Page 319 - Intro to Tensor Calculus

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since F i does not depend upon V 1 and f(~r, V, t) equals zero for V i equal to ±∞. The right-hand side of
equation (2.5.86) represents the integral of (D C f)φ over velocity space. This integral is zero because of
the symmetries associated with the right-hand side of equation (2.5.83). Physically, the integral of (D c f)φ
over velocity space must be zero since collisions with only scattering terms cannot increase or decrease the
number of particles per cubic centimeter in any element of phase space.
In equation (2.5.86) we write the velocities V 1i in terms of the mean velocities (u, v, w) and random
velocities (U r ,V r ,W r )with

V 11 = U r + u, V 12 = V r + v, V 13 = W r + w (2.5.90)

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or V 1 = V r + V with V 1 = V r + V = V since V r = 0 (i.e. the average random velocity is zero.) For
future reference we write equation (2.5.86) in terms of these random velocities and the material derivative.
Substitution of the velocities from equation (2.5.90) in equation (2.5.86) gives
3
∂(nφ) ∂ ∂ ∂ X F i ∂φ
+ n(U r + u)φ + n(V r + v)φ + n(W r + w)φ − n =0 (2.5.91)
∂t ∂x ∂y ∂z m ∂V 1i
i=1
or
∂(nφ) ∂ ∂ ∂
+ nuφ + nvφ + nwφ
∂t ∂x ∂y ∂z
3 (2.5.92)
∂ ∂ ∂ X F i ∂φ
+ nU r φ + nV r φ + nW r φ − n =0.
∂x ∂y ∂z m ∂V 1i
i=1
Observe that
+∞
ZZZ
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nuφ = uφf(~r, V, t)dV x dV y dV z = nuφ (2.5.93)
−∞
and similarly nvφ = nvφ, nwφ = nwφ. This enables the equation (2.5.92) to be written in the form
∂φ ∂φ ∂φ ∂φ
n + nu + nv + nw
∂t ∂x ∂y ∂z

∂n ∂ ∂ ∂
+ φ + (nu)+ (nv)+ (nw) (2.5.94)
∂t ∂x ∂y ∂z
3
∂ ∂ ∂ X F i ∂φ
+ nU r φ + nV r φ + nW r φ − n =0.
∂x ∂y ∂z m ∂V 1i
i=1
The middle bracketed sum in equation (2.5.94) is recognized as the continuity equation when multiplied by
m and hence is zero. The moment equation (2.5.86) now has the form

3
Dφ ∂ ∂ ∂ X F i ∂φ
n + nU r φ + nV r φ + nW r φ − n =0. (2.5.95)
Dt ∂x ∂y ∂z m ∂V 1i
i=1
Note that from the equations (2.5.86) or (2.5.95) one can derive the basic equations of fluid flow from
continuum mechanics developed earlier. We consider the following special cases of the Maxwell transfer
equation.