= Uniform potential flow around a fixed sphere
{title2=$\phi=U(r+a^3/(2r^2))\cos\theta$}
For a fixed impermeable <sphere> of radius $a$ and incident <uniform flow> $U\mathbf e_z$, the <velocity potential> is the displayed expression and the <Stokes streamfunction> is $\Psi=(U/2)(r^2-a^3/r)\sin^2\theta$. Differentiation gives $u_r=U(1-a^3/r^3)\cos\theta$ and $u_\theta=-U(1+a^3/(2r^3))\sin\theta$. Thus the <no-penetration boundary condition> holds, the poles are <stagnation points>, and the equatorial surface speed is $3U/2$. The disturbance is a <potential dipole> decaying as $r^{-3}$ in <velocity>. The <sphere theorem for a Stokes streamfunction> constructs this solution from $\Psi_0=Ur^2\sin^2\theta/2$.
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