A new proposal of power series method to solve the Navier-Stokes equations: application contexts and perspectives
Authors: 
Paola Lecca, Angela ReAuthors Info & Claims
Pages 21 - 35
Abstract
In this work we investigate the possibility of expressing the solution of the Navier-Stokes equations as a power series. Although there have been many studies on this subject, it is still much debated. The existence of such a solution and its uniqueness are debated, especially in the general case of a non-stationary fluid in two or three dimensions. The greater the complexity of the fluid dynamical phenomenon under consideration, the more controversial and therefore uncertain are the deductions about the existence and uniqueness of the solution as a power series. Here, we ask some crucial questions on the matter and try to give an answer from the point of view of applied mathematics and numerical analysis. In particular, by way of example, we construct the Navier-Stokes equations for a compressible, multicomponent and non-stationary fluid from elementary principles of conservation of mass, momentum and energy, and we introduce a bump function model for the presence of different components and/or different phases in the fluid. The bump function is a function of space, time and physical characteristics of the components (and/or phases) such as density. We present a method to calculate it and discuss about its uniqueness. We show that under certain conditions on the bump function, and the forces acting on and in the fluid, the power series solution is unique. We finally discuss the advantages and the limitations of a solution in power series, concluding that, although plagued by limitations, it is a viable way forward even in highly complex case studies.
References
[1]
1992. 14. On the possibility of rarefaction shock waves. In Selected Works of Yakov Borisovich Zeldovich, Volume I, Rashid Alievich Sunyaev (Ed.). Princeton University Press, Princeton, 152–154.
[2]
Ladyzhenskaia O. A.2014. The mathematical theory of viscous incompressible flow. Martino Publishing, Mansfield Centre, CT.
[3]
William A. Adkins and Mark G. Davidson. 2012. Ordinary Differential Equations. Springer New York. https://doi.org/10.1007/978-1-4614-3618-8
[4]
Manuchehr Aminian, Francesca Bernardi, Roberto Camassa, Daniel M. Harris, and Richard M. McLaughlin. 2016. How boundaries shape chemical delivery in microfluidics. Science 354, 6317 (Dec. 2016), 1252–1256. https://doi.org/10.1126/science.aag0532
[5]
Evgenii S. Baranovskii. 2020. Strong Solutions of the Incompressible Navier–Stokes–Voigt Model. Mathematics 8, 2 (Feb. 2020), 181. https://doi.org/10.3390/math8020181
[6]
Franck Boyer. 2013. Mathematical tools for the study of the incompressible Navier-Stokes equations and related models. Springer, New York London.
[7]
J.U Brackbill, D.B Kothe, and C Zemach. 1992. A continuum method for modeling surface tension. J. Comput. Phys. 100, 2 (June 1992), 335–354. https://doi.org/10.1016/0021-9991(92)90240-y
[8]
Didier Bresch and Pierre-Emmanuel Jabin. 2018. Global existence of weak solutions for compressible Navier–Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor. Annals of Mathematics 188, 2 (Sept. 2018), 577–684. https://doi.org/10.4007/annals.2018.188.2.4
[9]
Xiaojing Cai and Quansen Jiu. 2008. Weak and strong solutions for the incompressible Navier–Stokes equations with damping. J. Math. Anal. Appl. 343, 2 (July 2008), 799–809. https://doi.org/10.1016/j.jmaa.2008.01.041
[10]
Nitish B Chandrasekaran, Bertrand Mercier, and Piero Colonna. 2021. Formation of rarefaction shockwaves in non-ideal gases with temperature gradients. In Proceedings of the 3rd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion and Power. Springer International Publishing, Cham, 20–25.
[11]
Hi Jun Choe and Hyunseok Kim. 2003. Strong Solutions of the Navier–Stokes Equations for Nonhomogeneous Incompressible Fluids. Communications in Partial Differential Equations 28, 5-6 (Jan. 2003), 1183–1201. https://doi.org/10.1081/pde-120021191
[12]
Shirshendu Chowdhury and Sylvain Ervedoza. 2019. Open loop stabilization of incompressible Navier–Stokes equations in a 2d channel using power series expansion. Journal de Mathématiques Pures et Appliquées 130 (Oct. 2019), 301–346. https://doi.org/10.1016/j.matpur.2019.01.006
[13]
Shirshendu Chowdhury, Sylvain Ervedoza, and Jean-Pierre Raymond. 2014. Power series expansion method and application to incompressible Navier-Stokes equations in two dimensions. Researchgate document (10 2014).
[14]
Valdir Monteiro dos Santos Godoi. 2016. MS Windows NT Kernel Description. https://www.yumpu.com/en/document/view/56011879/solutions-for-euler-and-navier-stokes-equations-in-powers-of-time. Accessed: 2021-09-30.
[15]
R Paul Drake. 2006. Shocks and Rarefactions. In Shock Wave and High Pressure Phenomena. Springer Berlin Heidelberg, 107–167.
[16]
Feireisl Eduard, Antonìn Novotný, and Yongzhong Sun. 2011. Suitable Weak Solutions to the Navier-Stokes Equations of Compressible Viscous Fluids. Indiana University Mathematics Journal 60, 2 (2011), 611–31. http://www.jstor.org/stable/24903433
[17]
P. A. Egelstaff and S. S. Wang. 1972. Density Dependent Potentials in Simple Liquids. Canadian Journal of Physics 50, 20 (Oct. 1972), 2461–2463. https://doi.org/10.1139/p72-325
[18]
Renato Fiorenza. 2016. Hölder and locally Hölder continuous functions. The linear spaces . In Frontiers in Mathematics. Springer International Publishing, 1–35. https://doi.org/10.1007/978-3-319-47940-8_1
[19]
Flow Science, Inc.[n.d.]. Flow-3D, Solving the World’s Toughest CFD Problems. https://www.flow3d.com/resources/cfd-101/numerical-issues/artificial-and-numerical-viscosities/. Accessed: 2022-04-15.
[20]
Jiří Fürst, M. Huněk, and K. Kozel. 1996. Numerical solution of the Euler and Navier-Stokes equations. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 76 (01 1996).
[21]
Vincent Giovangigli. 2011. Multicomponent flow modeling. Science China Mathematics 55, 2 (Dec. 2011), 285–308. https://doi.org/10.1007/s11425-011-4346-y
[22]
Vincent Guinot. 2005. An Approximate Two-Dimensional Riemann Solver for Hyperbolic Systems of Conservation Laws. J. Comput. Phys. 205, 1 (may 2005), 292–314. https://doi.org/10.1016/j.jcp.2004.11.006
[23]
H. Hasimoto. 1959. On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. Journal of Fluid Mechanics 5, 02 (Feb. 1959), 317. https://doi.org/10.1017/s0022112059000222
[24]
Lijun Hu, Li Yuan, and Kunlei Zhao. 2020. Development of accurate and robust genuinely two-dimensional HLL-type Riemann solver for compressible flows. Computers & Fluids 213(2020), 104719. https://doi.org/10.1016/j.compfluid.2020.104719
[25]
Leo P. Kadanoff. 2002. A New Kind of Science A New Kind of Science Stephen Wolfram Wolfram Media, Champaign, Ill., 2002. $44.95 (1197 pp.). ISBN 1-57955-008-8. Physics Today 55, 7 (July 2002), 55–56. https://doi.org/10.1063/1.1506752
[26]
Jae-Myoung Kim. 2022. 3D Navier-Stokes equations of power law type with damping. Archiv der Mathematik 118, 3 (Jan. 2022), 323–335. https://doi.org/10.1007/s00013-021-01684-z
[27]
Randall J LeVeque. 2002. Preface. In Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, xvii–xx.
[28]
Dong Li and Yakov Sinai. 2008. Blow ups of complex solutions of the 3D Navier–Stokes system and renormalization group method. Journal of the European Mathematical Society (2008), 267–313. https://doi.org/10.4171/jems/111
[29]
Jinkai Li. 2017. Local existence and uniqueness of strong solutions to the Navier–Stokes equations with nonnegative density. Journal of Differential Equations 263, 10 (Nov. 2017), 6512–6536. https://doi.org/10.1016/j.jde.2017.07.021
[30]
Steven J. Lind, Benedict D. Rogers, and Peter K. Stansby. 2020. Review of smoothed particle hydrodynamics: towards converged Lagrangian flow modelling. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, 2241 (2020), 20190801. https://doi.org/10.1098/rspa.2019.0801 arXiv:https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0801
[31]
Mingshuo Liu, Xinyue Li, and Qiu-Lan Zhao. 2019. Exact solutions to Euler equation and Navier–Stokes equation. Zeitschrift für angewandte Mathematik und Physik 70 (02 2019), 43. https://doi.org/10.1007/s00033-019-1088-0
[32]
Boqiang Lü, Rong Zhang, and Xin Zhong. 2019. Global existence of weak solutions to the compressible quantum Navier-Stokes equations with degenerate viscosity. J. Math. Phys. 60, 12 (Dec. 2019), 121502. https://doi.org/10.1063/1.5127797
[33]
David Maltese, Martin Michálek, Piotr B. Mucha, Antonin Novotný, Milan Pokorný, and Ewelina Zatorska. 2016. Existence of weak solutions for compressible Navier–Stokes equations with entropy transport. Journal of Differential Equations 261, 8 (Oct. 2016), 4448–4485. https://doi.org/10.1016/j.jde.2016.06.029
[34]
Samy Merabia and Ignacio Pagonabarraga. 2007. Density dependent potentials: Structure and thermodynamics. The Journal of Chemical Physics 127, 5 (Aug. 2007), 054903. https://doi.org/10.1063/1.2751496
[35]
Piotr B. Mucha and Milan Pokorný. 2010. WEAK SOLUTIONS TO EQUATIONS OF STEADY COMPRESSIBLE HEAT CONDUCTING FLUIDS. Mathematical Models and Methods in Applied Sciences 20, 05 (May 2010), 785–813. https://doi.org/10.1142/s0218202510004441
[36]
R Nagle. 2012. Fundamentals of differential equations. Pearson Education, Boston.
[37]
A R Niknam, M Hashemzadeh, B Shokri, and M R Rouhani. 2009. Rarefaction shock waves and Hugoniot curve in the presence of free and trapped particles. Phys. Plasmas 16, 12 (Dec. 2009), 122109.
[38]
Antonin Novotny and Hana Petzeltová. 2017. Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stability, and Longtime Behavior. In Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer International Publishing, 1–165. https://doi.org/10.1007/978-3-319-10151-4_76-1
[39]
A. E. Perry and M. S. Chong. 1986. A series-expansion study of the Navier–Stokes equations with applications to three-dimensional separation patterns. Journal of Fluid Mechanics 173 (1986), 207–223. https://doi.org/10.1017/S0022112086001143
[40]
Katherine R. Pielemeier and Joseph M. Powers. [n.d.]. Anomalous Waves in Non-Ideal Detonation Dynamics. https://doi.org/10.2514/6.2022-0396 arXiv:https://arc.aiaa.org/doi/pdf/10.2514/6.2022-0396
[41]
G. Prodi. 1959. Un teorema di unicità per le equazioni di Navier-Stokes. Annali di Matematica 48 (1959), 173–182. https://doi.org/10.1007/BF02410664
[42]
Osborne Reynolds. 1900. Papers on Mechanical and Physical Subjects. Nature 62, 1602 (July 1900), 243–244. https://doi.org/10.1038/062243a0
[43]
Francis Ribaud. 2002. A remark on the uniqueness problem for the weak solutions of Navier-Stokes equations. Annales de la Faculté des sciences de Toulouse : Mathématiques Ser. 6, 11, 2 (2002), 225–238. http://www.numdam.org/item/AFST_2002_6_11_2_225_0/
[44]
James C. Robinson. 2020. The Navier–Stokes regularity problem. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, 2174 (June 2020), 20190526. https://doi.org/10.1098/rsta.2019.0526
[45]
Fabien Salmon. 2018. Analytical solution of the incompressible Navier-Stokes equations. https://www.yumpu.com/en/document/view/56011879/solutions-for-euler-and-navier-stokes-equations-in-powers-of-time. hal01802032v2f.
[46]
F. Salmon. 2019. Analytical solution of the incompressible Navier-Stokes equations. https://doi.org/10.48550/ARXIV.1901.10532
[47]
J Scheffel. 2001. On analytical solution of the Navier-Stokes equations.
[48]
Kleiton A. Schneider, José M. Gallardo, Dinshaw S. Balsara, Boniface Nkonga, and Carlos Parés. 2021. Multidimensional approximate Riemann solvers for hyperbolic nonconservative systems. Applications to shallow water systems. J. Comput. Phys. 444(2021), 110547. https://doi.org/10.1016/j.jcp.2021.110547
[49]
Markus Scholle, Florian Marner, and Philip H. Gaskell. 2020. Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances. Water 12, 5 (April 2020), 1241. https://doi.org/10.3390/w12051241
[50]
Paul Seeburger. 2022. CalcPlot3D. https://c3d.libretexts.org/. Accessed: 2022-02-29.
[51]
James Serrin. 1963. The initial value problem for the Navier-Stokes equations. Nonlinear Probl., Proc. Sympos. Madison 1962, 69-98 (1963).
[52]
Hamed Shahmohamadi and Mahdi Mohammadpour. 2014. A Series Solution for Three-Dimensional Navier-Stokes Equations of Flow near an Infinite Rotating Disk. World Journal of Mechanics 04, 05 (2014), 117–127. https://doi.org/10.4236/wjm.2014.45014
[53]
Cholmin Sin. 2021. The Existence of Strong Solution for Generalized Navier-Stokes Equations with p(x)-Power Law under Dirichlet Boundary Conditions. Advances in Mathematical Physics 2021 (Dec. 2021), 1–11. https://doi.org/10.1155/2021/6755411
[54]
Yakov Sinai. 2005. Power Series for Solutions of the 3 $${\mathcal D}$$ -Navier-Stokes System on R 3. Journal of Statistical Physics 121, 5-6 (Dec. 2005), 779–803. https://doi.org/10.1007/s10955-005-8670-x
[55]
B. Smith, T. Hauschild, and J.M. Prausnitz. 1992. Effect of a density-dependent potential on the phase behaviour of fluids. Molecular Physics 77, 6 (Dec. 1992), 1021–1031. https://doi.org/10.1080/00268979200102971
[56]
H. Sohr and Wolf von Wahl. 1984/85. On the Singular Set and the Uniqueness of Weak Solutions of the Navier-Stokes Equations.Manuscripta mathematica 49 (1984/85), 27–60. http://eudml.org/doc/155034
[57]
Himali Somaweera, Shehan O. Haputhanthri, Akif Ibraguimov, and Dimitri Pappas. 2015. On-chip gradient generation in 256 microfluidic cell cultures: simulation and experimental validation. The Analyst 140, 15 (2015), 5029–5038. https://doi.org/10.1039/c5an00481k
[58]
Roger Temam. 1984. Navier-Stokes equations : theory and numerical analysis. Elsevier Science, The Netherlands.
[59]
Sudarshan Tiwari and Jörg Kuhnert. 2007. Modeling of two-phase flows with surface tension by finite pointset method (FPM). J. Comput. Appl. Math. 203, 2 (June 2007), 376–386. https://doi.org/10.1016/j.cam.2006.04.048
[60]
E.F. Toro. 2016. Chapter 2 - The Riemann Problem: Solvers and Numerical Fluxes. In Handbook of Numerical Methods for Hyperbolic Problems, Rémi Abgralland Chi-Wang Shu (Eds.). Handbook of Numerical Analysis, Vol. 17. Elsevier, 19–54. https://doi.org/10.1016/bs.hna.2016.09.015
[61]
Eleuterio F Toro. 2009. Riemann solvers and numerical methods for fluid dynamics (3 ed.). Springer, Berlin, Germany.
[62]
Masao Yamazaki. 2018. Existence and Uniqueness of Weak Solutions to the Two-Dimensional Stationary Navier–Stokes Exterior Problem. Journal of Mathematical Fluid Mechanics 20, 4 (Sept. 2018), 2019–2051. https://doi.org/10.1007/s00021-018-0397-y
[63]
D. Zeidan. 2011. The Riemann problem for a hyperbolic model of two-phase flow in conservative form. International Journal of Computational Fluid Dynamics 25, 6 (2011), 299–318. https://doi.org/10.1080/10618562.2011.590800 arXiv:https://doi.org/10.1080/10618562.2011.590800
[64]
Dia Zeidan, Chi Kin Chau, and Tzon-Tzer Lu. 2021. On the characteristic Adomian decomposition method for the Riemann problem. Mathematical Methods in the Applied Sciences 44, 10 (2021), 8097–8112. https://doi.org/10.1002/mma.5798 arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/mma.5798
[65]
D. Zeidan, E. Romenski, A. Slaouti, and E. F. Toro. 2007. Numerical study of wave propagation in compressible two-phase flow. International Journal for Numerical Methods in Fluids 54, 4 (2007), 393–417. https://doi.org/10.1002/fld.1404 arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/fld.1404
[66]
A.V. Zhirkin. 2016. Existence and properties of the Navier–Stokes equations. Cogent Mathematics 3, 1 (June 2016), 1190308. https://doi.org/10.1080/23311835.2016.1190308
[67]
Shiqi Zhou. 2008. Phase behavior of density-dependent pair potentials. The Journal of Chemical Physics 128, 10 (March 2008), 104511. https://doi.org/10.1063/1.2888977
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- A new proposal of power series method to solve the Navier-Stokes equations: application contexts and perspectives
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June 2022
137 pages
ISBN:9781450396233
DOI:10.1145/3545839
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ICoMS 2022: 2022 5th International Conference on Mathematics and Statistics
June 17 - 19, 2022
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