= Solution
Assume all rings have the same <circulation> $\Gamma$ and orientation, and that there is no imposed background flow. <Vortex lines> describe the direction of <vorticity>, not the direction of <velocity>. For an incompressible <fluid>, the <velocity> induced by a thin <vortex filament> has the same Biot-Savart <integral> form as the <magnetic field> of a current filament:
$$
\mathbf u(\mathbf x)=\frac{\Gamma}{4\pi}\oint
\frac{d\boldsymbol\ell\times(\mathbf x-\mathbf x')}{|\mathbf x-\mathbf x'|^3}.
$$
A closely spaced coaxial array of <vortex rings> is therefore analogous to a solenoid. The <velocity field> passes \b[along the common axis through the rings and returns outside a finite array]. Its orientation reverses if every <circulation> is reversed. In a long array, far from its ends and vortex cores, the <velocity field of a dense coaxial vortex-ring array> is approximately
$$
\boxed{\mathbf u\simeq\frac{\Gamma}{l}\widehat{\mathbf z}\quad(\rho<r),
\qquad \mathbf u\simeq0\quad(\rho>r).}
$$
The exterior field is exactly zero only in the ideal continuous infinitely long limit, with the background <velocity> set to zero. In a finite array the <streamlines> spread out at one end, run back outside, and converge into the other. A large number of closely spaced rings alone does not guarantee that the total array <length> greatly exceeds $r$; if it does not, the end and return flows remain substantial. Different unspecified ring circulations would require superposing their actual signed contributions.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2017/paper-2-2017-vortex-array.png]
{title=Induced velocity through and around a finite array of co-oriented vortex rings}
{description=Original Biot-Savart calculation for 81 equal rings with spacing 0.1 radii and a small numerical core smoothing. Streamlines lie in a plane containing the common axis; the grey markers sample intersections with the vortex rings, rather than fluid trajectories. Flow returns outside the finite array.}
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