PX285 - L3 - potential flows involving cylinders and aerofoils

neglecting vorticity, and considering an irrotational and incompressible flow, with potential given by: $\nabla^{2} ϕ = 0$
$\nabla^{2} ϕ$ is linear, with linear boundary conditions
a sum of solutions is also a solution
considering a sum of uniform flow and potential dipole

ϕ = U_{0} x + \frac{a \cos θ}{r}

considering cylindrical coordinates are used, with the origin at the dipole:

ϕ = U_{0} r (l + \frac{R^{2}}{r^{2}}) \cos θ

where, $R^{2} = a / u_{0}$ , $u_{0}$ is the speed at a large distance from dipole

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

the stagnation points: $u_{r} (r, θ) = 0$ and $u_{θ} (r, θ) = 0$ at $r = R$ and $θ = 0, π$

there is no flow through the surface
hence, it can be considered rigid, ie: $u_{⊥} = 0$
instead of the dipole, a cylinder of radius, R, can be put

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

\begin{matrix} u_{r} = 0 \\ u_{θ} = - 2 U_{0} \sin θ \\ u_{θ, m a x} = 2 U_{0} at θ = \frac{π}{2}, - \frac{π}{2} \end{matrix}

$u_{θ} \neq 0$ is not consistent with the condition that $\vec{u} = 0$ at $r = R$
this will be revisited when viscosity is considered

calculating the force experienced by the cylinder from the flow
determining the pressure on the surface of the cylinder using bernoulli's principle at a large distance, $x \to - \infty$ :

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

the net force from the flow on the cylinder:

d \vec{S} = (\cos θ, \sin θ) R d θ

\begin{aligned} F_{x} & = - \int P (d \vec{S})_{x} \\ = - \int_{0}^{2 π} P_{s} (θ) R \cos θ d θ \\ = (P_{0} + ρ \frac{U_{0}^{2}}{2}) R \int_{0}^{2 π} \cos θ d θ \\ = 0 \end{aligned}

similarly, the force in $\hat{y} :$

F_{y} = \int_{0}^{2 π} P_{s} R \sin θ d θ = 0

therefore, the cylinder does not experience any net force:

| \vec{F} | = 0

the failure to satisfy the boundary condition and the net force at the boundary being zero is called the d'Alembert paradox, which is resolved by accounting for viscosity and turbulence