Particle in Cyclone - height of pipe that will keep the particle pressed against the inside wall

I’m trying to determine the required radius, $R$, and length of a hollow cylinder, $L$, to keep a particle that enters tangentially with an initial velocity $V_{t,i}$ at the top of the cylinder "stuck" to the wall of the cylinder (see figure below). This of course is similar to a traditional cyclone but without the cone at the bottom. I think I’ve been able to derive most of the kinetics equations of motion but I’m unsure how to proceed to determine the distance at which the particle moves away and separates from the wall or just begins to fall completely vertical.

As far as what I have so far, I’ve derived the sum of forces equations, but I may have simplified them to much. I’m assuming buoyant and frictional forces are negligible (correct me if that’s a bad assumption).
$$
Sigma F=F_D+F_C+F_g=ma
$$
Where $F_D$ is the drag force, $F_C$ is the centrifugal force, and $F_g$ is the force due to gravity.

From Stokes’ Law, if the particle has a radius $r_p$ viscosity $mu$ radial velocity $V_r$ and a density of $rho_p$ the drag force is given by:
$$
F_D=-6pi r_pmu V_r
$$

Similarly, the centrifugal force, once derived is given by:
$$
F_D=frac{4}{3} frac{pi rho_p r_p^3 V_t^2}{r}
$$

where $V_t$ is the tangential velocity.

The gravitational force is:
$$
F_g=mg= frac{pi d_p^3 g rho}{6}
$$

Summing together I get:

$$
Sigma F=frac{dV_t}{dt}=frac{4}{3} frac{pi rho_p r_p^3 V_t^2}{r}-6pi r_pmu V_r+frac{pi d_p^3 g rho}{6}
$$

This is a first-order nonlinear ordinary differential equation and has a solution that is very ugly. I’m still not sure how to relate any of this to the vertical distance, however my thought was that the particle will begin to fall away from the cylinder wall once the drag force is greater then the centrifugal force. If I can determine where that occurs I theoretically would be able to determine the height needed to keep the particle against the wall. However, I’m at a standstill. Any help would be greatly appreciated.