2.4 Two-fluid equations 39Let us now take the second moment of the Vlasov equation. Unlike the zeroth and
first moments, here the dimensionality of the system enters explicitlyso the more general
pressure definition given by Eq. (2.26) will be used. Multiplying the Vlasov equation by
mσv^2 / 2 and integrating over velocity gives
∂
∂t∫
mσv^2
2fσdNv+∂
∂x·
∫
mσv^2
2vfσdNv+qσ∫
v^2
2∂
∂v
·(E+v×B)fσdNv
=
∑
α∫
mσ
v^2
2CσαfσdNv. (2.29)We now consider each term of this equation separately as follows:
- The time derivative term becomes
∂
∂t
∫
mσv^2
2fσdNv=∂
∂t∫
mσ(v′+uσ)^2
2fσdNv′=∂
∂t(
NPσ
2+
mσnσu^2 σ
2)
.
- Again usingv=v′+uσthe space derivative term becomes
∂
∂x·
∫
mσv^2
2vfσdNv=∇·(
Qσ+2 +N
2
Pσuσ+mσnσu^2 σ
2uσ)
.
whereQσ=∫
mσv′^2
2v′fσdNvis called theheatflux.- On integrating by parts, the acceleration term becomes
qσ∫
v^2
2∂
∂v
·[(E+v×B)fσ]dNv=−qσ∫
v·Efσdv=−qσnσuσ·E.- The collision term becomes (using Eq.(2.16))
∑
α∫
mσv^2
2Cσαfσdv=∫
α=σmσv^2
2Cσαfσdv=−(
∂W
∂t)
Eσα.
where(∂W/∂t)Eσαis the rate at which speciesσcollisionally transfers energy to
speciesα.
Combining the above four relations, Eq.(2.29) becomes∂
∂t(
NPσ
2+
mσnσu^2 σ
2)
+∇·
(
Qσ+2 +N
2
Pσuσ+mσnσu^2 σ
2uσ)
−qσnσuσ·E=−(
∂W
∂t)
Eσα.
(2.30)
This equation can be simplified by invoking two mathematical identities, the first of which
is
∂
∂t(
mσnσu^2 σ
2)
+∇·
(
mσnσu^2 σ
2uσ)
=nσ(
∂
∂t+uσ·∇)
mσu^2 σ
2=nσd
dt(
mσu^2 σ
2