# Viscous Flow past a cylinder

I'm trying to solve the incompressible, viscous and small Reynolds number flow past a cylinder. At the surface of the cylinder ($$r=R$$) the velocity is zero and at infinity it is $$v_0 {\vec e}_y$$ where the infinite cylinder extents in the z-direction. I will restrict the problem to two dimensions and write $$\vec{v}=v_r \vec{e}_r + v_\phi \vec{e}_\phi \, .$$ The equations to be solved then are \begin{align} {\rm div} \vec{v} = \frac{1}{r} \partial_r r v_r + \frac{1}{r} \partial_\phi v_\phi &=0 \tag{continuity} \\ -\nabla p + \eta \nabla^2 \vec{v} &= 0 \tag{momentum} \end{align} where $$\eta$$ is the viscosity and $$p$$ the pressure. The continuity equation implies that \begin{align} rv_r &= \partial_\phi \psi(r,\phi) \\ v_\phi &= -\partial_r \psi(r,\phi) \end{align} with some unknown function $$\psi$$. This can be written as $$\vec{v} = \nabla \times \left(\psi \vec{e}_z\right)$$ and plugging into the momentum equation, it becomes after taking the curl $$\nabla \times \nabla \times \nabla \times \nabla \times (\psi \vec{e}_z)=\Delta^2 (\psi \vec{e}_z) = 0 \tag{1}$$ where the Laplacian was converted to the double curl using $${\rm div}\vec{v}=0$$ or vice versa.

The solution for $$\psi$$ as $$r \rightarrow \infty$$ can be obtained from integrating $$v_r = \frac{1}{r} \partial_\phi \psi \sim v_0 \sin(\phi)$$ and $$v_\phi = -\partial_r \psi \sim v_0 \cos(\phi)$$ and reads $$\psi \sim -v_0 r \cos(\phi) \qquad {\rm as} \qquad r \rightarrow \infty \, .$$

Therefore I expected the function $$\psi$$ to be of the form $$\psi(r,\phi)=f(r)\cos(\phi) \, .$$ Plugging into the Equation (1) and using a CAS, a solution can be found $$f(r)=c_1 r^3 + \frac{c_2}{r} + c_3 r + c_4 r \log(r) \, .$$

Now the problem is that I must have $$c_1=c_4=0$$ in order to match the asymptotics and also $$c_3=-v_0$$, but that leaves only 1 integration constant for the two boundary conditions $$v_r(R)=v_\phi(R)=0$$, which is impossible. So does this mean the product ansatz is not valid or what is happening here? Interestingly this ansatz works for a flow past a sphere in spherical coordinates, so I'm a bit stuck.

PS: Actually I just found out, that apparently this is called "Stokes paradox".