On the control volume arbitrariness in the Navier--Stokes equation

TL;DR

This paper develops a continuum theory exploring how arbitrary control volume shapes affect the Navier--Stokes equations, leading to insights on boundary conditions and thermodynamics.

Contribution

It introduces a generalized control volume approach that accounts for non-smooth surfaces, extending the derivation of Navier--Stokes-$\alpha\beta$ equations and boundary conditions.

Findings

Demonstrates the impact of control volume shape on fluid equations.

Provides an alternative derivation of natural boundary conditions.

Links control volume considerations to thermodynamics in fluid flow.

Abstract

We present a continuum theory to demonstrate the implications of considering general tractions developed on arbitrary control volumes where the surface enclosing it lacks smoothness. We then tailor these tractions to recover the Navier--Stokes-$\alpha\beta$ equation and its thermodynamics. Consistent with the surface balances postulated to propose this theory, we provide an alternative approach to derive the natural boundary conditions.