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Exact solutions of fluid equations on a sphere

Sun-Chul Kim, Habin Yim

TL;DR

This work investigates exact stationary solutions of the Navier–Stokes and Euler equations on the surface of a sphere by employing a stream-function formulation with $ψ(θ,φ)$ and vorticity $ω$. The main finding is that, under the vanishing convection (basic) condition, the only nontrivial solution on the sphere is the basic two-antipodal-vortex configuration, and there are no additional nontrivial solutions arising from a functional relation between $ψ$ and $ω$ (i.e., no $ψ=f(ω)$ or $ω=g(ψ)$ solutions). The analysis uses a reduction to axisymmetric dependence and a further transformation to polar-like variables $χ = \log\tan(θ/2)$ to show that any such relation collapses to the basic case, revealing a strong rigidity imposed by the sphere’s positive curvature and compactness. The results underscore the scarcity of exact, common NS/Euler solutions on curved, compact manifolds and point to geometric constraints as a key factor in the solvability landscape.

Abstract

Exact solutions of both the Navier-Stokes and Euler equations are found on the surface of a sphere. Under the assumption of a vanishing convection term, the flow of two oppositely rotating point vortices at the poles turns out to be the unique common solution.

Exact solutions of fluid equations on a sphere

TL;DR

This work investigates exact stationary solutions of the Navier–Stokes and Euler equations on the surface of a sphere by employing a stream-function formulation with and vorticity . The main finding is that, under the vanishing convection (basic) condition, the only nontrivial solution on the sphere is the basic two-antipodal-vortex configuration, and there are no additional nontrivial solutions arising from a functional relation between and (i.e., no or solutions). The analysis uses a reduction to axisymmetric dependence and a further transformation to polar-like variables to show that any such relation collapses to the basic case, revealing a strong rigidity imposed by the sphere’s positive curvature and compactness. The results underscore the scarcity of exact, common NS/Euler solutions on curved, compact manifolds and point to geometric constraints as a key factor in the solvability landscape.

Abstract

Exact solutions of both the Navier-Stokes and Euler equations are found on the surface of a sphere. Under the assumption of a vanishing convection term, the flow of two oppositely rotating point vortices at the poles turns out to be the unique common solution.
Paper Structure (4 sections, 2 theorems, 30 equations)

This paper contains 4 sections, 2 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Solutions depending only on $\theta$
  3. Solutions of $\psi= f(\omega)$ and $\omega=g(\psi)$
  4. Concluding remark

Key Result

Theorem 1

There is no nontrivial solution except the basic solution satisfying the functional relation $\psi = F(\omega)$ with a smooth single-valued function $F$ on a sphere.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2