  The steady, coaxial flow in which two immiscible, incompressible fluids move
past each other in a cylindrical tube has a continuum of possibilities due to
the arbitrariness of the interface between the fluids. By invoking the presence
of surface tension to at least restrict the shape of any interface to that of a
circular arc or full circle, we consider the following question: which flow
will maximise the exchange when there is only one dividing interface Gamma?
Surprisingly, the answer differs fundamentally from the better-known
co-directional two-phase flow situation where an axisymmetric (concentric)
core-annular solution always optimises the flux. Instead, the maximal flux
state is invariably asymmetric either being a `side-by-side' configuration
where Gamma starts and finishes at the tube wall or an eccentric core-annular
flow where Gamma is an off-centre full circle in which the more viscous fluid
is surrounded by the less viscous fluid. The side-by-side solution is the most
efficient exchanger for a small viscosity ratio beta less than or equal to 4.60
with an eccentric core-annular solution optimal otherwise. At large beta, this
eccentric solution provides 51% more flux than the axisymmetric core-annular
flow which is always a local minimiser of the flux.
