Analytical solutions are presented for longitudinal flow along a superhydrophobic annular
pipe where one wall, either the inner or outer, is a fully no-slip boundary while the other
is a no-slip wall decorated by a rotationally symmetric pattern of no-shear longitudinal
stripes. Formulas are given for the effective slip length associated with laminar flow along
the superhydrophobic pipe and the friction properties are characterized. It is shown
how these new solutions generalize two solutions to mixed no-slip/no-shear boundary
value problems due to Philip [J. Appl. Math. Phys., 23, (1972)] for flow in a singlewalled superhydrophobic pipe and a superhydrophobic channel. This is done by providing
alternative representations of Philip’s two solutions, including a useful new formula for
the effective slip length for his channel flow solution. For a superhydrophobic annular
pipe with inner-wall no-shear patterning there is an optimal pipe bore for enhancing
hydrodynamic slip for a given pattern of no-shear stripes. These optimal pipes have
a ratio of inner-outer pipe radii in the approximate range 0.5–0.6 and depending only
weakly on the geometry of the surface patterning. Boundary point singularities are found
to be deleterious to the slip suggesting that, in designing slippery pipes, maximizing the
size of uninterrupted no-shear regions is preferable to covering the same surface area with
a larger number of smaller no-shear zones. The results add to a list of analytical solutions
to mixed boundary value problems relevant to modelling superhydrophobic surfaces