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

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-$αβ$ 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.

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