On the steady motion of Navier-Stokes flows past a fixed obstacle in a three-dimensional channel under mixed boundary conditions

TL;DR

This paper investigates the existence, uniqueness, and force bounds of steady viscous incompressible flows past an obstacle in a 3D channel under mixed boundary conditions, providing explicit criteria based on inflow data.

Contribution

It offers explicit bounds on inflow velocity ensuring steady flow existence and uniqueness, and derives formulas for drag and lift forces with geometric bounds.

Findings

Explicit bounds on inflow velocity for flow existence and uniqueness

Volume integral formulas for drag and lift forces

Upper bounds on forces based on domain geometry

Abstract

We analyze the steady motion of a viscous incompressible fluid in a three-dimensional channel containing an obstacle through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet, together with the standard no-slip assumption on the obstacle and on the remaining walls of the domain. Explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity through the Bogovskii formula. A quantitative analysis of the forces exterted by the fluid over the obstacle constitutes the main application of our results: by deriving a volume integral formula for the drag and lift, explicit upper bounds on these forces are given in terms of the geometrical constraints of the domain.