= Two-boundary mass split of a ring-fed disk
{title2=$\dot M_{\rm in}/\dot M=(\sqrt{R_{\rm out}}-\sqrt{R_0})/(\sqrt{R_{\rm out}}-\sqrt{R_{\rm in}})$}
For steady injection into a <Keplerian disk> between two zero-<torque> boundaries, <conservation of mass> gives $\dot M=\dot M_{\rm in}+\dot M_{\rm out}$. <Conservation of angular momentum> gives $\dot M h_0=\dot M_{\rm in}h_{\rm in}+\dot M_{\rm out}h_{\rm out}$, where $h(R)=\sqrt{GMR}$. Eliminating the outward mass flux gives the displayed inward fraction. This fraction does not depend on the viscosity law: the two boundary fluxes must carry away the injected mass and angular momentum. Equivalently, continuity of the <viscous torque in an accretion disk> joins $\nu\Sigma\sqrt R=\dot M_{\rm in}(\sqrt R-\sqrt{R_{\rm in}})/(3\pi)$ inside the source to $\dot M_{\rm out}(\sqrt{R_{\rm out}}-\sqrt R)/(3\pi)$ outside.
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