fσ
∂t+
∂
∂x·(vfσ) +∂
∂v·(afσ)]
dv=∑
α∫
vCσα(fσ)dv. (2.20)This may be re-arranged in a more tractable form by: