Page 478 - Electromagnetics

Figure A.5: Derivation of the Helmholtz transport theorem.

Using the notation

∂ ∂ ∂
u ·∇ = u x + u y + u z
∂x ∂y ∂z
we can write
DF ∂F
= + (u ·∇)F. (A.60)
Dt ∂t
This is the material derivative of a vector field F when u describes the motion of a
physical material. Similarly, the total derivative of a vector field is
dF ∂F
= + (v ·∇)F
dt ∂t
where v is arbitrary.

The Helmholtz and Reynolds transport theorems

We choose the intuitive approach taken by Tai [190] and Whitaker [214]. Consider
an open surface S(t) moving through space and possibly deforming as it moves. The
velocity of the points comprising the surface is given by the vector field v(r, t). We are
interested in computing the time derivative of the flux of a vector field F(r, t) through
S(t):
d
ψ(t) = F(r, t) · dS
dt S(t)

F(r, t + t) · dS − F(r, t) · dS
S(t+ t) S(t)
= lim . (A.61)
t→0 t
Here S(t + t) = S 2 is found by extending each point on S(t) = S 1 through a displacement
v t, as shown in Figure A.5. Substituting the Taylor expansion

∂F(r, t)
F(r, t + t) = F(r, t) + t +· · ·
∂t

© 2001 by CRC Press LLC