Others

# Compressible Navier-Stokes Equations: Cylindrical Passages and General Dynamics of Surfaces

Subjects: Analysis View times: 210
Created by: Terry Moschandreou
(ADDENDUM)- SEE PAGE 8,9 below for proof that the viscosity terms for compressible flow vanish for a quadratic form for vorticity.

A new approach to solve the compressible Navier-Stokes equations in cylindrical co-ordinates using Geometric Algebra is proposed. This work was recently initiated by corresponding author of this current work,and in contrast  due to a now complete geometrical analysis, particularly, two dimensionless parameters are now introduced whose correct definition depends on the scaling invariance of the N-S equations and the one parameter $\delta$ defines an equation in density which can be solved for in the tube, and a geometric Variational Calculus approach showing that the total energy of an existing wave vortex in the tube is made up of kinetic energy by vortex movement and internal energy produced by the friction against the wall of the tube. Density of a flowing gas or vapour varies along the length of the tube due to frictional losses along the tube implying that there is a pressure loss and a corresponding density decrease. After reducing the N-S equations to a single PDE, it is here proven that a Hunter-Saxton wave vortex exists along the wall of the tube due to a vorticity argument. The reduced problem shows finite-time blowup as the two parameters $\delta$ and $\alpha$ approach zero. Finally we propose a CMS (Calculus of Moving Surfaces)âinvariant variational calculus to analyze general dynamic surfaces of Riemannian 2-Manifolds in $\mathbb{R}^3$. Establishing fluid structures in general compressible flows and analyzing membranes in such flows for example flows with dynamic membranes immersed in fluid (vapour or gas) with vorticity as, for example, in the lungs there can prove to be a strong connection between fluid and solid mechanics.

https://doi.org/10.3390/math7111060[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][18][20][18][21][18][22][18][23][18][24][18][25][18][26][18][27][18][28][18][29][18][30][18][31][18][32][18][33][18][34][35][36][18][37][18][18][38][18][39][40][41][18][18][42][18][18][43][18][44][18][45][18][46][18][18][47][18][48][18][18][49][50][51][18][18][18][18][18][18][18][18][18][18][18][52][18][53][54][55]

## References

1. \begin{thebibliography}{999}
2. L. M. Pereira; J. S. Perez-Guerrero; R. M. Cotta; Integral transformation of the Navier-Stokes equations in cylindrical geometry. Computational Mechanics 1998, 21, 60-70, 10.1007/s004660050283.
3. Ruixian Cai; Some Explicit Analytical Solutions of Unsteady Compressible Flow. Journal of Fluids Engineering 1998, 120, 760-764, 10.1115/1.2820735.
4. R. W. MacCormack; A Numerical Method for Solving the Equations of Compressible Viscous Flow. AIAA Journal 1982, 20, 1275-1281, 10.2514/3.51188.
5. Arthur Piquet; Boubakr Zebiri; Abdellah Hadjadj; Mostafa Safdari Shadloo; A parallel high-order compressible flows solver with domain decomposition method in the generalized curvilinear coordinates system. International Journal of Numerical Methods for Heat & Fluid Flow 2019, , , 10.1108/hff-01-2019-0048.
6. Shrinivas G. Apte; Brian H. Dennis; Pseudo Compressible Mixed Interpolation Finite Element Method for Solving Three Dimensional Navier-Stokes Equations. Volume 8: 22nd Reliability, Stress Analysis, and Failure Prevention Conference; 25th Conference on Mechanical Vibration and Noise 2013, , , 10.1115/detc2013-13484.
7. \bibitem{Feistauer} Feistauer, M.; Felcman, J.; Straskraba, I. \textit{Mathematical and Computational Methods for Compressible Flow}; {Clarendon Press-Oxford Science Publications: Oxford, England}, 2003.%Please add city, country of the publisher
8. \bibitem{Vadasz} Vadasz, P. {Rendering the Navier-Stokes equations for a compressible fluid into the Schr\"{o}dinger Equation for quantum mechanics}. \emph{Fluids} \textbf{2016}, \emph{1}, 18.
9. D. G. Ebin; Motion of a Slightly Compressible Fluid. Proceedings of the National Academy of Sciences 1975, 72, 539-542, 10.1073/pnas.72.2.539.
10. Y. L. Wang; P. A. Longwell; Laminar flow in the inlet section of parallel plates. AIChE Journal 1964, 10, 323-329, 10.1002/aic.690100310.
11. Yanping Ran; Jing Li; Pullback attractors for non-autonomous reaction–diffusion equation with infinite delays in C γ , L r ( Ω ) $C_{\gamma,L^{r}(\Omega)}$ or C γ , W 1 , r ( Ω ) $C_{\gamma,W^{1,r}(\Omega)}$. Boundary Value Problems 2018, 2018, 99, 10.1186/s13661-018-1017-8.
12. Kuppalapalle Vajravelu; Ronald Li; Mangalagama Dewasurendra; Joseph Benarroch; Nicholas Ossi; Ying Zhang; Michael Sammarco; K.V. Prasad; Effects of second-order slip and drag reduction in boundary layer flows. Applied Mathematics and Nonlinear Sciences 2018, 3, 291-302, 10.21042/amns.2018.1.00022.
13. V.A. Bazhenov; O.S. Pogorelova; T.G. Postnikova; Intermittent transition to chaos in vibroimpact system. Applied Mathematics and Nonlinear Sciences 2018, 3, 475-486, 10.2478/amns.2018.2.00037.
14. \bibitem{Gibbs} Gibbs, J.W.; Wilson, E.B. \textit{Vector Analysis}; Charles Scribner's Sons: New York, NY, USA, 1901.
15. Kundeti Muralidhar; Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics. Mathematics 2015, 3, 781-842, 10.3390/math3030781.
16. \bibitem{Muller} Muller, C. {\emph{Foundations of the Mathematical Theory of Electromagnetic Waves}}; Springer: New York, NY, USA, 1969; pp. 339--341.
17. S. Ulrych; Gravitoelectromagnetism in a complex Clifford algebra. 2006, , , .
18. \bibitem{Nelson} Nelson, D.R. \textit{Statistical Mechanics of Membranes and Surfaces}, 2nd ed.; World Scientific: London, England, {2004}.%Please add publisher: city, country
19. \bibitem{CMS} Grinfeld, P. \textit{Introduction to Tensor Analysis and the Calculus of Moving Surfaces}; Springer: New York, NY, USA,~2010.
20. M. F. McMullin; The molecular basis of disorders of the red cell membrane.. Journal of Clinical Pathology 1999, 52, 245-248, 10.1136/jcp.52.4.245.
21. A Iolascon; E Miraglia Del Giudice; C Camaschella; Molecular pathology of inherited erythrocyte membrane disorders: hereditary spherocytosis and elliptocytosis.. Haematologica 1992, 77, , .
22. G. Falk; Paul Fatt; Linear electrical properties of striated muscle fibres observed with intracellular electrodes. Proceedings of the Royal Society of London. Series B. Biological Sciences 1964, 160, 69-123, 10.1098/rspb.1964.0030.
23. Monika M. M. Henszen; Michael Weske; Stephan Schwarz; Cees W. M. Haest; Bernhard Deuticke; Electric field pulses induce reversible shape transformation of human erythrocytes.. Molecular Membrane Biology 1998, 14, 195-204, 10.3109/09687689709048182.
24. \bibitem{EMRadHumanExpm} Durney, C.H.; Johnson, C.C.; Barber, P.W.; Massoudi, H.; Iskander, M.F.; Allen, S.J.; Mitchell, J.C. {Descriptive summary: Radiofrequency radiation dosimetry handbook}-2nd Edition. \emph{Radio Sci.} \textbf{1979}, \emph{14}, 5--7.
25. Jemal Guven; Laplace pressure as a surface stress in fluid vesicles. Journal of Physics A: Mathematical and General 2006, 39, 3771-3785, 10.1088/0305-4470/39/14/019.
26. R Capovilla; Jemal Guven; Stresses in lipid membranes. Journal of Physics A: Mathematical and General 2002, 35, 6233-6247, 10.1088/0305-4470/35/30/302.
27. A.R. Kavalov; I.K. Kostov; A.G. Sedrakyan; Dynamics of Dirac and Weyl fermions on a two-dimensional surface. Physics Letters B 1986, 175, 331-334, 10.1016/0370-2693(86)90865-8.
28. Jaap. Bos; Robert. Tijssen; M. Emile. Van Kreveld; Determination of the dissociation temperature of organic micelles by microcapillary hydrodynamic chromatography. Analytical Chemistry 1989, 61, 1318-1321, 10.1021/ac00188a004.
29. David V. Svintradze; Moving Manifolds in Electromagnetic Fields. 2016, , , .
30. Terry E. Moschandreou; A Method of Solving Compressible Navier Stokes Equations in Cylindrical Coordinates Using Geometric Algebra. 2018, , , 10.20944/preprints201812.0081.v2.
31. \bibitem{Doran} Doran, C.; Lasenby, A. \textit{Geometric Algebra for Physicists}; {Cambridge University Press}, Cambridge, UK, 2003.%Please add city, country of the publisher
32. J. T. Beale; T. Kato; A. Majda; Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Communications in Mathematical Physics 1984, 94, 61-66, 10.1007/bf01212349.
33. Keith C. Afas; Extending the Calculus of Moving Surfaces to Higher Orders. 2018, , , .
34. Vladimir G. Ivancevic; Tijana T. Ivancevic; Undergraduate Lecture Notes in De Rham-Hodge Theory. 2008, , , .
35. \bibitem{Deriglazov} Deriglazov, A. \textit{Classical Mechanics}; Springer: Berlin, Germany, 2010.
36. Keith C. Afas; Normal Calculus on Moving Surfaces. 2018, , , 10.20944/preprints201806.0148.v1.
37. Ou-Yang Zhong-Can; W. Helfrich; Instability and Deformation of a Spherical Vesicle by Pressure.. Physical Review Letters 1988, 60, 1209-1209, 10.1103/physrevlett.60.1209.2.
38. Robert L. Bryant; A duality theorem for Willmore surfaces. Journal of Differential Geometry 1984, 20, 23-53, 10.4310/jdg/1214438991.
39. \bibitem{SqCurvBio} Toda, M.; Athukoralage, B. \textit{Geometry of Biological Membranes and Willmore Energy}; Simos, T., Psihoyios, G., Tsitouras, C., Eds.; American Institute of Physics Conference Series; {American Institute of Physics Conference: New York, NY, USA}, 2013; Volume 1558, pp. 883--886.
40. \bibitem{Gonzalez} Gonzalez, O.; Stuart, A.M. \textit{Kinematics}; Cambridge Texts in Applied Mathematics; {Cambridge University Press, Cambridge, UK,} 2008; pp. 112--166.%Please add city, country of the publisher
41. Pavel Grinfeld; A Better Calculus of Moving Surfaces. A Better Calculus of Moving Surfaces 2012, , , 10.7546/jgsp-26-2012-61-69.
42. \bibitem{GravBook} Misner, C.W.; Thorne, K.S.; Wheeler, J.A. \textit{Gravitation}, 1st ed.; Physics Series; W. H. Freeman: San Fransisco, CA, USA, 1973.
43. Physics, the human adventure: from Copernicus to Einstein and beyond. Choice Reviews Online 2002, 39, 39-39, 10.5860/choice.39-4029.
44. Pere Roura; Thermodynamic derivations of the mechanical equilibrium conditions for fluid surfaces: Young’s and Laplace’s equations. American Journal of Physics 2005, 73, 1139, 10.1119/1.2117127.
45. Alexander Laugier; Jozsef Garai; Derivation of the Ideal Gas Law. Journal of Chemical Education 2007, 84, 1832, 10.1021/ed084p1832.
46. \bibitem{FundPhysics} Halliday, D.; Resnick, R.; Walker, J. \textit{Fundamentals of Physics}, 9th ed.; Wiley: New York, NY, USA, {2011}. %Please add publisher: city, country
47. \bibitem{Morisson}Morisson, F.A. \textit{Compressible Fluids}; {Michigan Technological University: Houghton, MI, USA,} 2004, pp. 94--98.%Please add city, country of the publisher
48. %Please take notice that reference 47--49 were not cited in main text
49. Zhouping Xin; Blowup of smooth solutions to the compressible Navier‐Stokes equation with compact density. Communications on Pure and Applied Mathematics 1998, 51, 229-240, 10.1002/(sici)1097-0312(199803)51:3<229::aid-cpa1>3.3.co;2-k.
50. John K. Hunter; Ralph Saxton; Dynamics of Director Fields. SIAM Journal on Applied Mathematics 1991, 51, 1498-1521, 10.1137/0151075.
51. H.J. Deuling; W. Helfrich; Red blood cell shapes as explained on the basis of curvature elasticity.. Biophysical Journal 1976, 16, 861-868, 10.1016/s0006-3495(76)85736-0.
52. %\bibitem{ExtCMS} K.C. Afas, \textit{Extending the calculus of moving surfaces to higher orders.},ArXiv e-prints (June 2018).
53. %\bibitem{Henszen} M.M. Henszen, M. Weske, S.Schwarz, C.W. Haest, and B. Deuticke,\textit{Electric Field Pulses induce reversible shape transformation of human erythrocytes}, Molecular Membrane Biology 14 (1997), 195-204.
54. \end{thebibliography}