User:Egm6321.f10.team4.Yoon/Mtg34

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_04_14_51_31.djvu | Mtg 34:]] Thu, 4 Nov 10

= Heat conduction on a sphere (cont'd) =

[[media: 2010_11_04_14_51_31.djvu | Page 34-1]]
Cartesian coordinates in terms of spherical coordinates:

Infinitesimal length in spherical coordinates
Now general curvilinear coordinates.

Laplace operator in spherical coordinates
For spherical coordinates:

$$\displaystyle h_1h_2h_3 = r^2\cos \theta $$

Axisymetric problems: No dependence on $$\displaystyle \varphi$$ (longitude).

[[media: 2010_11_04_14_51_31.djvu | Page 34-3]]
cf. King 2003 p.45 (in which the mathematical physics convention is used).

Unit sphere


Problem: Find $$\displaystyle \Psi(r,\theta)$$ such that $$\displaystyle \Delta \Psi = 0$$ and

Seperation of variables
(See [[media: 2010_09_02_13_55_50.djvu | Mtg25]]: Vibration of beam, Euler equation with constant coefficient $$\displaystyle \equiv $$ Euler L4-ODE-CC)

Ansatz:

[[media: 2010_11_04_14_51_31.djvu | Page 34-4]]
Use[[media: 2010_11_04_14_51_31.djvu | Eqn(4) p.34-3 in Eqn(1) p.34-3]]

Simplify $$\displaystyle \frac{1}{r^2} $$ and divide by $$\displaystyle R\Theta $$

Separated equations
2 seperated equations:

The L2-ODE-VC (2) is an Euler (homogeneous) L2-ODE-VC, see [[media: 2010_10_14_14_56_10.djvu | Eqn(2) p.25-3]].