Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 347 / 2 / b

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- Solution b

Solution

In a Keplerian accretion disk,

u_{ϕ} = R Ω = (GM / R)^{1/2}

. Substitution into the steady azimuthal Navier--Stokes equation and cancellation of the common Keplerian factors gives

Σ u_{R} \frac{d}{d R} (R^{2} Ω) = \frac{1}{R} \frac{d}{d R} (ν Σ R^{3} \frac{d Ω}{d R}) .

Since

R^{2} Ω \propto R^{1/2}

and

d Ω/ d R = - 3Ω/ (2 R)

, this reduces to the standard viscous drift formula

u_{R} = - \frac{3}{Σ R ^{1/2}} \frac{d}{d R} (ν Σ R^{1/2}) .

The minus sign describes inward drift when

ν Σ R^{1/2}

increases outward.

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