Variational resolution of outflow boundary conditions for incompressible Navier-Stokes

TL;DR

This paper extends the WIDE variational approach to non-Newtonian incompressible Navier-Stokes equations with outflow boundary conditions, providing a theoretical basis for boundary conditions like Navier-slip and outflow conditions.

Contribution

It introduces a variational method for non-Newtonian fluids with outflow boundaries, deriving natural boundary conditions from minimization, including Navier-slip and outflow conditions.

Findings

Extended WIDE approach to non-Newtonian fluids with power-law index r≥11/5

Derived natural boundary conditions at outflows and inflows from variational principles

Provided theoretical explanation for boundary conditions like Navier-slip and outflow conditions

Abstract

This paper focuses on the so-called Weighted Inertia-Dissipation-Energy (WIDE) variational approach for the approximation of unsteady Leray-Hopf solutions of the incompressible Navier-Stokes system. Initiated in [56], this variational method is here extended to the case of non-Newtonian fluids with power-law index $r\geq11/5$ in three space dimensions and large nonhomogeneous data. Moreover, boundary conditions are not imposed on some parts of boundaries, representing, e.g., outflows. Correspondingly, natural boundary conditions arise from the minimization. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition $-\frac12|\boldsymbol{v}|^2\boldsymbol{n}+\mathbb{T}\boldsymbol{n}=0$. This provides the first theoretical explanation for the onset of such boundary conditions.