





Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Solutions to the cylindrical laplace equation, a partial differential equation used in engineering and physics to describe various phenomena such as heat transfer, potential flow, and electrostatics. The solutions are presented for both homogeneous and non-homogeneous boundary conditions, with eigenfunctions and eigenvalues determined for radial and vertical directions.
Typology: Slides
1 / 9
This page cannot be seen from the preview
Don't miss anything!
Larry Caretto Mechanical Engineering 501B Seminar in EngineeringSeminar in Engineering AnalysisAnalysis February 16, 2009
2
Overview
3
Review Superposition
x = 0 x = L
y = H
y = 0
u = 0 u = u (^) E(y)
u = uN (x)
u = 0 4
Review Superposition Solution
x = 0 x = L
y = H
y = 0
u = 0 u = 0
u = uN (x)
u = 0
x = 0 x = L
y = H
y = 0
u = 0 u = uE(y)
u = 0
u = 0
5
Review General Superposition
6
Review x and y Swap
x = 0 x = L
y = H
y = 0
u = 0 u = 0
u = uN (x)
u = 0
x = 0 x = L
y = H
y = 0
u = 0 u = uE(y)
u = 0
u = 0
7
Review x-y Swap Solution
H u xy B y x n n n
= (^) ∑ n κ (^) n κn κ =^ π
∞ = 1
2 (, ) sin( )sinh( )
L u xy C x y n n n n n n
= (^) ∑ λ λ λ =^ π
∞ (^1) = 1 (, ) sin( )sinh( )
∫ (^ )
= • L N n
n
n
u x xdx
HL
C
0
()sin
sinh
2
λ
∫ (^ )
= • H E n
n
n
u y ydy
LH
B
0
( )sin
sinh
2
κ
κ
∑[^ ]
= + 1
( , ) sin( )sinh( ) sin( )sinh( ) n
uxy Cn λnx λny Bn κny κnx
8
Review Coordinate Transform
x = 0 x = L
y = H
y = 0
u = 0 u = 0
u = uN (x)
u = 0
x = 0 x = L
y = H
y = 0
u = 0 u = 0
u = 0
u = u (^) S(x)
9
Review y ← H – y Solution
n
∞ = 1
2 (, ) sin( )sinh ( )
L u xy C x y n n n
= (^) ∑ n λn λn λ =^ π
∞ = 1
1 (, ) sin( )sinh( )
= ∫ ( )
L N n n
Cn (^) HL u x xdx 0
()sin sinh
(^2) λ λ
= ∫ ( )
L S n n
Bn (^) HL u x xdx 0
()sin sinh
(^2) λ λ 10
Cylindrical Laplace
L
z = 0
R
u(r,L) = uN (r)
2
∂
z
u r
u r r r
11
What do We Expect?
2
2
u Rz u z
z
u r
u r r r = R
u Rz u z
t
u r
u r r r = R
α
What do We Expect? II
0 (, 0 ) 0 ( , ) ( )
2
2 ur urL u r z
u r
u r r r N
2 2
2 ux uxH u x y
u x
u = = = N ∂
19 20
21
2 2
− 1 = =− λ dz
dZz dr Zz
rdPr dr
d rPr
( ) (^2) rP(r) 0 P(r) CI 0 (r) DK 0 (r ) dr
rdPr dr
d −λ = = λ + λ
( 2 ) (^2) () 0 () sin( ) cos( ) 2 Zz Zz A z B z dz
dZz+λ = = λ + λ
2
∂
z
u r
u r r r
( ) (^2) rP(r) 0 P(r) CI 0 (r) DK 0 (r ) dr
rdPr dr
d −λ = = λ + λ
( 2 ) (^2) () 0 () sin( ) cos( ) 2 Zz Zz A z B z dz
dZz+λ = = λ + λ
23
2 (^2) + −x + y= dx
dy x dx
d y
2
λ
ν
Since J ν ~ x ν , I ν and K ν are real
24
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 x
Kn(x)
n = 0 n = 1n = 4
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5 6 7 x
In(x)
n = 0n = 1 n = 4 All Kn become infinite as x approaches 0
I 0 (0) = 1
25
Boundary Conditions
u rz C zI r m m m
(, ) m sin( m ) ( m) ( 2 1 ) 2 0
0
π = (^) ∑ λ λ λ = +
∞
= (^26)
Boundary Condition at r = R
∑
∞
=
0
( ) ( , ) sin( ) 0 ( ) m
∫[^ ]
∫
m m
L m R m
0
2 0
0
L m m 2
λ =( 2 + 1 )^ π
27
Boundary Condition at r = R II
π π
π
λ
λ λ λ λ
( 2 1 )
2 2
( 2 1 )
cos(^221 ) 1
sin( ) cos( ) cos( )^1 (^00)
⎜⎝⎛^ + ⎟⎠⎞− =−
=− =− −
n
LU L
n
n U
zUdz U z U L m
m
L m
L m m
[ sin( )] 2 sin( 2 )cos( ) 2 sin( 2 )cos( ) 2 0 0
z^2 dz x z z L L L L m
m m
L m
L m m m ⎥ = − = ⎦
⎤ ⎢⎣ =⎡^ −
λ λ λ λ λ λ
0
28
Boundary Condition at r = R III
∞ = (^) ⎟ ⎠
⎜ ⎞ ⎝
⎟ ⎠
⎜ ⎞ ⎝
⎟ ⎛^ + ⎠
⎜ ⎞ ⎝
(^00)
0
2 ( 2 1 ) (^21 )
2
( 2 1 ) 2
sin(^21 ) ( ,)^4 n L n I n R
L
I n r L
n z urz U π
π π
π
4 ( ) 2
( 2 1 )
2
( ) sin( )
sin( )
0 0 0
2 0
0 n I R
U I RL
n
LU
I R z dz
zUdz C m L m m m
L m m π λ λ
π λ λ
λ
= = + =
L m n 2
λ =( 2 + 1 )^ π
29 30
Exercise
0
(, 0 ) (, ) 0 ( , ) () (^1 ) 2 =
∂
+∂ ∂
∂ ∂
∂ uisfiniteat r
ur urL uRz u z z
u r
r u r r
R
37
Results for uR(z) = U
⎟⎟⎠
⎞ ⎜⎜⎝
⎛ +
⎟⎟⎠
⎞ ⎜⎜⎝
L
R R K n R
L
R R I n R F i
i n 0 (^00)
0 (^00) ( 2 1 )
( 2 1 )
π
π ⎟ ⎠
⎜ ⎞ ⎝
⎟− ⎛^ + ⎠
⎜ ⎞ ⎝
= ⎛^ + L FK n R L G I n R n 0 0 n 0 0
( 2 1 )π ( 2 1 ) π
∞ = +
⎥⎦
⎤ ⎢⎣
⎡ ⎟⎟⎠
⎞ ⎜⎜⎝ − ⎛^ + ⎟⎟⎠
⎞ ⎜⎜⎝ ⎟ ⎛^ + ⎠ ⎜ ⎞ ⎝
0
0 (^00) 0 (^00) ( 2 1 )
( 2 1 ) ( 2 1 ) 2 sin(^21 ) (,) 4 n n
n n G
L
R R FK n r L
R R I n r L
n z
U
urz
π π π
π
∞ = +
⎥⎦
⎤ ⎢⎣
⎡ (^) ⎟ ⎠
⎜ ⎞ ⎝
⎟− ⎛^ + ⎠
⎜ ⎞ ⎝
⎟ ⎛^ + ⎠
⎜ ⎞ ⎝
0
0 0 ( 2 1 )
sin(^21 ) (^21 ) (^21 ) ( ,)^4 n (^) n
n n G
L
FK n r L
I n r L
n z
urz U
π π π
π
⎟⎟⎠
⎞ ⎜⎜⎝
0
0 0
, , , R
R L
R L
z R
f r U
u (^) i 38
39
Another Hollow Cylinder
2
2 2
2 (^2) ∂ =
z
u u r r
u r
0
40
Separation of Variables
(^2) () ()
() 1 ()
(^1) = − =− λ dz
dZz dr Zz
rdPr dr
d rPr
( ) (^2) rP(r) 0 P(r) CJ 0 (r) DY 0 (r ) dr
rdPr dr
d +λ = = λ + λ
( 2 ) (^2) () 0 () sinh( ) cosh( ) 2 Zz Zz A z B z dz
dZz −λ = = λ + λ
41
Boundary Conditions
0 0 0 0
0 0
Boundary Conditions
0 0 0
0 0 0 0 0 0 0
i
i i i J R Y R
Det
0
0 0 0 0
J αm i αm αm αm i
43
Finding Eigenvalues = f(Ri /R 0 )
-0.
-0.
-0.
-0.
0.
0.
0.
0 20 40 60 80 100
radius ratio = 0. radius ratio = 0.5radius ratio = 0.
( ) ( ) 0 (^0 ) ⎟⎟= ⎠
⎞ ⎜⎜⎝ − ⎛ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ R Y J Y R R J R i αm i αm αm αm
44
Boundary Conditions
0 0 0 0
0 0
45
Boundary Conditions II
∑^ [^ ]
= − 1 0 0 0 0 0 0
( , ) sinh( ) ( ) ( ) ( ) ( ) m m^ m m m m m
urz C λzY λR J λr J λRY λr
( )
()
( ) ( )
() ( ) ( )
0 0 0 0 0 0 0 0
0 0 0
0 0 0
Y R J r J R Y r J R
Pr D
DY r J R
Pr DY R J r
m m m m m
m m
m m
λ λ λ λ λ
λ λ
λ λ
=− −
=− +
46
Boundary Condition at z = L
sinh( ) 2 [ ( ) ( )]
( ) () ( )
sinh( ) ( )
() ( )
02 02 0
0 2 0
0 2
0
0
0
0
L J R J R
J R ru rP rdr
L rP r dr
ru rP rdr C m m i m
R R
m m i N m R R
m m
R R
N m m i
i
i λ λ λ
πλ λ λ
λ λ
λ
−
= =
= = 1
( ) (, ) sinh( ) 0 ( ) m
uN r urL Cm λmLP λmr
47
sinh( )[ ( ) ( )]
( )
sinh( ) 2 [ ( ) ( )]
( ) ( )^2 [ ( ) ( )]
sinh( ) 2 [ ( ) ( )]
( ) ( )
0 0 0
0
02 02 0
(^20) 0 2 0 0 0
02 02 0
0 2 0
0
L J R J R
UJ R
L J R J R
J R U J R J R J R
L J R J R
J R rUP rdr C
m mi m
mi
m mi m
m mi m mi mi m
m mi m
R
m m mi Ri m
λ λ λ
π λ
λ λ λ
πλ λ πλ λ λ λ
λ λ λ
πλ λ λ
= +
−
⎥⎦
⎤ ⎢⎣
= −
∑
∞
(^1 )
0 0
m (^) m i m
m i m m
m
Integrals from Carslaw and Jaeger, Conduction of Heat in Solids, Oxford, 1959.
48
∞ = (^) ⎟⎟+ ⎠
⎞ ⎜⎜⎝
⎛
⎟⎟⎠
⎞ ⎜⎜⎝
⎛ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛
⎟⎟⎠
⎞ ⎜⎜⎝
⎛
⎟⎟⎠
⎞ ⎜⎜⎝
(^10) (^00)
(^0000)
0
0 sinh ( )
(,) sinh m m i m
m i m
m
m J RR J
R P r R J R
R
L
L
z R
L
U
urz α α
α α
α
α π
∑
∞
(^1 )
0 0
m (^) m i m
m i m m
m
U(r,z)/U depends on z/L, r/R 0 , Ri/R 0 , and L/R