  We construct viscous fluid flow sourced by a force dipole embedded in a
cylindrical fluid membrane, coupled to external embedding fluids. We find
analytic expressions for the flow, in the limit of infinitely long and thin
tubular membranes. We utilize this solution to formulate the in-plane dynamics
of a pair of pusher-type dipoles along the cylinder surface. We find that a
mutually perpendicular dipole pair generically move together along helical
geodesics. Since the cylindrical geometry breaks the in-plane rotational
symmetry of the membrane, there is a difference in flows along the axial and
transverse directions of the cylinder. This in turn leads to anisotropic
hydrodynamic interaction between the dipoles and is remarkably different from
flat and spherical fluid membranes. In particular, the flow along the compact
direction of the cylinder has a local rigid rotation term (independent of the
angular coordinate but decays along the axis of the cylinder). Due to this
feature of the flow, we observe that the interacting dipole pair initially
situated along the axial direction exhibits an overall drift along the compact
angular direction of the tubular fluid membrane. We find that the drift for the
dipole pair increases linearly with time. Our results are relevant for
non-equilibrium dynamics of motor proteins in tubular membranes arising in
nature, as well as in-vitro experiments (25).
