Exact solutions for restricted incompressible Navier–Stokes equations with Dirichlet boundary conditions

Abstract

Abstract This paper exposes how to obtain a relation that have to be hold for all free-divergence velocity fields that evolve according to Navier–Stokes equations. However, checking the violation of this relation requires a huge computational effort. To circumvent this problem it is proposed an additional ansatz to free-divergence Navier–Stokes fields. This makes available six degrees of freedom which can be tuned. When they are tuned adequately, it is possible to find finite L 2 norms of the velocity field for volumes of R 3 and for t ∈ [ t 0 , ∞ ) . In particular, the kinetic energy of the system is bounded when the field components u i are class C 3 functions on R 3 × [ t 0 , ∞ ) that hold Dirichlet boundary conditions. This additional relation lets us conclude that Navier–Stokes equations with no-slip boundary conditions have not unique solution. Moreover, under a given external force the kinetic energy can be computed exactly as a funtion of time.