The linear term $-(3/2)\Omega_Kx\mathbf e_y$ is the local <Keplerian shear> in the <shearing sheet>. The remaining constant
$$
V=r_0[\Omega(r_0)-\Omega_K(r_0)]
$$
is the gas's azimuthal velocity relative to the local Keplerian frame. The outwardly decreasing midplane pressure found in part (a)(iii) makes the gas <Sub-Keplerian>, so
$$
\boxed{V<0},
\qquad
|V|\sim\frac{c_s^2}{r_0\Omega_K}
\sim\Omega_Kr_0\left(\frac H{r_0}\right)^2.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 2 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-321/2/b/i/solution