The problem of formulation of appropriate boundary conditions for periodic flow fields is investigated. Sufficient conditions are established for the uniqueness of solutions for the creeping flow of Newtonian fluids through periodically constricted tubes. A perturbation solution is developed for the Newtonian periodic flow problem for an arbitrary boundary shape with sufficiently small axial variations in tube radius. The nonuniqueness of solutions to periodic flow problems when an insufficient number of periodic boundary conditions are imposed is illustrated.