  Biological swimmers frequently navigate in geometrically restricted media. We
study the prescribed-stroke problem of swimmers confined to a planar viscous
membrane embedded in a bulk fluid of different viscosity. In their motion,
microscopic swimmers disturb the fluid in both the membrane and the bulk. The
flows that emerge have a combination of two-dimensional (2D) and
three-dimensional (3D) hydrodynamic features, and such flows are referred to as
quasi-2D. The cross-over from 2D to 3D hydrodynamics in a quasi-2D fluid is
controlled by the Saffman length, a length scale given by the ratio of the 2D
membrane viscosity to the 3D viscosity of the embedding bulk fluid. We have
developed a computational and theoretical approach based on the boundary
element method and the Lorentz reciprocal theorem to study the swimming of
microorganisms for a range of values of the Saffman length. We found that a
flagellum propagating transverse sinusoidal waves in a quasi-2D membrane can
develop a swimming speed exceeding that in pure 2D or 3D fluids, while the
propulsion of a two-dimensional squirmer is slowed down by the presence of the
bulk fluid.
