= Elastic secondary circulation around a rotating sphere
{title2=$\Psi=H(r)\sin^2\theta\cos\theta$}
For a sphere of radius $a$ rotating with <angular velocity> $\Omega$, the primary <Stokes flow> has $v_\phi=K\sin\theta/r^2$, $K=\Omega a^3$. In the weak, inertialess <second-order fluid> correction, put $c=\psi_1+2\psi_2$. The <stream function> in the title has
$$
H(r)=\frac{cK^2}{4\mu a^3}\left[1-3(a/r)^2+2(a/r)^3\right],\quad
v_r=\frac{H}{r^2}(3\cos^2\theta-1),\quad v_\theta=-\frac{H'}r\sin\theta\cos\theta.
$$
Both meridional components vanish at $r=a$ and decay at infinity. Indeed, $E_{r\phi}=-3K\sin\theta/(2r^3)$ gives $[\nabla\times\nabla\cdot(2cE^2)]_\phi=72cK^2\sin\theta\cos\theta/r^8$, and the curl of the momentum equation reduces to $(d^2/dr^2-6/r^2)^2H=72cK^2/(\mu r^7)$. For $c>0$ the <secondary flow> enters towards the equator and leaves near the poles; $c<0$ reverses it and $c=0$ removes this correction. The meridional circulation is even in the sign of sphere rotation. Inertial secondary flow is excluded from this calculation.
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