On Circulation-Preserving Complex-Lamellar Motions with Steady Streamlines

Abstract

Although the circulation-preserving property is defined in terms of a material description of fluid flow, an equivalent condition, referring to the existence of the acceleration potential, presents a differential equation for the flow velocity in the Eulerian description:

$$ \frac{\partial }{{\partial t}}curl{\mkern 1mu} v{\mkern 1mu} - {\mkern 1mu} curl{\mkern 1mu} \left( {v{\mkern 1mu} \times {\mkern 1mu} curl{\mkern 1mu} v} \right) = {\mkern 1mu} 0. $$

This equation can be useful, when affiliated with other defining assumptions of purely geometrical types, for investigating the nature of the field of flow in specific classes of circulation-preserving motions. In the present analysis, we introduce additional assumptions requiring the streamlines (but generally not the velocity magnitude) to be steady and to form a congruence of curves normal to a family of surfaces (complex-lamellar flows with steady streamlines). These assumptions by themselves can be used to extract from the preceding equation a simple relation, valid at each instant and along each vortex line, between the flow speed and the local spacing of the normal surfaces of the velocity vector (see the Lemma of Section 2). In case a steady vortex line C exists in the flow region, more significant results can be obtained concerning the velocity and vorticity magnitudes as well as the nature of streamlines and vortex lines upon the stream surface emanating from the curve C (see the Theorem and the Corollary of Section 3). If all vortex lines are steady, then the conclusions of the present analysis are valid on each surface composed of streamlines and vortex lines, and therefore valid in the entire flow region.