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Periodic Navier Stokes Equations: PNS system within & external to embedded spheres of the 3-Torus: The mapping function at a 90 degree corner

Terry E. Moschandreou

Abstract

In this paper a solution for the third component of velocity for the Periodic Navier Stokes equations at the corners outside the sphere embedded in each cell of the lattice of $\mathbb{T}^3$ is presented.

Periodic Navier Stokes Equations: PNS system within & external to embedded spheres of the 3-Torus: The mapping function at a 90 degree corner

Abstract

In this paper a solution for the third component of velocity for the Periodic Navier Stokes equations at the corners outside the sphere embedded in each cell of the lattice of is presented.

Paper Structure

This paper contains 4 sections, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Incompressible flow on $\mathbb{T}^3$
  3. The Mapping function developed at a corner
  4. Conclusions

Figures (2)

  • Figure 1: The tilting of vorticity. Suppose that the vorticity, ${\omega}$ is initially directed horizontally, as in the lower figure, so that $\boldsymbol{\omega}_3={\boldsymbol{\omega}}^z$ , its vertical component, is zero. The material lines, and therefore also the vortex lines, are tilted by the positive vertical velocity $w=v_3$ thus creating a non-zero vertically oriented vorticity.(Figure from "Essentials of Atmospheric and Oceanic Dynamics"-Geoffrey K.Vallis)
  • Figure 2: Two parametric representations of curves $x=x(t_1)$ and $y=y(t_1)$ forming a right angle at the vertex of a cell in $\mathbb{T}^3$. Two dimensional mapping function.