 Dear Community

I am solving the steady Stokes flow past a sphere problem in a hemispherical annulus (2D),

with the following governing equations for the stream function (\psi) and the vorticity (w):

w r\sin\theta+E^2\psi=0\\ \nabla^2 w-\dfrac{w}{r^2\sin^2\theta}=0

where

\nabla^2\equiv\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2\dfrac{\partial}{\partial r}\right)+\dfrac{1}{r^2\sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right)\\[10pt] E^2\equiv\dfrac{\partial^2}{\partial r^2} + \dfrac{\sin\theta}{r^2}\dfrac{\partial}{\partial \theta}\left(\dfrac{1}{\sin\theta}\dfrac{\partial}{\partial \theta}\right)

and variational formulation

\int\int\Biggl[\left(\dfrac{\partial w}{\partial x}\dfrac{\partial v}{\partial x}+\dfrac{\partial w}{\partial y}\dfrac{\partial v}{\partial y}\right)-\dfrac{v}{r\sin\theta}\dfrac{\partial w}{\partial y}+\dfrac{wv}{r^2\sin^2\theta}\Biggr]dxdy=0\\[5pt] \int\int\Biggl[\left(\dfrac{\partial \psi}{\partial x}\dfrac{\partial u}{\partial x}+\dfrac{\partial \psi}{\partial y}\dfrac{\partial u}{\partial y}\right)+\dfrac{u}{r\sin\theta}\dfrac{\partial \psi}{\partial y}-wr\sin\theta u\Biggr]dxdy=0

To check that I have set up the variational form correctly, I imposed the known solutions to the stream function and vorticity

\psi_{\text{exact}}=U\sin^2\theta\left[\dfrac{r^2}{2}-\dfrac{3r}{4}+\dfrac{1}{4r}\right]\\ w_{\text{exact}}=-\dfrac{3U\sin\theta}{2r^2}

as Dirichlet boundary conditions and observed that the computed error between the numerical result and the analytical one is of \mathcal{O}(10^{-7}) for both the vorticity and the stream function. This convinces me of the correctness of the variational formulation.

However, when I try to solve the problem without relying on manufactured solutions, and using the following boundary conditions,