Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 54 / 1 / b

Put $F=\bar{\nu }\Sigma$. In steady state, $r^{1/2}{\partial }_r(r^{1/2}F)$ is constant, soThe inward mass accretion rate and outward viscous torque in an accretion disk are

For a zero-torque inner boundary condition at finite $r_{\mathrm{in}}$,With positive finite inner kinematic viscosity, $\Sigma$ tends to zero there. At fixed nonzero inward mass flux, the corresponding $v_r=-\dot{M}/(2\pi r\Sigma )$ becomes large, signaling the breakdown of the nearly circular thin disc approximation in the inner transition region.

For no central accretion, $A=0$ andThis nonaccreting constant-torque disk has a finite inner stress and normally a nonzero inner surface density. An external inner torque supplies the angular momentum transported to an outer sink, although mass does not flow. If both this zero-mass-flux condition and a zero inner torque are imposed, $B=0$; there is no nontrivial positive-viscosity steady disk satisfying both.