Hodograph method of flow on two-dimensional manifold

For some special flows,especially the potential flow in a plane,using the hodograph method has obvious advantages.Realistic flows have a stream surface,namely,a two-dimensional manifold,on which the velocity vector of the flow lies on its tangent space.By introducing a stream function and a potential function,we establish the hodograph method for potential flows on a surface using the tensor analysis.For the derived hodograph equation,we obtain a characteristic equation and its characteristic roots,from which we can classify the type of the second-order hodograph equation.Moreover,we give some examples for special surfaces.