Integral Theorems and Second Order Operators 1st Order Integral Theorems Volume R with Surface S • Gradient theorem ndS

d • Divergence theorem

Open Surface S with Perimeter C • ClthCurl theorem ndS

• Stokes’ theorem

2 The Gradient Theorem Finite Volume R Begin with the definition of grad: Surface S 1 grad  Lim ndS τ0  τ ΔS Sum over all the d in R:

d

nidS

di+1

ni+1dS di

3 Assumptions in Gradient Theorem

4 S Flow over a 1 fin ite w ing

S1

S2

S=SS = S1 +S+ S2

R is the volume of fluid

enclosed between S1 and S2

p is not defined inside the p d  pndS wing so the wing itself   must be excluded from the R S integral 5 Alternative Definition of the Curl

e

Perimeter Ce Area  ds



1  e.curlA  Lim A.ds  Lim Ce 0    0  Ce 6 Stokes’ Theorem Finite Surface S Begin with the alternative definition of curl, choosing the With Perimeter C direction e to be the outward normal to the surface n: 1 n. A  Lim A.ds n  0   C Sum over all the d in S: e d

dsi+1 d dsi i+1

di 7 Stokes´ Theorem and Velocity

• Apply Stokes ´ Theorem to a velocity field

 V.ndS  V.ds S  C • Or, in terms of vorticity and circulation

Ω.ndS  V.ds  C S  C • What about a closed surface?  Ω.ndS  0 S 8 Assumptions of Stokes´ Theorem

9 Flow over a finit e wi ng  V.ndS  V.ds S  C

C

S

Wing with circulation must trail vorticity. Always. 10 Vector Operators of Vector PdProducts

( ) =  +     .( A) = .A+.A    ( A) =  A +  A          (A.B) = (A. )B + (B.A) + A(  B) + B(  A)       .(A B) = B. A - A. B           (A B) = A( .B) + (B. )A - B( .A) - (A. )B

11 Convective Operator

        (A. ) =  Ax  Ay  Az   x y z    A.(  )

V . = change in density in direction of V, multiplied by magnitude of V         (A. )B =  Ax  Ay  Az B  x y z  1       = 2 [(A.B)- A(  B)- B(  A)      - (A B)+ A( .B)- B( .A)]

12 Second Order Operators  2 2  2 .  2    The Laplacian, may also be x2 y 2 z 2 applie d to a vec tor fie ld.

(.A)

 A  (.A)  2A

• So, any vector differential equation of the form B=0   0 can be solved identically by writing B=. • We say B is irrotational. • We refer to  as the scalar potential.

• So, any vector differential equation of the form .B=0 . A  0 can bldidtillbitibe solved identically by writing B=A. • We say B is solenoidal or incompressible. • We refer to A as the vector potential. 13 2nd Order Integral Theorems

 • Green’s theorem (1st form)   2   Volume R     d   dS with Surface S R  n S ndS

• Green’s theorem (2nd form)  d      2  2 d   - dS R  S n n

These are both re-expressions of the divergence theorem.

14 Helmholz Decomposition Theorem

• Any vector field may be expressed as the sum of a gradient vector field and a curl vector field. B   A  

Vector Potential Scalar Potential

15