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Previous: Infinite body with a
RS01 Solid sphere, 0 < r < b, with G = 0 (Dirichlet) at r = b.
a. Best convergence for
(t - )/b2 small:
GRS01(r, t r,
) |
= |
|
|
|
|
x exp
-
- exp
-
|
|
b. Best convergence for
(t - )/b2 large:
GRS01(r, t r,
) = exp
- m2(t - )/b2sin(m)sin(m) |
|
RS02 Solid sphere, 0 < r < b, with
G/r = 0
(Newmann) at r = b.
a. Approximate relation for
(t - )/b2
0.022 (here B2 = - 1):
GRS02(r, t r,
) |
= |
exp
-
- exp
-
|
|
|
|
+exp
-
|
|
|
|
- expB2
+ B22
|
|
|
|
x erfc
+
|
|
b. Best convergence for
(t - )/b2 large:
GRS02(r, t r,
) |
= |
+ exp
- (t - )/b2 |
|
|
|
x sin()sin() |
|
The eigenvalues are given by the positive roots of
cot
= 1 |
|
RS03 Solid sphere, 0 < r < b, with
kG/r + hG = 0 (convection) at r = b.
a. Approximate relation for
(t - )/b2
0.022 (here
B2 = hb/k - 1):
GRS03(r, t r,
) |
= |
exp
-
- exp
-
|
|
|
|
+exp
-
|
|
|
|
- expB2
+ B22
|
|
|
|
x erfc
+
|
|
b. Best convergence for
(t - )/b2 large:
GRS03(r, t r,
) |
= |
exp
- (t - )/b2 |
|
|
|
x sin()sin() |
|
where
B2 = hb/k - 1 and where the eigenvalues are given by the positive
roots of
cot
= - B2 |
|
Next: Hollow Sphere, transient 1-D.
Up: Radial-spherical coordinates. Transient 1-D.
Previous: Infinite body with a
Kevin D. Cole
2002-12-31