PX285 - L4a - lift and circulation - magnus effect

considering that the cylinder is rotating around its axis, which would drag the fluid in the direction of rotation due to viscosity
modelling it as:

ϕ = U_{0} r (1 + \frac{R^{2}}{r^{2}}) \cos θ - \frac{K}{2 π} θ

where, the negative term is the free vortex due to clockwise rotation

\begin{matrix} u_{r} = \frac{\partial ϕ}{\partial r} = U_{0} (1 - \frac{R^{2}}{r^{2}}) \cos θ \\ u_{θ} = \frac{1}{r} \frac{\partial ϕ}{\partial θ} = - U_{0} (1 + \frac{R^{2}}{r^{2}}) \sin θ - \frac{K}{2 π r} \end{matrix}

at the surface of the cylinder, $r = R :$

\begin{matrix} u_{r} (r = R) = 0 ⟹ rigid wall \\ u_{θ} (r = R) = - 2 U_{0} \sin θ - \frac{K}{2 π R} \end{matrix}

the pressure around the cylinder, $P_{s} (θ) :$

\begin{matrix} P_{0} + \frac{1}{2} ρ U_{0}^{2} = P_{s} + \frac{1}{2} ρ U_{θ}^{2} \\ P_{s} (θ) = P_{0} + \frac{ρ U_{0}^{2}}{2} - \frac{ρ}{2} [\frac{K^{2}}{4 π R^{2}} + 4 U_{0}^{2} \sin^{2} θ + \frac{4}{2} \frac{U_{0} K}{π R} \sin θ] \end{matrix}

the force:

\begin{matrix} \vec{F} = - \int P_{s} (θ) d \vec{S} \\ F_{x} = - \int_{0}^{2 π} P_{s} (θ) \cos θ R d θ = 0 \\ F_{y} = - \int_{0}^{2 π} P_{s} (θ) \sin θ R d θ = ρ U_{0} K \end{matrix}

there is no drag (ie. force in the $x$ direction)
the force is perpendicular to the velocity, which can be the lift force:

\vec{L} = ρ {\vec{U}}_{0} \times \vec{K}

where, $\vec{K}$ is the circulation vector

this is called the magnus effect
in the fluid's frame of reference:

\vec{L} = ρ \vec{K} \times {\vec{U}}_{o b j}

where, ${\vec{U}}_{o b j}$ is the velocity of a moving object with respect to the stationary fluid

in the vicinity of the boundary, the no-slip condition applies as a boundary condition
the flow moves with the cylinder
from bernoulli's principle: $u_{t} > u_{b}$ , and if $P_{s}^{(t)} > P_{s}^{(b)}$ , then there is lift