#12 (10/02/2023)

Divergence theorem (Gauss theorem)

Remember the definition of the divergence operator for 2-D as

div u
lim
s→ 0 

(⎜)



S 
n ·u dl

s
,
(1)
where n is the normal to the boundary, ∆s is the surface area of the object, and ∂S is the boundary of the object domain. This definition has an advantage over other definitions as it is independent of the coordinate system.
Equation (1) is approximately written as


(⎜)



S 
n·u dl ∼ ∇·us.
(2)
If Eq.(2) is applied to the two regions below,
gausstheorem.jpg
it follows


(⎜)



s1 
n·u dl
 ∼ 
∇·us1,
(3)

(⎜)



s2 
n·u dl
 ∼ 
∇·us2,
(4)
which can be added together to yield


(⎜)



∂(s1 + s2) 
n·u dl ∼ 

∇·u dS.
(5)
By repeating this for many small cells and taking limit of ∆si → 0, Eq.(5) becomes


(⎜)



S 
n ·u dl =



S 
∇·u dS.
(6)
This is called the Gauss divergence theorem.

Verification

gauss_ex.jpg

u =


x2 y
x + y



.
(LHS) On the boundary, one can set
x
=
a cos θ
(7)
y
=
a sin θ
(8)
and
On the boundary, one can set
nx
=
cos θ,
(9)
ny
=
sin θ,
(10)
so

n·u = a3 cos3  θsin θ+a sinθ(cos θ+sin θ),
(11)
and


(⎜)



S 
n·u dl
=



0 
( a3 cos3  θsin θ+a sinθ(cos θ+ sin θ) )a dθ
=
a2 π1
(12)
(RHS) Use the polar coordinate system, i.e.
x = r cosθ,     y = r sin θ,     dS = r dr dθ.
As

∇·u
=
2 x y + 1,
=
2 r2 cosθsin θ+ 1,
(13)
so





S 
∇·u dS
=




S 
(2 x y + 1) dx dy
=



0 

a

0 
(2 r2 cos θsin θ+1) r dr dθ
=
a2 π.
(14)

Alternative interpretation of Gauss theorem

gauss_1d.jpg
The fundamental theorem of calculus postulates that integrations and differentiations are reciprocal to each other, i.e.


b

a 
f ′(x) dx = [ f ]ab.
(15)
The right hand side of Eq.(15) can be written as
[ f ]ab
=
f |x=a + f |x=b
=
(−1) ×f |x=a + (+1) ×f |x=b
=
na  f |x = a + nb  f |x = b
=
(n f) |x=a + (n f) |x=b
=


boundary 
n f.
(16)
So Eq.(15) can be written as


b

a 
f ′(x) dx =

boundary 
n f.
(17)
Equation (17) can be extended to 2-D as





S 
f  dS =
(⎜)



S 
n f dl.
(18)
Note that both ∇ and n are vectors. If we set f = u (vector), Eq.(18) can be written as





S 
∇·u dS =
(⎜)



S 
n ·u dl,
(19)
which is the Gauss divergence theorem.
Note that most of the usage of the divergence theorem is to convert a boundary integral that contains the normal to the boundary into a volume (area) integral by replacing the normal (n) by a nabla (∇) to be placed in front of the expression.



(⎜)



S 
ndl =



S 
∇…dS.
(20)

Integration by parts

Integrating the both sides of

( u v )′ = u ′ v + u v′,
(21)
yields


b

a 
( u v ) ′d x =
b

a 
 uv dx +
b

a 
 u vdx,
(22)
or

[ u v ]ab =
b

a 
 uv dx +
b

a 
 u vdx,
(23)
or

b

a 
uv dx = [ uv]ab
b

a 
uvdx.
(24)
Equation (24) is the 1-D formula for integration by parts.
Equation (24) can be extended to 2D as




S 
u v dS =
(⎜)



S 
u v n dl



S 
uv dS.
(25)
Example (Green's identity)





S 
(∇u ·∇v + uv) dS =
(⎜)



S 
u v

n
dl.
(26)
Proof:
Rewrite the right hand side of Eq.(26) as

(⎜)



S 
uv ·n d l,
(27)
which is a boundary integral with n. Therefore, according to the Gauss theorem, it follows


(⎜)



S 
uv ·n d l
=




S 
∇(uv ) d S
=




S 
( ∇u ·∇v + u ∇·∇v) d S
=




S 
( ∇u ·∇v + uv) d S .
(28)
Examples
  1. Heat conduction
    The balance of energy is stated as





    S 
    ρCp T

    t
    dS =
    (⎜)



    S 
    (−n) ·h dl,
    (29)
    where ρ is the mass density, Cp is the specific heat, h is the heat flux across the boundary of the control surface.
    Using the Gauss theorem, the right hand side of Eq.(29) becomes


    (⎜)



    S 
    (−n) ·h dl = −



    S 
    ∇·h dS,
    (30)
    so it follows




    S 
    ρCp T

    t
    dS = −



    S 
    ∇·h dS,
    (31)
    or
    ρCp T

    t
    + ∇·h = 0.
    (32)
    Using Fourier's law,
    h = − kT,
    (33)
    where k is the thermal conductivity, Eq.(32) becomes
    ρCp T

    t
    = ∇·(kT).
    (34)
  2. Equilibrium equation in static elasticity
    The balance of force for continua is stated as


    (⎜)



    S 
    t dl+



    S 
    b dS=0,
    (35)
    where t is the surface traction force and b is the body force. The surface traction force t is the contribution of the stress tensor, σ, in the direction of n as
    t = σ·n,
    (36)
    so Eq.(35) becomes

    (⎜)



    S 
    σ·n dl+



    S 
    b dS=0,
    (37)
    or




    S 
    ∇·σdS +



    S 
    b dS=0,
    (38)
    or

    ∇·σ+ b = 0,
    (39)
    which is known as the equation of equilibrium. For small elastic deformation,
    σ = Cu,
    (40)
    where C is the elastic modulus and u is the displacement, Eq.(39) becomes
    ∇·( Cu) + b=0,
    (41)
    which is known as the Navier's equation 2.
  3. Green's second identity




    S 
    (uvvu) dS =
    (⎜)



    S 

    u v

    n
    v u

    n

    dl.
    From the first Green's identity,



    S 
    (∇u ·∇v + uv) dS
    =

    (⎜)



    S 
    u v

    n
    dl,
    (42)



    S 
    (∇v ·∇u + vu) dS
    =

    (⎜)



    S 
    v u

    n
    dl.
    (43)
    Subtracting Eq.(43) from Eq.(42) gives the second Green's identity. This identity is used to derive solutions to Poisson's equation (∆u = −ρ).


Footnotes:

1
Integrate with respect to
from to .

Enter a^4 Cos[x]^3 Sin[x] +a^2 Sin[x]Cos[x]+a^2 Sin[x]^2, x, 0, 2 Pi (you can copy and paste with the mouse.)

2 Each quantity is a tensor.


File translated from TEX by TTH, version 4.03.
On 01 Oct 2023, 15:13.