Navier-Stokes equations in one and two dimensions

The Navier-Stokes equations are an important tool in understanding and describing fluid flow. We investigate different formulations of the incompressible Navier-Stokes equations in the one-dimensional case along an axis and in the two-dimensional case in a circular pipe without swirl. For the one-dimensional case we show that the velocity approximations are remarkably accurate and we suggest that understanding this simple axial behaviour is an important starting point for further exploration in higher dimensions. The complexity of the boundary is then increased with the two-dimensional case of fluid flow through the cross section of a circular pipe, where we investigate two separate formulations of the Navier-Stokes equations and observe their differences. The first twodimensional formulation exhibits an auxiliary field which differs from the velocity by a gauge transformation. We are then able to eliminate the ambiguity related to the pressure boundary condition in the traditional formulation since the gauge freedom lets us assign specific and simple boundary conditions for both the auxiliary field and the gauge field. The latter two-dimensional formulation considers external forces acting on the fluid and resembles a more traditional approach to solving the Navier-Stokes equations. The two-dimensional results are then discussed and found to correspond with fluid mechanics theory given the initial conditions as well as the boundary conditions of the systems