  Starting from the standard three-dimensional (3D) Gross-Pitaevskii equation
(GPE) and using a variational approximation, we derive an effective
one-dimensional nonpolynomial Schr\"odinger equation (1D-NPSE) governing the
axial dynamics of atomic Bose-Einstein condensates (BECs) under the action of a
singular but physically relevant funnel-shaped transverse trap, i.e., an
attractive 2D potential $\sim-1/r$ (where $r$ is the radial coordinate in the
transverse plane), in combination with the repulsive self-interaction. Wave
functions of the trapped BEC are regular, in spite of the potential's
singularity. The model applies to a condensate of particles (small molecules)
carrying a permanent electric dipole moment in the field of a uniformly charged
axial thread, as well as to a quantum gas of magnetic atoms pulled by an axial
electric current. By means of numerical simulations, we verify that the
effective 1D-NPSE provides accurate static and dynamical results, in comparison
to the full 3D GPE, for both repulsive and attractive signs of the intrinsic
nonlinearity.
