Sıkıştırılamaz bir sıvıyı varsayan Navier – Stokes denklemlerinin kesin çözümlerinin analizi

Yıl 2024, Cilt: 13 Sayı: 2, 467 - 472, 15.04.2024

Nesij Ünal Yahya Öz Tugrul Oktay

https://doi.org/10.28948/ngumuh.1370615

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

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• D. J. Acheson, Elementary Fluid Dynamics. Oxford University Press, Oxford, 1990.
• B. Bilalov, T. Gasymov and A. Guliyeva, On the solvability of the Riemann boundary value problem in Morrey--Hardy classes. Turkish Journal of Mathematics, 40, 5, 14, 1085 – 1101, 2016. https://doi.org/10.3906/mat-1507-10
• C. X. Li and S. Liang Wu, Eigenvalue distribution of relaxed mixed constraint preconditioner for saddle point problems. Hacettepe Journal of Mathematics and Statistics, 45, 6, 1705 – 1718, 2016. https://doi.org/10.15672/HJMS.20164515686
• A. Pınarbaşı and M. İmal, Nonisothermal channel flow of a non-newtonian fluid with viscous heating. International Communications in Heat and Mass Transfer, 29, 8, 1099 – 1107, 2002. https://doi.org/10.1016/s0735-1933(02)00438-4
• F. Reetz, T. Kreilos and T. M. Schneider, Exact invariant solution reveals the origin of self-organized oblique turbulent-laminar stripes. Nature Communications, 10, 2277, 2019. https://doi.org/10.1038/s41467-019-10208-x
• M. Scholle, P. H. Gaskell and F. Marner, Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications. Journal of Mathematical Physics, 59, 4, 2018. https://doi.org/10.1063/1.5031119
• N. Uygun, Effect of hall current on the MHD fluid flow and heat transfer due to a rotating disk with olauniform radial electric field. Hacettepe Journal of Mathematics and Statistics, 44 (6), 1445 – 1462, 2015. https://doi.org/10.15672/HJMS.2015449675
• N. Ünal, Y. Öz and T. Oktay, Conceptual design modeling by the novel aircraft conceptual design and analysis system (ACDAS). Aircraft Engineering and Aerospace Technology, 95, 5, 799 – 813, 2023. https://doi.org/10.1108/AEAT-02-2022-0056
• A. Urichuk, Y. Oez, A. Klümper and J. Sirker, The spin Drude weight of the XXZ chain and generalized hydrodynamics. SciPost Physics, 6, 1, 005, 2019. https://doi.org/10.21468/SciPostPhys.6.1.005
• J. D. Gibbon, The three-dimensional Euler equations: Where do we stand?. Physica D: Nonlinear Phenomena, 237, 14–17, 1894 – 1904, 2008. https://doi.org/10.1016/j.physd.2007.10.014
• H. Gümral, Lagrangian description, symplectization, and Eulerian dynamics of incompressible fluids. Turkish Journal of Mathematics, 40, 5, 925 – 940, 2016. https://doi.org/10.3906/mat-1410-38
• V. I. Arnold, S. F. Shandarin and Y. B. Zeldovich, The large scale structure of the universe I. General properties. Geophysical & Astrophysical Fluid Dynamics, 20, 1 – 2, 111 – 130, 1982. https://doi.org/10.1080/03091928208209001
• A. M. Polyakov, Turbulence without pressure. Physical Revıew E, 52, 6, 6183 – 6188, 1995. https://doi.org/10.1103/PhysRevE.52.6183
• F. Gaitan, Finding flows of a Navier–Stokes fluid through quantum computing. NPJ Quantum Information, 6, 61, 2020. https://doi.org/10.1038/s41534-020-00291-0
• J. Ou and J. Chen, Hypersonic Aerodynamics of Blunt Plates in Near-Continuum Regime by Improved Navier–Stokes Model. AIAA Journal, 58, 9, 4037 – 4046, 2020. https://doi.org/10.2514/1.J059333
• Liu, M., Li, X. and Zhao, Q. Exact solutions to Euler equation and Navier–Stokes equation. Z. Angew. Math. Phys, 70, 43, 1 – 13, 2019. https://doi.org/10.1007/s00033-019-1088-0
• T. Pedergnana and et al., Explicit unsteady Navier–Stokes solutions and their analysis via local vortex criteria. Physics of Fluids, 32, 4, April 2020. https://doi.org/10.1063/5.0003245
• Prosviryakov, E.Y. new class of exact solutions of Navier–Stokes equations with exponential dependence of velocity on two spatial coordinates. Theor Found Chem Eng, 53, 107 – 114, 2019. https://doi.org/10.1134/S0040579518060088
• M.J. Zhang and W.D. Su, Exact solutions of the Navier-Stokes equations with spiral or elliptical oscillation between two infinite planes. Physics of Fluids, 25, 7, 2013. https://doi.org/10.1063/1.4813629
• Y. Öz, Rigorous investigation of the Navier–Stokes momentum equations and correlation tensors. AIP Advances, 11, 5, 055009, 2021. https://doi.org/10.1063/5.0050330
• E. Varley and B. R. Seymour, Applications of exact solutions to the Navier–Stokes equations: free shear layers. Journal of Fluid Mechanics, 274, 267 – 291, 2006. https://doi.org/10.1017/S0022112094002120
• Y. Öz, Novel exact solutions to Navier-Stokes momentum equations describing an incompressible fluid. Turkish Journal of Mathematics, 46, 8, 3192 – 3200, 2022. https://doi.org/10.55730/1300-0098.3327
• V. Trkal, A remark on the hydrodynamics of the viscous fluids. Journal for the Cultivation of Mathematics and Physics, 48, 3, 302 – 311, 1919. https://doi.org/10.21136/CPMF.1919.109099
• M. Henon, On the numerical computation of Poincaré maps. Physica D: Nonlinear Phenomena, 5, 2 – 3, 412 – 414, 1982. https://doi.org/10.1016/0167-2789(82)90034-3
• W. Tucker, Computing accurate Poincaré maps. Physica D: Nonlinear Phenomena, 171, 3, 127 – 137, 2002. https://doi.org/10.1016/S0167-2789(02)00603-6
• P. D. Huck, N. Machicoane and R. Volk, Lagrangian acceleration timescales in anisotropic turbulence. Physical Review Fluids, 4, 6, 064606, 2019. https://doi.org/10.1103/PhysRevFluids.4.064606
• K. Whitehead and R. Gray, Generation and development of a viscous vortex ring. 10th Aerospace Sciences Meeting, San Diego, USA, 2012. https://doi.org/10.2514/6.1972-151