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Description of Fluids

  • Achim FeldmeierEmail author
Chapter
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Part of the Theoretical and Mathematical Physics book series (TMP)

Abstract

In fluid mechanics, infinitesimal masses \({\mathrm d}M\), called fluid parcels, occupy infinitesimal volumes \({\mathrm d}V\) that have well-defined positions \(\vec {r}(t)\) in some fixed frame of reference, similar to point masses in classical point mechanics. The fluid parcels fill some connected volume in space homogeneously (tearing fluids are not treated here), where space is the Euclidean vector space \(\mathbb {R}^3\).

References

  1. Batchelor, G.K. 1952. The effect of homogeneous turbulence on material lines and surfaces. Proceedings of the Royal Society of London A 213: 349.CrossRefADSMathSciNetzbMATHGoogle Scholar
  2. Batchelor, G.K. 1967. An introduction to fluid dynamics. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  3. Bateman, H. 1944. Partial differential equations of mathematical physics. New York: Dover.zbMATHGoogle Scholar
  4. Benedetto, D., and M. Pulvirenti. 1992. From vortex layers to vortex sheets. SIAM Journal on Applied Mathematics 52: 1041.CrossRefADSMathSciNetzbMATHGoogle Scholar
  5. Bott, R., and L.W. Tu. 1982. Differential forms in algebraic topology. New York: Springer.CrossRefzbMATHGoogle Scholar
  6. Broer, L.J.F. 1974. On the Hamiltonian theory of surface waves. Applied Scientific Research 29: 430.CrossRefzbMATHGoogle Scholar
  7. Case, K.M. 1961. Hydrodynamic stability and the inviscid limit. Journal of Fluid Mechanics 10: 420.CrossRefADSMathSciNetzbMATHGoogle Scholar
  8. Chorin, A.J., and J.E. Marsden. 1990. A mathematical introduction to fluid mechanics. New York: Springer.CrossRefzbMATHGoogle Scholar
  9. Clebsch, A. 1859. Ueber die Integration der hydrodynamischen Gleichungen. Journal für die reine und angewandte Mathematik 56: 1.MathSciNetGoogle Scholar
  10. Cocke, W.J. 1969. Turbulent hydrodynamic line stretching: Consequences of isotropy. Physics of Fluids 12: 2488.CrossRefADSMathSciNetzbMATHGoogle Scholar
  11. Dussan, V.E.B. 1976. On the difference between a bounding surface and a material surface. Journal of Fluid Mechanics 75: 609.CrossRefADSzbMATHGoogle Scholar
  12. Flanders, H. 1963. Differential forms with applications to the physical sciences. New York: Academic Press; New York: Dover (1989).Google Scholar
  13. Flanders, H. 1967. Differential forms. In Studies in global geometry and analysis, ed. S.-S. Chern, 27. Washington: Mathematical Association of America.Google Scholar
  14. Friedrichs, K.O. 1934. Über ein Minimumproblem für Potentialströmungen mit freiem Rande. Mathematische Annalen 109: 60.CrossRefMathSciNetzbMATHGoogle Scholar
  15. Hargreaves, R. 1908. A pressure integral as kinetic potential. Philosophical Magazine Series 6, 16: 436.Google Scholar
  16. Herglotz, G. 1985. Vorlesungen über die Mechanik der Kontinua. Leipzig: B.G. Teubner.CrossRefzbMATHGoogle Scholar
  17. Huang, K. 1963. Introduction to statistical physics. New York: Wiley.Google Scholar
  18. Hundhausen, A.J. 1972. Coronal expansion and solar wind. Berlin: Springer.CrossRefGoogle Scholar
  19. Jackiw, R. 2002. Lectures on fluid dynamics. New York: Springer.CrossRefGoogle Scholar
  20. Jänich, K. 2001. Vector analysis. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  21. Kotschin, N.J., I.A. Kibel, and N.W. Rose. 1954. Theoretische Hydromechanik, vol. I. Berlin: Akademie-Verlag.Google Scholar
  22. Kraichnan, R.H. 1974. On Kolmogorov’s inertial-range theories. Journal of Fluid Mechanics 62: 305.CrossRefADSzbMATHGoogle Scholar
  23. Lamb, H. 1932. Hydrodynamics. Cambridge: Cambridge University Press; New York: Dover (1945).Google Scholar
  24. Lichtenstein, L. 1929. Grundlagen der Hydromechanik. Berlin: Springer.zbMATHGoogle Scholar
  25. Lighthill, J. 1986. An informal introduction to theoretical fluid mechanics. Oxford: Clarendon Press.zbMATHGoogle Scholar
  26. Love, A.E. 1901. Hydrodynamik. In Encyklopädie der mathematischen Wissenschaften, vol. 4–3, 48. Leipzig: Teubner (digital at GDZ Göttingen).Google Scholar
  27. Luke, J.C. 1967. A variational principle for a fluid with a free surface. Journal of Fluid Mechanics 27: 395.CrossRefADSMathSciNetzbMATHGoogle Scholar
  28. Markushevich, A.I. 1965. Theory of functions of a complex variable, vol. I. Englewood Cliffs: Prentice-Hall.zbMATHGoogle Scholar
  29. Meyer, R.E. 1971. Introduction to mathematical fluid dynamics. New York: Wiley; Mineola: Dover (2007).Google Scholar
  30. Miles, J.W. 1977. On Hamilton’s principle for surface waves. Journal of Fluid Mechanics 83: 153.CrossRefADSMathSciNetzbMATHGoogle Scholar
  31. Milne-Thomson, L.M. 1968. Theoretical hydrodynamics, 5th ed. London: Macmillan; New York: Dover (1996).Google Scholar
  32. Norman, M.L., and K.-H. Winkler. 1986. 2-D Eulerian hydrodynamics with fluid interfaces, self-gravity and rotation. In NATO advanced research workshop on astrophysical radiation hydrodynamics, ed. K.-H. Winkler and M.L. Norman, 187. Dordrecht: D. Reidel Publishing Co.Google Scholar
  33. Orszag, S.A. 1970. Comments on ‘Turbulent hydrodynamic line stretching: Consequences of isotropy’. Physics of Fluids 13: 2203.CrossRefADSMathSciNetGoogle Scholar
  34. Prager, W. 1961. Einführung in die Kontinuumsmechanik. Basel: Birkäuser Verlag.CrossRefzbMATHGoogle Scholar
  35. Prosperetti, A. 2011. Advanced mathematics for applications. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  36. Proudman, J. 1953. Dynamical oceanography. London: Methuen & Co.Google Scholar
  37. Rosenhead, L. 1954. Introduction—The second coefficient of viscosity: A brief review of fundamentals. Proceedings of the Royal Society of London A 226: 1.ADSMathSciNetGoogle Scholar
  38. Saffman, P.G. 1995. Vortex dynamics. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  39. Saffman, P.G., and G.R. Baker. 1979. Vortex interactions. Annual Reviews in Fluid Mechanics 11: 95.CrossRefADSzbMATHGoogle Scholar
  40. Schaefer, H. 1967. Das Cosserat-Kontinuum. Zeitschrift für angewandte Mathematik und Mechanik 47: 485.CrossRefADSzbMATHGoogle Scholar
  41. Sierpinski, W. 1958. Cardinal and ordinal numbers. Polska Akademia Nauk. Monografie matematyczne, vol. 34. Warszawa.Google Scholar
  42. Stokes, G.G. 1845. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Transactions of the Cambridge Philosophical Society 8: 287 (1849); in Mathematical and physical papers by G.G. Stokes, vol. 1, 75. Cambridge: Cambridge University Press (1880).Google Scholar
  43. Stone, J.M., and M.L. Norman. 1992. ZEUS-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. I. The hydrodynamic algorithms and tests. Astrophysical Journal Supplement Series 80: 753.CrossRefADSGoogle Scholar
  44. Tanaka, M., and S. Kida. 1993. Characterization of vortex tubes and sheets. Physics of Fluids A 5: 2079.CrossRefADSGoogle Scholar
  45. Truesdell, C. 1952. The mechanical foundations of elasticity and fluid dynamics. Journal of Rational Mechanics and Analysis 1: 125.MathSciNetzbMATHGoogle Scholar
  46. Truesdell, C. 1966. The elements of continuum mechanics. Berlin: Springer.zbMATHGoogle Scholar
  47. Truesdell, C., and R.G. Muncaster. 1980. Fundamentals of Maxwell’s kinetic theory of a simple monatomic gas. New York: Academic Press.Google Scholar
  48. Truesdell, C., and W. Noll. 1965. The non-linear field theories of mechanics. In Handbuch der Physik = Encyclopedia of Physics, vol. 2/3–3, 1, ed. S. Flügge and C. Truesdell. Berlin: Springer.Google Scholar
  49. Weatherburn, C.E. 1924. Advanced vector analysis. London: G. Bell & Sons.zbMATHGoogle Scholar
  50. Whitham, G.B. 1965. A general approach to linear and non-linear dispersive waves using a Lagrangian. Journal of Fluid Mechanics 22: 273.CrossRefADSMathSciNetGoogle Scholar
  51. Whitham, G.B. 1967. Non-linear dispersion of water waves. Journal of Fluid Mechanics 27: 399.CrossRefADSMathSciNetzbMATHGoogle Scholar
  52. Winkler, K.-H., M.L. Norman, and D. Mihalas. 1984. Adaptive-mesh radiation hydrodynamics. I. The radiation transport equation in a completely adaptive coordinate system. Journal of Quantitative Spectroscopy & Radiative Transfer 31: 473.CrossRefADSGoogle Scholar
  53. Wright, M.C.M. 2006. Green function or Green’s function? Nature Physics 2: 646.CrossRefADSGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Physik und AstronomieUniversität PotsdamPotsdamGermany

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