Boundary conditions

For $\mathbf{v}^{(T)}=\mathbf{0}$ on a spherical surface, $\mbox{$\mathcal T$}[\mathbf{v}]$ must be uniform over the surface. Since any function of $r$ can be subtracted from $\mbox{$\mathcal T$}[\mathbf{v}]$ without affecting $\mathbf{v}^{(T)}$, this constant can be taken as zero without loss of generality.

For $\mbox{\boldmath$\hat{r}$}\cdot\mathbf{v}^{(P)}=0$ on a spherical surface,

$${\mathcal L}^2\mbox{$\mathcal P$}[\mathbf{v}] = 0,\qquad\mbox{on surface}.$$ (B.17)

Since the only regular solutions of this equation are functions only of $r$ (Padmavathi et al. 1998, Lemma 2), and any such function can be subtracted from $\mbox{$\mathcal P$}[\mathbf{v}]$ without affecting $\mathbf{v}^{(P)}$, we can require $\mbox{$\mathcal P$}[\mathbf{v}]=0$ on the surface.

For a poloidal velocity field to satisfy the no-slip condition, $\partial \mbox{$\mathcal P$}/\partial r$ must be uniform over the surface. Again, there is no loss of generality in requiring it to vanish.