Solutions to the Navier-Stokes equation set for spiral pipes
The research presented herein embodies three subject area specifically aimed at the investigation and application of the spiral geometry. These areas are: the derivation of a spiral coordinate system in E2; the formulation of a new metric suitable for spiral pipe structures; the numerical simulation of an incompressible viscous fluid flowing through spiral pipe structures. The spiral coordinate system is first derived and then proven admissible using differential geometry. Validation is achieved using the spiral coordinate system as an alternative transformation for mapping from Cartesian to Polar coordinates for the solution domain of the general wave equation from a square to a circular elastic membrane. Problems associated with curves that do not possess natural-parameterisation in terms of arc-length, and as such cannot use the standard form of the Frenet-Serrat formulae, are solved with the derivation of a generalized metric. This metric is presented and proven for use on an arbitrary shaped pipe of class 'n' and is especially suited for spiral pipe structures. The associated Christoffel symbols of the second kind are also derived and presented in association with the generalized metric for use with the tensorial form of the Navier-Stokes and continuity equations. Finally, the spiral coordinates system is extended into E3 for two types of pipe; the spiral conic and the spiral parabolic. The Continuity and Navier-Stokes equations are numerically solved for an incompressible viscous Newtonian fluid for these pipe structures with various inlet conditions and geometric constraints. A correlation is made with these solutions and solutions found for the helical pipe structure, the nearest equivalent to the spiral found in the open literature.