Application of Adaptive Quadrature to Axi-symmetric Vortex Sheet Motion

Studies of the formation of ne structures on free-surfaces in liquids, such as curvature singularities or interface pinching, demand that the motion of the interface must be computed very accurately. Boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one. On the other hand, the boundary integrals are singular, and their accurate evaluation can prove quite challenging. In two dimensional motion, the interface is a just a curve. When this curve is closed or periodic, the singularity in the integrand may be removed and the trapeziodal rule may be applied with spectral accuracy. Unfortunately, the nature of the singularity in the integrand for three dimenional motion is much more di cult to treat. In this paper, we present an accurate adaptive quadrature to compute the motion of a vortex sheet in axi-symmetric ow. The technique is based on a vector-potential formulation which o ers some computational advantages over other methods based on the Biot-Savart Integral. Direct numerical computations show that our technique is much more accurate and e cient than existing techniques. 3