= Formal viscous inspiral time in Schwarzschild spacetime
{title2=$\Delta t=(m^2/\nu)\mathcal T(r_1/(3m))$}
For constant <kinematic viscosity> $\nu$, use the phenomenological <viscous torque in an accretion disk>, <specific angular momentum> $h=rc\sqrt{m/(r-3m)}$, and $h_0=\sqrt{12}mc$ at the <innermost stable circular orbit>. Neglect radial kinetic motion in the clock relation and neglect differential light travel time. The <Schwarzschild time> to drift formally from $r_1\geq6m$ to $6m$ is the displayed expression, with
$$
\mathcal T(X)=3X^2+6X+6\log(X-1)-8(X-1)^{3/2}-24\sqrt{X-1}+8.
$$
Indeed $\mathcal T(2)=0$ and $\mathcal T'(X)=6X[X-2\sqrt{X-1}]/(X-1)\geq0$ for $X\geq2$. The <zero-torque inner boundary condition> makes the formal drift speed diverge as $r\downarrow6m$, so the nearly circular approximation must give way to the <plunging region of a black-hole accretion disk> sufficiently close to that orbit. This formula describes the stipulated approximation, not an exact relativistic flow solution.
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