We analyze experimentally and theoretically the flows that develop around the core of a +1 disclination placed at the center of a freely suspended ferroelectric smectic-C^{★} film subjected to a flow of ethanol. We show that the c[over ⃗] director partially winds under the action of the Leslie chemomechanical effect by forming an imperfect target and that this winding is stabilized by flows which are induced by the Leslie chemohydrodynamical stress. We show moreover that there is a discrete set of solutions of this type. These results are explained in the framework of the Leslie theory for chiral materials. This analysis confirms that the Leslie chemomechanical and chemohydrodynamical coefficients are of opposite signs and of the same order of magnitude to within a factor of 2 or 3. A method for measuring the velocity field is also proposed, which does not require seeding the film with particles that can disturb the flows.