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Up: Radial-spherical coordinates. Transient 1-D.
Previous: Solid Sphere, transient 1-D.
RS11 Hollow sphere, a < r < b, with G = 0 (Dirichlet) at r = a and G = 0 (Dirichlet) at r = b.
a. Best convergence for (t - ) small:
GRS11(r, t r,
) |
= |
exp
-
|
|
|
|
+exp
-
|
|
b. Best convergence for (t - ) large:
GRS11(r, t r,
) |
= |
|
|
|
|
x exp
-
sin(m)sin(m) |
|
RS12 Hollow sphere, a < r < b, with G = 0 (Dirichlet) at r = a and
G/r = 0 (Neumann) at r = b.
a. Best convergence for
(t - )/(b - a)2 < 0.022:
GRS12(r, t r,
) |
= |
exp
-
|
|
|
|
-exp
-
+ exp
-
|
|
|
|
- expB2
+ B22
|
|
|
|
x erfc
+
|
|
b. Best convergence for (t - ) large:
GRS12(r, t r,
) |
= |
exp
-
|
|
|
|
sin()sin() |
|
where the eigenvalues are given by positive roots of
cot
= - H2 |
|
and where
H2 = B2R2; B2 = - 1; R2 = (b - a)/b.
RS13 Hollow sphere, a < r < b, with G = 0 (Dirichlet) at r = a and
kG/r + hG = 0 (convection) at r = b.
a. Best convergence for
(t - )/(b - a)2 < 0.022
(note
B2 = h2b/k - 1):
GRS13(r, t r,
) |
= |
exp
-
|
|
|
|
-exp
-
+ exp
-
|
|
|
|
- expB2
+ B22
|
|
|
|
x erfc
+
|
|
b. Best convergence for (t - ) large:
GRS13(r, t r,
) |
= |
exp
-
|
|
|
|
sin()sin() |
|
where the eigenvalues are given by positive roots of
cot
= - H2 |
|
and where
H2 = B2R2; B2 = h2b/k - 1; R2 = (b - a)/b.
Next: Laplace Equation. Steady Heat
Up: Radial-spherical coordinates. Transient 1-D.
Previous: Solid Sphere, transient 1-D.
Kevin D. Cole
2002-12-31