A new proposal of power series method to solve the Navier-Stokes equations: application contexts and perspectives

In this work we investigate the possibility of expressing the solution of the Navier-Stokes equations as a power series. Although there have been many studies on this subject, it is still much debated. The existence of such a solution and its uniqueness are debated, especially in the general case of a non-stationary fluid in two or three dimensions. The greater the complexity of the fluid dynamical phenomenon under consideration, the more controversial and therefore uncertain are the deductions about the existence and uniqueness of the solution as a power series. Here, we ask some crucial questions on the matter and try to give an answer from the point of view of applied mathematics and numerical analysis. In particular, by way of example, we construct the Navier-Stokes equations for a compressible, multicomponent and non-stationary fluid from elementary principles of conservation of mass, momentum and energy, and we introduce a bump function model for the presence of different components and/or different phases in the fluid. The bump function is a function of space, time and physical characteristics of the components (and/or phases) such as density. We present a method to calculate it and discuss about its uniqueness. We show that under certain conditions on the bump function, and the forces acting on and in the fluid, the power series solution is unique. We finally discuss the advantages and the limitations of a solution in power series, concluding that, although plagued by limitations, it is a viable way forward even in highly complex case studies.