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Consider a ball of radius $R$, fully immerged in an infinite incompressible fluid. We will suppose that the density of the fluid is equal to the density of the ball so that the ball is neutrally buoyant (the gravity force exactly compensates the buoyancy force).

Suppose that the fluid flow comes from a potential: $\vec{u}=\nabla\Phi$. Since the fluid is assumed to be incompressible, we obtain the Laplace equation $\Delta\Phi=0$. The boundary conditions are $\Phi=0$ at infinity, and $$-\nabla\Phi\cdot\vec{n}=(\vec{\Omega}\times\vec{r})\cdot\vec{n}=(\vec{\Omega}\times R\vec{e}_r)\cdot\vec{e}_r=0,$$ $\vec{\Omega}$ being the angular velocity of the ball (in the ball coordinates), $\vec{r}$ is a point on the surface of the ball and $\vec{n}$ is the outward-pointing normal at that point.

I would like to find the potential $\Phi$. We can assume that $\Phi=\vec{\Omega}\cdot\vec{\chi}$ for some vector field $\vec{\chi}$. Because of the second boundary condition, I assume further that each component of $\vec{\chi}$ only depends on $\theta$ and $\varphi$ (spherical coordinates) and then separate the variables to find that $\Phi$ can be expressed as an infinite combination of sinusoidal functions multiplied by Legendre polynomials. However there are too much constants to be determined.

I have seen in Physics books that instead of following this general idea, they can find a solution quite directly (like selecting the right spherical harmonics and combining them) and then determine the remaining constant using the boundary condition. However, in this case, since the normal velocity is always zero, I always end up finding the zero solution.

Can somebody provide some directions for finding the potential $\Phi$ for this problem?

Thank you!

[Disclaimer: it's not an homework exercise, I am trying to build an example for a Mathematics article, I don't know much about Physics.]

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  • $\begingroup$ In potential flow, the no-slip boundary condition does not have to be satisfied at the surface of the sphere. So the solution is the same as for potential flow past a non-rotating sphere. $\endgroup$ – Chet Miller Dec 14 '16 at 0:04
  • $\begingroup$ A more simplified problem that might help is lifting flow over a cylinder. The solution is obtained by superimposing a uniform flow with a doublet to obtain flow over a cylinder, and then adding a vortex of strength $\Gamma$. The superposition is a consequence of the linear Laplace equation for the potential. The resulting stream function can be readily obtained in any classical aerodynamics text. This might help you get the ball rolling, no pun intended. Additionally, there is no "no-slip condition" since the flow is inviscid and in fact the "body" of the solution is a streamline of the flow. $\endgroup$ – TRF Dec 14 '16 at 0:10
  • $\begingroup$ This might belong more on this site as here I think it falls under the homework-like category. $\endgroup$ – heather Dec 17 '16 at 2:36
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The Laplace equation you gave with the stated boundary conditions does not have a unique solution. This is as it should be, since there is an infinite number of flows that satisfy the condition that the flow be tangential to the sphere, just as in the 2-D case of flow around an infinite cylinder. What you are looking for seems to be a flow that also matches the tangential velocity of the sphere. This is possible for the spinning cylinder in 2-D, but I'm not sure a potential flow with this property exists.

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