The azimuthal momentum equation with $u_y=-3\Omega x/2$ is
$$
-\frac32\Omega u_x=-2\Omega u_x
+\frac1\Sigma\frac\partial{\partial x}
\left(-\frac32\Omega\nu\Sigma\right).
$$
Hence the slow radial drift is
$$
\boxed{u_x=-\frac3\Sigma\frac{\partial(\nu\Sigma)}{\partial x}}.
$$
Substitution into <mass conservation>, $\partial_t\Sigma+\partial_x(\Sigma u_x)=0$, gives the <nonlinear diffusion equation>
$$
\boxed{\frac{\partial\Sigma}{\partial t}
=3\frac{\partial^2(\nu\Sigma)}{\partial x^2}}.
$$
It describes slow axisymmetric viscous spreading while the velocity remains close to the imposed Kepler shear. It omits epicyclic and acoustic waves, rapid transients, self-gravity, nonaxisymmetric structure, and edge dynamics for which the assumed shear and secular force balance fail.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-321/2/a/solution