Next: Radial-spherical coordinates. Transient 1-D.
Up: Cylindrical Coordinates. Transient 1-D.
Previous: Solid cylinder transient 1-D.
R11 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and r = b.
GR11(r, t r,
) |
= |
exp
- (t - )/a2 |
|
|
|
x |
|
where R(r) |
= |
J0()Y0() - Y0()J0() |
|
with eigenvalues given by
J0()Y0(b/a) - Y0()J0(b/a) = 0.
R12 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and
G/r = 0 (Neumann)at r = b.
GR12(r, t r,
) |
= |
exp
- (t - )/a2 |
|
|
|
x R(r) R(r) |
|
where R(r) |
= |
J0()Y1() - Y0()J1() |
|
with eigenvalues given by
J0()Y1(b/a) - Y0()J1(b/a) = 0.
R13 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and
kG/r + hG = 0 (convection) at r = b.
GR13(r, t r,
) |
= |
exp
- (t - )/a2 |
|
|
|
x R(r) R(r) |
|
where R(r) |
= |
S0J0() - V0Y0() |
|
with V0 |
= |
J1() + BJ0() |
|
S0 |
= |
- Y1() + BY0() |
|
B |
= |
ha/k |
|
Eigenvalues are given by
S0J0() - V0Y0() = 0
.
R21 Hollow cylinder a < r < b, with
G/r = 0 (Dirichlet) at r = a and G = 0 (Neumann) at r = b.
GR21(r, t r,
) |
= |
exp
- (t - )/a2 |
|
|
|
x R(r) R(r) |
|
where R(r) |
= |
J0()Y0() - Y0()J0() |
|
with eigenvalues given by
J1()Y0(b/a) - Y1()J0(b/a) = 0.
R22 Hollow cylinder a < r < b, with
G/r = 0 (Dirichlet) at r = a and G = 0 (Neumann) at r = b.
GR22(r, t r,
) |
= |
+ exp
- (t - )/a2 |
|
|
|
x R(r) R(r) |
|
where R(r) |
= |
J0()Y1() - Y0()J1() |
|
with eigenvalues given by
J1()Y1(b/a) - Y1()J1(b/a) = 0.
R23 Hollow cylinder a < r < b, with
G/r = 0 (Neumann) at r = a and
kG/r + hG = 0 (convection) at r = b.
GR23(r, t r,
) |
= |
exp
- (t - )/a2 |
|
|
|
x R(r) R(r) |
|
where R(r) |
= |
S0J0() - V0Y0() |
|
with V0 |
= |
J1() + BJ0() |
|
S0 |
= |
- Y1() + BY0() |
|
B |
= |
ha/k |
|
Eigenvalues are given by
S0J1() - V0Y1() = 0
.
R31 Hollow cylinder a < r < b, with
- kG/r + hG = 0 (convection) at r = a and G = 0 (Dirichlet) at r = b.
GR31(r, t r,
) |
= |
exp
- (t - )/a2 |
|
|
|
x R(r) R(r) |
|
where R(r) |
= |
J0()Y0() - Y0()J0() |
|
with U0 |
= |
- J1() + BJ0() |
|
W0 |
= |
- Y1() + BY0() |
|
B |
= |
ha/k |
|
Eigenvalues are given by
U0Y0(b/a) - W0J0(b/a) = 0.
R32 Hollow cylinder a < r < b, with
- kG/r + hG = 0 (convection) at r = a and
G/r = 0 (Neumann) at r = b.
GR32(r, t r,
) |
= |
exp
- (t - )/a2 |
|
|
|
x R(r) R(r) |
|
where R(r) |
= |
J0()Y1() - Y0()J1() |
|
with U0 |
= |
- J1() + BJ0() |
|
W0 |
= |
- Y1() + BY0() |
|
B |
= |
ha/k |
|
Eigenvalues are given by
U0Y1(b/a) - W0J1(b/a) = 0.
R33 Hollow cylinder a < r < b, with
- kG/r + h1G = 0 (convection) at r = a and
kG/r + h2G = 0
(convection) at r = b.
GR33(r, t r,
) |
= |
exp
- (t - )/a2 |
|
|
|
x R(r) R(r) |
|
where R(r) |
= |
S0J0() - V0Y0() |
|
with S0 |
= |
- Y1() + B2Y0() |
|
with U0 |
= |
- J1() - B1J0() |
|
V0 |
= |
- J1() + B2J0() |
|
W0 |
= |
- Y1() + B1Y0() |
|
B1 |
= |
; B2 = |
|
Eigenvalues are given by
S0U0 - V0W0 = 0.
Next: Radial-spherical coordinates. Transient 1-D.
Up: Cylindrical Coordinates. Transient 1-D.
Previous: Solid cylinder transient 1-D.
Kevin D. Cole
2002-12-31