ON STEADY FLOWS WITH LAMB SURFACES

Lamb surfaces, which contain both stream lines and vorticity lines, ar e shown to exist for any smoothly-deforming continuum whose Eulerian v orticity is steady and is proportional to the curl of the acceleration vector. These two conditions are realized in steady Rows with materia l vorticity lines, for which Lamb surfaces provide a complete topologi cal classification in terms of tori and cylinders when the Row domain is bounded. They are also met by steady Bows with zero helicity everyw here, for which vorticity lines are material and stream lines are geod esics on the Lamb surfaces. In a Lagrangian description, these latter Lamb surfaces become cylinders on which both vorticity lines and strea m lines are geodesics. For any Lamb surface, a natural system of coord inates can be defined in terms of any independent pair of smooth scala r functions that are uniform on vorticity lines. The evolution of the two coordinates is shown to be governed by a form of Hamilton's equati ons that generalizes the Lamb-Stuart-Mezic-Wiggins equations of motion for circulation-preserving fluid hows. Trajectories in the phase plan e of the two coordinates satisfy a variational principle based on the Poincare-Cartan relative integral invariant. (C) 1997 Elsevier Science Ltd.