Σ = \int_{- \infty}^{\infty} ρ d z, F = 2 π r \int_{- \infty}^{\infty} ρ u_{r} d z .

Vertical integration of mass conservation eliminates the surface term because

ρ u_{z} \to 0

, and axisymmetry eliminates the azimuthal derivative. Hence

\partial_{t} Σ + \frac{1}{2 π r} \partial_{r} F = 0 .

For

u_{ϕ} ≃ r Ω (r)

h = r u_{ϕ} = r^{2} Ω

. Multiply the azimuthal momentum equation by

r

, integrate vertically and azimuthally, and define the viscous torque in an accretion disk

G = - 2 π r^{2} \int_{- \infty}^{\infty} Π_{r ϕ} d z .

Subtracting

h

F \frac{d h}{d r} + \frac{d G}{d r} = 0 .

For an axisymmetric circular flow,

Π_{r ϕ} = μ r d Ω/ d r

. If

\overset{ν}{ˉ} Σ = \int_{- \infty}^{\infty} μ d z,

then

G = - 2 π \overset{ν}{ˉ} Σ r^{3} \frac{d Ω}{d r} .

Combining the two conservation laws gives

\partial_{t} (2 π r Σ h) + \partial_{r} (F h + G) = 0 .

The first term is the local rate of change of angular momentum per radial interval,

F h

is outward advective angular-momentum flux, and

G

is outward stress-carried angular-momentum flux.

Solved by gpt-5.6-sol high.

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