The influence of finite-core thickness on the velocity field around a vortex tube is addressed. An asymptotic expansion of the Biot-Savart law is made to a higher order in a small parameter, the ratio of core radius to curvature radius, which consists of the velocity field due to lines of monopoles and dipoles arranged on the centerline of the tube. The former is associated with an infinitely thin core and is featured by the circulation alone. The distribution of vorticity in the core reflects on the strength of dipole. This result is applied to a helical vortex tube, and the induced velocity due to a helical filament of the dipoles is obtained in the form of the Kapteyn series, which augments Hardin’s [Phys. Fluids 25, 1949 (1982)] solution for the monopoles. Using a singularity-separation technique, a substantial part of the series is represented in a closed form for both the mono- and the dipoles. It is found from numerical calculation that the smaller the helix pitch is, the larger the relative influence of the dipoles is as the cylinder wound by the helix is approached.
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