The stability of three-dimensional time-periodic flows with ellipsoidal stream surfaces

Most steady flows with constant vorticity and elliptical streamlines are known to be unstable. These, and certain axisymmetric time-periodic hows, can be analysed by Floquet theory. However, Floquet theory is inapplicable to other time-periodic hows that yield disturbance equations containing a quasi-periodic, rather than periodic, function. A practical method for surmounting this difficulty was recently given by Bayly, Holm & Lifschitz. Employing their method, we determine the stability of a class of three-dimensional time-periodic flows: namely, those unbounded hows with fixed ellipsoidal stream surfaces and spatially uniform but time-periodic strain rates. Corresponding, but bounded, hows are those within a fixed ellipsoid with three different principal axes. This is perhaps the first exact stability analysis of non-reducibly three-dimensional and time-dependent flows. Though the model has some artificial features, the results are likely to shed light on more complex systems of practical interest.