Analysis & validation of exact solutions to Navier-Stokes equation in connection with quantum fluid dynamics

Abstract

Navier-Stokes equation is considered to be the fundamental equation governing the dynamics of fluids. However, exact analytical solutions to this equation could only be obtained for highly idealized flows. More general exact solutions, if found, would be greatly beneficial since they would further our understanding of fluids and could be used to improve current computational solvers.
In the recent years, a general solution to the Navier-Stokes equation for incompressible fluids has been presented based on a quantum mechanics formulation known as quantum fluid dynamics (QFD). QFD predicts the presence of a velocity potential in the motion of quantum particles. This result was used to argue the presence of a velocity potential in fluid motion. The objective of the current study is to verify the validity of this general solution mathematically and conceptually. Through direct substitution, the mathematical validity of the general solution was confirmed provided a velocity potential function exists for the fluid motion. However, the general solution failed to produce the right solutions even for simple flows. The reason for this apparent failure requires further analysis. Conceptually, the existence of a velocity potential function for fluid motion cannot be argued from QFD as was done in the papers. Although not applicable to conventional fluids, the solution may find applications to the seemingly inviscid superfluids which are believed to possess a velocity potential function.