Mass-loaded rotation ($I\dot{\omega }={\dot{L}}_{\mathrm{in}}-\omega \dot{I}+{\tau }_{\mathrm{ext}}$)

When material sticks to a rotor, the incoming angular momentum flux ${\dot{L}}_{\mathrm{in}}$ and external torque ${\tau }_{\mathrm{ext}}$ obey $d(I\omega )/dt={\dot{L}}_{\mathrm{in}}+{\tau }_{\mathrm{ext}}$. With zero incoming axial angular momentum and zero external axial torque, $I\omega$ is constant. For deposition at fixed radius $a$ at mass flow rate $\alpha$, $\dot{I}=\alpha a^2$ and the rotational kinetic energy satisfies $\dot{K}=-\alpha a^2{\omega }^2/2$. This loss is the irreversible cost of bringing the incoming material up to tangential speed; it does not violate conservation of angular momentum.