A Condition On Spherical Surfaces To Non-Existence Of Incompressible Velocity Fields

TL;DR

This paper establishes a geometric criterion based on surface area evolution for incompressible flows, indicating certain initial velocity configurations cannot evolve over time, with implications for Navier-Stokes solutions.

Contribution

It introduces a novel geometric condition involving surface area convexity to determine the non-existence of certain incompressible velocity fields in fluid dynamics.

Findings

Derived an inequality linking surface area convexity to flow evolution.

Identified initial velocity fields that satisfy or violate this inequality.

Proposed examples illustrating the applicability of the criterion.

Abstract

In an incompressible velocity field, the surface area of a volume varies with time, but volume remains unchanged. If incidentally the surface becomes spherical along time, the area reaches a local minimum, since sphere has the least area that surrounds a volume. So the area is a function of time that is locally convex at this point. When applied to an incompressible Navier--Stokes fluid, this property is used to work out an inequality that suggest a criterion to non-existence of initial configurations of velocity fields, revealing its impossibility to evolve with time. Three velocity fields are proposed as examples. One of them agrees the inequality, the other two violate it.