10.4 The Euler Equations of Divergence Type Functionals
659
ϕδ∇ · (u|a|a) = ϕ∇ · δ(u|a|a) = ∇ · [ϕδ(u|a|a)] − ∇ϕ · (|a|aδu + uaδ|a| + u|a|δa)
= ∇ · [ϕδ(u|a|a)] − ∇ϕ · (|a|aδu + uaδ|a| + u|a|δa)
= ∇ · [ϕδ(u|a|a)] − (|a|a · ∇ϕ)δu − (ua · ∇ϕ)
a
|a|
· δa − (u|a|∇ϕ) · δa
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
− |a|a · ∇ϕ
δu +
∂ F
∂ a
− (ua · ∇ϕ)
a
|a|
− u|a|∇ϕ
· δa + ∇ · [ϕδ(u|a|a)]
dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂u
− |a|a · ∇
∂ F
∂∇ · (u|a|a)
= 0
∂ F
∂ a
−
ua · ∇
∂ F
∂∇ · (u|a|a)
a
|a|
− u|a|∇
∂ F
∂∇ · (u|a|a)
= 0
(5)
The natural boundary condition is
∂ F
∂∇ · (u|a|a)
S
= 0
( 6 )
Example 10.4.9 The functional with the Laplace operator. Let the double inner
product of the tensors of second order conform to the law of parallel operation,
derive the Euler equations and natural boundary conditions of the functional
J =
V
F(u, a, ∇ · ∇(ua))dV
(1)
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂ a
· δa +
∂ F
∂∇ · ∇(ua)
· δ∇ · ∇(ua)
dV
(2)
Putting b =
∂ F
∂∇·∇(ua)
, the integral of the last term in Eq. (2) can be written as
b · δ∇ · ∇(ua) = b · ∇ · ∇δ(ua) = ∇ · [∇δ(ua) · b] − ∇b : ∇δ(ua)
= ∇ · [∇δ(ua) · b] − ∇ · [∇b · δ(ua)] + (∇ · ∇b) · (uδa + aδu)
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
+ a · ∇ · ∇b
δu +
∂ F
∂ a
+ u∇ · ∇b
· δa + ∇ · [∇δ(ua) · b] − ∇ · [∇b · δ(ua)]
dV
(4)