An Analytical Framework for Imposition of a Rigid Immersed Surface on the Incompressible Navier--Stokes Equations

TL;DR

This paper introduces a novel analytical framework that reformulates the incompressible Navier-Stokes equations using time dilation to precisely impose rigid solid boundaries, demonstrated through exact solutions of classical flows.

Contribution

It develops a modified Navier-Stokes formulation with penalization terms enabling exact modeling of rigid bodies within fluid flows.

Findings

Modified equations satisfy key physical boundary conditions.

Exact solutions for classical flows with embedded rigid bodies.

Framework effectively models solid-fluid interactions.

Abstract

The incompressible Navier-Stokes equations are re-formulated to involve an arbitrary time dilation; and in this manner, the modified Navier-Stokes equations are obtained which have some penalization terms in the right hand side. Then, the solid rigid bodies are modeled as the regions where time is dilated infinitely. The physical and mathematical properties of the modified equations and the penalization terms are investigated, and it is shown that the modified equations satisfy the no-slip, no-diffusion, no-advection, and no-pressure coupling conditions. The modified equations can be used in exact imposition of the solid rigid bodies on the incompressible Navier-Stokes equations. To show the capability of the modified equations, three classical exact solutions of the Navier-Stokes equations, that is, the Stokes first problem, the plane stagnation point flow, and the stokes flow over a sphere are re-solved exactly, this time in the presence of a solid region.