= Solution
The standard <Shakura--Sunyaev thin disk> assumes: a steady state; axial symmetry; a geometrically thin disk $H/R\ll1$; Newtonian, nearly Keplerian circular motion outside the inner edge; subsonic radial drift $|u_R|\ll u_\phi$; vertical <hydrostatic equilibrium>; negligible disk self-gravity; an optically thick, locally thermal spectrum; local radiative balance between viscous heating and cooling; and a local alpha-viscosity stress with a zero-torque inner boundary near the <innermost stable circular orbit>. These assumptions also exclude dynamically dominant winds and large-scale external torques from the standard solution.
Start from the <continuity equation>
$$
\partial_t\rho+\nabla\mathbin\cdot(\rho\mathbf u)=0.
$$
Integrate vertically, use axial symmetry and steadiness, and define the <surface density of a disk> $\Sigma=\int\rho\,dz$. Then
$$
\frac1R\frac d{dR}(R\Sigma u_R)=0,
$$
so the inward-positive <accretion rate> is
$$
\boxed{\dot m=-2\pi R\Sigma u_R=\text{constant}.}
$$
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