Unsure about vorticity boundary condition in steady Stokes flow past sphere

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


\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,

\psi=0\quad\text{at}\quad r=1;\\ \dfrac{\partial \psi}{\partial r}=0 \quad\text{at}\quad r=1;\\ \psi=\dfrac{Ur^2\sin^2\theta}{2} \quad\text{at}\quad r=R_o;\\ w=0 \quad\text{at}\quad r=R_o,

I get a null solution for the vorticity, i.e, the computed solution for the vorticity is zero throughout the domain. This is clearly not the case for creeping flow, where the vorticity is maximum at the surface of the inner sphere.

I would highly appreciate any insight on the selection of boundary conditions for this problem. @kamensky

Thank You
Warm Regards