A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation

R Meulens

doi:10.1063/5.0074083

Abstract

Repetitive curling of the incompressible viscid Navier–Stokes differential equation leads to a higher-order diffusion equation. Substituting this equation into the Navier–Stokes differential equation transposes the latter into the Korteweg–De Vries–Burgers-equation with the Weierstrass p-function as the soliton solution. However, a higher-order derivative of the studied variable produces the so-called N-soliton solution, which is comparable with the N-soliton solution of the Kadomtsev–Petviashvili equation. Experiments have made it clear that the system behaves like a coupled (an)harmonic oscillator on a discrete collapsed-state level. The streamlines obtained are derivatives of the error function as a function of the obtained Lax functional of the particle filaments dynamics induced by the (hypothetical) Calogero–Moser many-body system with elliptical potential and are the so-called Hermite functions. Hermite tried to introduce doubly periodic Hermite functions (the so-called Hermite problem) using coefficients related to the Weierstrass p-function. A solution-sensitive analysis of the incompressible viscid Navier–Stokes equation is performed using the Lamb vector. Cases with a meaningful potential-energy contribution require a particle interaction model with an N-soliton solution using a hierarchy-like solution of the Kadomtsev–Petviashvili equation. A three-soliton solution is emulated for the cylinder-wake problem. Our analytical results are put in perspective by comparison with two well-studied benchmark cases of fluid dynamics: the cylinder-wake problem and the driven-lid problem. The time-average velocity distribution (limit of streamline patterns) is consistent with published results and is enclosed in an asymmetrical lemniscate.

Highlights

ROADMAP TO SOLUTION OF INCOMPRESSIBLE VISCID NAVIER–STOKES DIFFERENTIAL EQUATIONTaking two curls of the vortex transport equation yields a diffusion equation for higher derivatives of vorticity vectors.The Navier–Stokes differential equation (d.e.) transposes to a Korteweg–De Vries–Burgers d.e
Repetitive curling of the incompressible viscid Navier–Stokes differential equation leads to a higher-order diffusion equation
A higher-order derivative of the studied variable produces the so-called N-soliton solution, which is comparable with the N-soliton solution of the Kadomtsev–Petviashvili equation

Summary

INTRODUCTION

ROADMAP TO SOLUTION OF INCOMPRESSIBLE VISCID NAVIER–STOKES DIFFERENTIAL EQUATION. Taking two curls of the vortex transport equation yields a diffusion equation for higher derivatives of vorticity vectors. The Euler–Cornu spirals obtained as solutions of the underlying Schrödinger equation explain the von Kármán vortex street enigma and are a diffraction pattern caused by the object in the cylinder-wake problem, which may be seen as a fixed external potential causing measurable and predictable singularities in the particle flow (see Fig. 13) Cornu originally used this concept to give a geometric explanation for the Fresnel diffraction for the so-called wave-phenomena. The Calogero–Moser manybody Hamiltonian system with elliptic particle interactions and the Burgers–Hopf equation are among the few known parts that glue the soliton KdV solutions ∣ψ∣2 and the Schrödinger map equation solutions ψ using the Lax functional ψ obtained and Madelung’s coupled hydrodynamical system with the amplitude of the wave proportional to its arclength. EXACT THEORETICAL SOLUTIONS SET IN PERSPECTIVE WITH NUMERICAL RESULTS OF KNOWN BENCHMARK CASES

Cylinder-wake problem

D D2 z-component x-component velocity y-component velocity

FINAL REMARKS AND FUTURE WORK