The curvature of material lines in chaotic cavity flows

Abstract

Material line folding is studied in two-dimensional chaotic cavity flows. Line folding is measured by the local curvature k=l×l′/‖l‖3, where l(q) is an infinitesimal vector in the tangential direction of the line, q is a coordinate along the line, and l′ is the derivative of l with respect to q. It is shown both analytically and numerically that folding is always accompanied by compression. The vector l′ plays a crucial role as a driving force for the stretching and folding processes. A material line is stretched when l′ is tangential to the line and it is folded when l′ is normal to the line. The spatial structure of the curvature field is computed numerically. The short-time structure of the curvature field is similar to the structure of unstable manifolds of periodic hyperbolic points, and closely resembles patterns observed in tracer mixing experiments and in stretching field computations. The long time structure of the field asymptotically approaches an entirely different time-independent structure. Probability density functions of curvature are independent of both time and initial conditions.