I know quite well what a dipole is and in general what multipole moments are (in the context of, for instance, electrodynamics). What I find myself confused by is something called a "force dipole" in fluid dynamics. I believe that it is the same thing as a "stresslet". There are many papers about micro-organisms at low Reynolds numbers which talk about the velocity fields around "force dipoles", and these fields are always dissimilar to what I understand as the field around a "dipole". In fact, the field looks more like that of a pure "quadrupole" to me. As an example, I am attaching a figure cut out from the paper "Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations" (S. E. Spagnolie and E. Lauga, Journal of Fluid Mechanics 700, 105-147 (2012)). To me it is only the field in part (b), here called as that of a "source dipole", which looks like a dipole field, not the one in part (a). Is this merely a matter of convention, that fluid dynamicists refer to as a "force dipole" what would generally be a called a quadrupole? Any light shed on the matter would be appreciated.
i searched for the exact same problem recently after a debate with one of my colleagues. In my opinion, you already gave the answer to your question yourself.
A source dipole is the flow field resulting from a sink and a source brought together. In a sink, all streamlines point radially inward to the singularity at the origin, in a source, all point radially outward. Here, you have the analogy to electrodynamics. Both source and sink fields are irrotational flows. the source dipole is solution to the laplace equation.
A force dipole on the other hand is the flow field resulting from two close point forces in opposite directions (Those flows are not irrotational, and are of special importance in the hydrodynamics of bacterial suspensions, as force-free self propelled swimmers might not exert forces monopoles on the fluid). Point forces are modeled by Delta-functions at some point in space in one definite direction. The resulting flow fields are fundamental solutions of the Stokes equations (Navier-Stokes equations for vanishing Reynolds number).
C. Pozridikis: introduction to theoretical and computational fluid dynamics
M. Doi and S. F. Edwards: the theory of polymer dynamics
Hope i could help and i am not wrong ! =)