In the above equations, ρ is the mass density, v is the velocity, ψ(= −GM/r) is the Newtonian potential (where r is the distance from the central black hole, M is the black hole mass, and G is the gravitational constant), pgas is the gas pressure, σ is the viscous stress tensor, e is the gas internal energy, and Fc is the thermal conduction. Here, d/dt ≡ ∂/∂t + v · ∇ represents the Lagrangian or comoving derivative. The equation of state of the ideal gas is considered as 






p


gas


=


γ
−
1


ρ
e


, with γ = 5/3 being the adiabatic index. In the purely HD limit, like our case, Fc is defined as
4






F


c


=
−
λ
∇
T
,


where λ is the thermal diffusivity and T is the gas temperature. We use spherical coordinates (r, θ, ϕ) to solve the above set of equations. In most previous semianalytical studies on hot accretion flow, they assumed that the system is in a steady state and axisymmetric, i.e., ∂/∂ϕ = ∂/∂t = 0. These assumptions imply that all physical quantities are independent of the azimuthal angle, ϕ, and the time, t. Following numerical simulations of hot accretion flow (e.g., Stone et al. 1999; Yuan et al. 2012a), we consider the following components of the viscous stress tensor:
5






σ


r
θ


=
ρ
ν


r



∂


∂
r










v


θ




r





+



1


r






∂


v


r




∂
θ





,




6






σ


r
ϕ


=
ρ
ν


r



∂


∂
r










v


ϕ




r







,




7






σ


θ
ϕ


=
ρ
ν





sin
θ


r






∂


∂
θ










v


ϕ




sin
θ







,


where ν is called the kinematic viscosity coefficient. By substituting all the above assumptions and definitions into Equations (1)–(3), we obtain following partial differential equations (PDEs) in spherical coordinates. Hence, the continuity equation is reduced to the following form
8







1




r


2








∂


∂
r







r


2


ρ


v


r




+



1


r
sin
θ






∂


∂
θ





ρ


v


θ


sin
θ


=
0
.


The three components of the momentum equation are as follows:
9







ρ




v


r





∂


v


r




∂
r



+





v


θ




r








∂


v


r




∂
θ



−


v


θ




−





v


ϕ


2




r










=

−



GM
ρ




r


2





−



∂


p


gas




∂
r



+



1


r
sin
θ






∂


∂
θ





sin
θ


σ


r
θ




,







10







ρ




v


r





∂


v


θ




∂
r



+





v


θ




r








∂


v


θ




∂
θ



+


v


r




−





v


ϕ


2




r



cot
θ







=

−



1


r






∂


p


gas




∂
θ



+



1




r


3








∂


∂
r







r


3




σ


r
θ




,







11







ρ




v


r





∂


v


ϕ




∂
r



+





v


θ




r






∂


v


ϕ




∂
θ



+





v


ϕ




r







v


r


+


v


θ


cot
θ









=




1




r


2








∂


∂
r







r


2




σ


r
ϕ




+



1


r
sin
θ






∂


∂
θ





sin
θ


σ


θ
ϕ









+




1


r







σ


r
ϕ


+


σ


θ
ϕ


cot
θ


.





The energy equation is written as
12







ρ




v


r





∂
e


∂
r



+





v


θ




r






∂
e


∂
θ





−





p


gas




ρ







v


r





∂
ρ


∂
r



+





v


θ




r






∂
ρ


∂
θ










=




∂


v


θ




∂
r





σ


r
θ


+



∂


v


ϕ




∂
r





σ


r
ϕ


+



1


r








∂


v


r




∂
θ



−


v


θ






σ


r
θ







+




1


r






∂


v


ϕ




∂
θ





σ


θ
ϕ


−





v


ϕ




r







σ


r
ϕ


+


σ


θ
ϕ


cot
θ







+




1




r


2








∂


∂
r







r


2


λ



∂
T


∂
r





+



1


r
sin
θ






∂


∂
θ








λ
sin
θ


r






∂
T


∂
θ





.
Citation: Yuan et al. 2012a
Citation function: Uses; Uses