= Solution
Denote the four terms by
$$
I=(\mathbf v_p\mathbin\cdot\nabla)\mathbf v_p,
\qquad
P=-\frac1\rho\nabla p,
\qquad
G=-\nabla\Phi,
\qquad
C=\Omega^2\mathbf R,
$$
so the radial or poloidal momentum equation is $I=P+G+C$.
i) In a thin <Keplerian accretion disk>, radial inertia and pressure are higher-order in $H/R$, leaving $G+C\simeq0$.
ii) In a nearly static stellar atmosphere, $I=C=0$ and <hydrostatic pressure> balance gives $P+G=0$.
iii) In pressureless gravitational collapse, rotation and pressure are negligible, so $I=G$; this is free fall.
iv) A <slim accretion disk> retains radial inertia and radial pressure together with gravity and centrifugal support, so all four terms generally survive: $I=P+G+C$.
v) A stationary geometrically thick disk or torus has negligible poloidal inertia but order-one pressure support, giving $P+G+C=0$.
vi) Nonrotating <Bondi accretion> has $C=0$ and $I=P+G$. In a highly supersonic Bondi--Hoyle limit the pressure term is also negligible, reducing this to ballistic $I=G$.
vii) A sub-Keplerian <advection-dominated accretion flow> has significant pressure support and radial inflow as well as rotation, so again $I=P+G+C$, with $C$ smaller than the Keplerian value and the remaining inward gravity balanced by $P$ and $I$.
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