Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional flows

Geometric structures of surfaces are formulated based on Caputo fractional derivatives. The Gauss frame of a surface with fractional order is introduced. Then, the non-locality of the fractional derivative characterizes the asymmetric second fundamental form. The mean and Gaussian curvatures of the surface are defined in the case of fractional order. Based on the fractional curvatures, incompressible two-dimensional flows are discussed. The stream functions are obtained from a fractional continuity equation. The asymmetric second fundamental form of stream-function surface is related to the path dependence of flux. Moreover, the fractional curvatures are calculated for the stream-function surfaces of uniform and corner flows. The uniform flow with fractional order is characterized by the non-vanishing mean curvature. The non-locality of corner flow is expressed by the mean and Gaussian curvatures with fractional order. In particular, the fractional order within the stream-function of corner flow determines the change of sign of Gaussian curvature. Therefore, the non-local property of incompressible flows can be investigated by the fractional curvatures.