656
10 Variational Problems of Functionals with Vector, Tensor …
Obviously, owing to ua = 0, so the two expressions of Eq. (8) can be transformed
to Eq. (6).
Example 10.4.5 The functional with the divergence of product of a scalar function and a vector. Derive the Euler equations and natural boundary condition of the
functional
J =
V
F(u, a, ∇ · f (u)a)dV
(1)
Solution Putting f = f (u), taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂ a
· δa +
∂ F
∂∇ · f a
δ∇ · f a
dV
(2)
Let ϕ =
∂ F
∂∇· f a
, b = δ( f a) = a f u δu + f δa, there is
ϕδ∇ · f a = ϕ∇ · b = ∇ · ϕb − ∇ϕ · b = ∇ · ϕb − ∇ϕ · (a f u δu + f δa)
= ∇ · ϕb − ( f u a · ∇ϕ)δu + f ∇ϕ · δa
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
− f u a · ∇ϕ
δu +
∂ F
∂ a
− f ∇ϕ
· δa + ∇ · ϕb
dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂u
− f u (u)a · ∇
∂ F
∂∇ · f (u)a
= 0
∂ F
∂ a
− f (u)∇
∂ F
∂∇ · f (u)a
= 0
(5)
The natural boundary condition is
∂ F
∂∇ · f (u)a
S
= 0
( 6 )
Example 10.4.6 The functional with the divergence of a scalar multiplied by the
cross product of two vectors. Derive the Euler equations and natural boundary
condition of the functional
J =
V
F(u, a, b, ∇ · (ua × b))dV
(1)
Solution Taking the variation to the functional, we get