Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 1 / b / i / Solution

With $\mathcal{G}=0$ and $\mathcal{T}=2\pi v_m{h}^{\prime }\Sigma$, the evolution equation reduces to the radial continuity equationor, for the mass per unit radius $M_r=2\pi r\Sigma$,The outward mass flux is $F_M=2\pi r{v}_m\Sigma$. If $\dot{M}>0$ denotes the inward accretion rate, steady mass conservation requires $F_M=-\dot{M}$. HenceThis is positive because the magnetic drift velocity satisfies $v_m<0$.