352 Chapter 5 Higher Dimensions and Other Coordinates
ifλis the second positive solution ofJ 5 / 2 (λ)= 0.
8.Solve the potential problem in the exterior of a sphere.
1
ρ^2[∂
∂ρ(
ρ^2 ∂u
∂ρ)
+^1
sin(φ)∂
∂φ(
sin(φ)∂u
∂φ)]
= 0 ,
R<ρ, 0 <φ<π,
u(R,φ)=f(φ), 0 <φ<π.9.L.M. Chiappetta and D.R. Sobel [Temperature distribution within a hemi-
sphere exposed to a hot gas stream,SIAM Review, 26 (1984): 575–577]
analyze the steady-state temperature in the rounded tip of a combustion-
gas sampling probe. The tip is approximately hemispherical in shape. Its
outer surface is exposed to hot gases at temperatureTG, and its base
is cooled by water at temperatureTWcirculating inside the probe. If
T(ρ, φ)is the temperature inside the tip, it should satisfy the condi-
tions
1
ρ^2[∂
∂ρ(
ρ^2 ∂T
∂ρ)
+^1
sin(φ)∂
∂φ(
sin(φ)∂T
∂φ)]
= 0 ,
0 <ρ<R, 0 <φ<π 2 ,
T(ρ, π/ 2 )=TW, 0 <ρ<R,k∂T
∂ρ(R,φ)=h[
TG−T(R,φ)]
, 0 <φ<π
2
together with boundedness conditions atρ=0andatφ=0.
The authors then change the variables to simplify the problem. Let
r=ρ/R,u(r,φ)=T(ρ, φ)−TW, and show that the problem forube-
comes
∂
∂ρ(
ρ^2 ∂∂ρu)
+sin^1 (φ)∂φ∂(
sin(φ)∂φ∂u)
= 0 , 0 <r< 1 , 0 <φ<π/ 2 ,
u(r,π/ 2 )= 0 , 0 <r< 1 ,
K∂u
∂r( 1 ,φ)+u( 1 ,φ)=D, 0 <φ<π/ 2 ,whereK=k/hRandD=TG−TW.
10.Solve the problem in Exercise 9. Hint: Use odd-indexed Legendre polyno-
mials to satisfy the boundary condition atφ=π/2.