Page 93 - Electromagnetics Handbook

Superposing (2.273) and (2.274) and substituting into (2.275) we have

∂
(ρ m v) dV + (ρ m v)v · dS = f dV + t dS. (2.276)
V ∂t S V S
If we define the dyadic quantity

¯
T k = ρ m vv
then (2.276) can be written as

∂

¯
(ρ m v) dV + ˆ n · T k dS = f dV + t dS. (2.277)
V ∂t S V S
This principle of linear momentum [214] can be interpreted as a large-scale form of
¯
conservation of kinetic linear momentum. Here ˆ n · T k represents the flow of kinetic mo-
mentum across S, and the sum of this momentum transfer and the change of momentum
within V stands equal to the forces acting internal to V and upon S.
The surface traction may be related to the surface normal ˆ n through a dyadic quantity
¯
T m called the mechanical stress tensor:
¯
t = ˆ n · T m .
With this we may write (2.277) as

∂
¯
¯
(ρ m v) dV + ˆ n · T k dS = f dV + ˆ n · T m dS
V ∂t S V S
and apply the dyadic form of the divergence theorem (B.19) to get

∂
¯
(ρ m v) dV + ∇· (ρ m vv) dV = f dV + ∇· T m dV. (2.278)
V ∂t V V V
Combining the volume integrals and setting the integrand to zero we have
∂
¯
(ρ m v) +∇ · (ρ m vv) = f +∇ · T m ,
∂t
which is the point-form equivalent of (2.277). Note that the second term on the right-
hand side is nonzero only for points residing on the surface of the body. Finally, letting
g denote momentum density we obtain the simple expression

¯ ∂g k
∇· T k + = f k , (2.279)
∂t
where

g k = ρ m v
is the density of kinetic momentum and

¯
f k = f +∇ · T m (2.280)
is the total force density.
Equation (2.279) is somewhat analogous to the electric charge continuity equation
(1.11). For each point of the body, the total outflux of kinetic momentum plus the time
rate of change of kinetic momentum equals the total force. The resemblance to (1.11)
is strong, except for the nonzero term on the right-hand side. The charge continuity

© 2001 by CRC Press LLC