Analysis of exact solutions of Navier – Stokes equations assuming an incompressible fluid

Abstract

Navier – Stokes results which assume incompressible fluids and divergence-free flows, i.e. ∇ ⃗∙u ⃗(x ⃗,t)=0, obtained analytically for three dimensions in the literature, are expanded. For this purpose, numerical analyses were performed. In related studies, especially time-dependant viscosities μ(t) were investigated. Therefore, Poincaré maps were obtained for μ(t)<1/t. The Navier-Stokes equations are fundamental partial differential equations that mathematically describe the motion of fluids. These equations establish the relationship between velocity, pressure and density of a fluid. The first equation, known as the momentum equation, determines how the velocity of a fluid changes over time. The second equation, called the mass equation, expresses how the density of the fluid changes. The third equation, the energy equation, calculates the energy changes within the fluid. The fourth equation relates thermodynamic properties such as temperature and pressure. These coupled equations are crucial for understanding fluid dynamics and engineering applications. In this context, different assumptions for the viscosity are considered. Three different cases were examined. These assumptions include a time-dependent upper limit, time-dependent lower limit and constant lower limit of viscosity. In addition, evidence is presented that certain streams of literature-related solutions of the Beltrami equation which are vector fields parallel to their own curl do not have chaotic streaklines. For constant viscosities, this proves that the dynamics of Trkalian (exponentially decreasing with the time) flows are not chaotic.

Keywords

• Non compresable flow
• Navier–Stokes
• Trkalyan flow
• Beltrami equation

Sıkıştırılamaz bir sıvıyı varsayan Navier – Stokes denklemlerinin kesin çözümlerinin analizi

Öz

Literatürde üç boyutta analitik olarak elde edilmiş olan sıkıştırılamaz ve ıraksamasız, yani ∇ ⃗∙u ⃗(x ⃗,t)=0 şeklindeki akışları varsayan Navier – Stokes sonuçları günden güne detaylandırılmakta ve genişletilmektedir. Bu amaç için nümerik incelemeler gerçekleştirilmiştir. Literatürde ilgili çalışmalarda, özellikle zamana bağlı viskoziteler μ(t) araştırılmaktadır. Bu sebepten kaynaklı olarak nümerik analizlerde μ(t)<1/t için Poincaré haritaları elde edilmiştir. Genel olarak, Navier-Stokes denklemleri, akışkanların hareketini matematiksel olarak tanımlayan temel kısmi diferansiyel denklemlerdir. Bu denklemler, bir akışkanın hızı, basıncı ve yoğunluğu arasındaki ilişkiyi açıklar. İlk denklem, momentum denklemi olarak bilinir ve akışkanın hızının zamanla nasıl değiştiğini belirler. İkinci denklem, kütle denklemi olarak adlandırılır ve akışkanın yoğunluğunun nasıl değiştiğini ifade eder. Üçüncü denklem, enerji denklemi olarak bilinir ve akışkanın enerji değişimini hesaplar. Son denklem ise, sıcaklık ve basınç gibi termodinamik özellikleri ilişkilendirir. Bu kuple denklemler, akışkan dinamiğini ve mühendislik uygulamalarını anlamak için önem arz etmektedir. Bu kapsamda, viskozite için farklı varsayımlar ele alınmaktadır. Üç farklı durum incelenmiştir. Bu varsayımlar; viskozitenin zamana bağlı üst limiti, zamana bağlı alt limiti ve sabit alt limiti kapsamaktadır. Ek olarak, literatür ile ilişkilendirilmiş olan Beltrami denkleminin çözümleri için, ki bu vektör alanları kendi rotasyonlarına paraleldir, belirli akışlarının düzensiz çıkış çizgilerine sahip olmadığına dair kanıtlar sunulmaktadır. Viskozitenin sabit olduğunda bu durum Trkalyan (yani zamana bağlı katlanarak zayıflayan) akışlarının dinamiği düzensiz olmadığını kanıtlamaktadır.

Anahtar Kelimeler

• Sıkıştırılamaz akış
• Navier–Stokes
• Trkalyan akışları
• Beltrami denklemi

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