A Note on Natural Coordinates and Frenet Frames

This paper discusses the relationship between natural coordinates in fluid mechanics and or-thogonal curvilinear coordinates. Since orthogonal curvilinear coordinates have some excellent mathemati-cal properties, natural coordinates can be applied more widely if they can be transformed to orthogonal cur-vilinear coordinates. Frenet formulas which describe the differential properties of natural coordinates were compared with the derivative formulas of orthogonal curvilinear coordinates to show that natural coordinates are not generally orthogonal curvilinear coordinates. A method was introduced to transform natural coordi-nates into orthogonal curvilinear coordinates by rotating the normal planes of the natural coordinates about the streamlines. The transformation is true as long as the natural coordinates satisfy several equations. Vor-ticity decomposition in the natural coordinates is used to show that these conditional equations are satisfied only if the streamlines are perpendicular to the vortexlines on every given point in the flow field. These equa-tions apply in both planar flows and axisymmetric flows without a circumferential velocity component, but do not apply in some 3-D flows such as Beltrami flow.