452 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
whereMbis the mass of the black hole core,Qis the heat source generated by the nuclear
burning, and
̃∆=^1
r^2 sinθ∂
∂ θ
(sinθ∂
∂ θ)+
1
αr^2∂
∂r
(r^2∂
∂r),
α=(
1 −
2 MbG
c^2 r)− 1
.
The boundary conditions of (7.4.23) become
(7.4.25) Pφ(Rs) =RsΩ 0 , Pφ(R 1 ) =R 1 Ω 1 , T(Rs) =T 0 , T(R 1 ) =T 1 ,
whereΩ 0 ,Ω 1 only depend onθ,T 0 ,T 1 are constants.
Make the translation
Pr→Pr, Pθ→Pθ, Pφ→Pφ+P ̃φ, T→T+T ̃, p→p+ ̃p,where(P ̃φ,T ̃, ̃p,ρ)is the solution of (7.4.24)-(7.4.25). Then the equations (7.4.7) are rewrit-
ten as
(7.4.26)
∂Pθ
∂ τ+
1
ρ(P·∇)Pθ=ν∆Pθ−P ̃φ
ρrsinθ∂Pθ
∂ φ+
2cosθP ̃φ
ρrsinθPφ−1
r∂p
∂ θ,
∂Pφ
∂ τ+
1
ρ(P·∇)Pφ=ν∆Pφ−1
ρr∂P ̃φ
∂ θPθ−P ̃φ
ρrsinθ∂Pφ
∂ φ−
1
ρ∂P ̃φ
∂rPr−
P ̃φ
ρrPr−cosθP ̃φ
ρrsinθPθ−1
rsinθ∂p
∂ φ,
∂Pr
∂ τ+
1
ρ(P·∇)Pr=ν∆Pr−P ̃φ
ρrsinθ∂Pr
∂ φ+
2 P ̃φ
ρ αrPφ−1
α∂p
∂r+β ρMbG
αr^2T,
∂T
∂ τ+
1
ρ(P·∇)T=κ ̃∆T−P ̃φ
ρrsinθ∂T
∂ φ−
1
ρdT ̃
drPr,divP= 0 ,with the boundary conditions
(7.4.27) Pr= 0 , Pφ= 0 ,
∂Pθ
∂r
= 0 , T=0 atr=Rs,R 1.2.Taylor instability.By the conservation of angular momentum andR 1 ≫Rs, the angular
momentumsΩ 0 andΩ 1 in (7.4.25) satisfy that
(7.4.28) Ω 0 ≫Ω 1 ,
This property leads to the instability of the rotating flow represented by the stationary solu-
tion:
(7.4.29) (Pr,Pθ,Pφ) = ( 0 , 0 ,P ̃φ),