Skip to main content

On the boundary layer in liquids

  • Chapter
  • 77 Accesses

Summary

The author discusses differential systems governing the motion of liquids having various equations of state. The resulting equations are ordinary parametric differential equations with two point boundary conditions. The problem of existence of solutions is discussed and the Iglisch proof is adjusted to prove the existence of the flow and of the temperature pattern.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   59.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Birkhoff, G.: Dimensional analysis of partial differential equations. Electrical Engineering, 67, 1185–1188, 1948.

    Article  Google Scholar 

  2. Birkhoff, G.: Hydrodynamics. Princeton University Press, 1950, Chapt. IV.

    MATH  Google Scholar 

  3. Falkner, V.M., and Skan, S. W.: Some approximate solutions of the boundary layer equations. Brit. A.R.C., R. and M. 1314, Apr. 1930, and Phil. Mag. 12, 865, 1931.

    Google Scholar 

  4. Goldstein, S.: Editor, Modern developments in fluid dynamics. Vol. I and II, Oxford Clarendon Press, 1938.

    Google Scholar 

  5. Goldstein, S.: A note on the boundary layer equations. Proc. Cambr. Philos. Soc. 35, 338, 1939.

    Article  ADS  Google Scholar 

  6. Hamel, G.: Integralgleichungen. Springer Verlag, Berlin, 1949.

    Book  MATH  Google Scholar 

  7. Hartree, D. R.: On an equation occuring in Falkner and Skan’s approximate treatment of the equations of the boundary layer. Proc. Cambr. Phil. Soc. 33, Part II, 223, 1937.

    Article  ADS  Google Scholar 

  8. Hiemenz, F.: Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Disser. Göttingen 1911, Dingl. Polytech. Jour. 326, Heft 21–26, 321–326, 1911.

    Google Scholar 

  9. Homann, F.: Der Einfluß großer Zähigkeit bei der Strömung um den Zylinder und um die Kugel. ZAMM, 16, 153, 1936.

    Article  MATH  Google Scholar 

  10. Iglisch, R.: Über das asymptotische Verhalten der Lösung einer nichtlinearen gewöhnlichen Differentialgleichung 3. Ordnung. Veröffentl. des Math. Instit. der Tech. Hochsch. Braunschweig, 1943.

    Google Scholar 

  11. Iglisch, R.: Elementarer Existenzbeweis für die Strömung in der laminaren Grenzschicht zur Potentialströmung Uu 1 x m mit m > 0 bei Absaugen und Ausblasen, ZAMM, 33, 4, 143–147, Apr. 1953.

    Article  MATH  MathSciNet  Google Scholar 

  12. Iglisch, R., and Grohne, D.: Die laminare Grenzschicht an der längsangeströmten ebenen Platte mit schrägem Absaugen und Ausblasen. Veröffentl. des Math. Instit. der Techn. Hochsch. Braunschweig, Bericht 1/45, 1945.

    Google Scholar 

  13. Janet, M.: Sur les systèmes d’équations aux dérivées partielles. Journal de Mathématiques pures et appliquées, 85, Ser. 8, 3, 65–151, 1920.

    Google Scholar 

  14. Kamke, E.: Differentialgleichungen. I. Gewöhnliche Differentialgleichungen. Akad. Ver-lagsges. Becker and Erler, Leipzig, 1943.

    Google Scholar 

  15. Kampier, E.: Physics of liquids and gases. Fiat Review of German Science, 1939–1946. Published by Office of Milit. Government for Germany, Field Inform. Agencies. Printed by I. W. Klemm, Wiesbaden, Germany, 1948.

    Google Scholar 

  16. v. Krzywoblocki, M. Z.: On the two dimensional laminar boundary layer equations for hypersonic flow in continuum and in rarefied gases. J. Aer. Soc. India, 5, 1, 1–13, 1953.

    Google Scholar 

  17. v. Krzywoblocki, M. Z.: On the fundamentals of the boundary layer theory. J. Franklin Institute, 255, 4, 289–300, April, 1953.

    Article  Google Scholar 

  18. v. Krzywoblocki, M. Z.: Possible particular solutions of the laminar boundary layer equations along a flat plate in hypersonic flow in continuum and in rarefied gases. J. Aer. Soc. India, 5, 2, 23–35, 1953.

    Google Scholar 

  19. v. Krzywoblocki, M. Z.: On the generalized theory of the laminar, two-dimensional boundary layer along a flat plate in continuum and slip flow regimes. Bulletin de la Société Mathématique de Grèce (to be published).

    Google Scholar 

  20. Mangier, W.: Die „ähnlichen” Lösungen der Prandtlschen Grenzschichtgleichungen, ZAMM, 23, 243, 1943.

    Google Scholar 

  21. Morgan, A. J. A.: The reduction by one of the number of independent variables in some systems of partial differential equations. Quart. J. Math. Oxford Second Series, 3, 250–259, Dec. 1952.

    Article  ADS  MATH  Google Scholar 

  22. Pohlhausen, E.: Der Wärmeaustausch zwischen festen Körpern und Flüssigkeiten mit kleiner Wärmeleitung. ZAMM, 1, 115, 1921.

    Article  MATH  Google Scholar 

  23. Rellich, F.: Über Lösungen nichtlinearer Differentialgleichungen. Festschrift zur Feier des 200 jähr. Bestehens der Akad. d. Wissensch. in Göttingen, Math. Phys. Klasse, 168–174, 1951.

    Google Scholar 

  24. Riquier, C.: Les systèmes d’équations aux dérivées partielles. Gauthier-Villars, Paris, 1910, XVII + 590.

    Google Scholar 

  25. Riquier, C: La méthode des fonctions majorantes et les systèmes d’équations aux dérivées partielles. Mémorial des Sciences Mathématiques, Fascicule 32, 1928, 63 pp.

    Google Scholar 

  26. Ritt, J. F.: Differential equations from the algebraic standpoint. Amer. Math. Soc-Collogquium Public. Vol. 14, New York, 1932.

    Google Scholar 

  27. Schlichting, H.: Grenzschicht-Theorie. Verlag G. Braun, Karlsruhe, 1951.

    MATH  Google Scholar 

  28. Thomas, J. M.: Note on a differential equation, Annals of Math. 28, 240–244, 1926.

    Google Scholar 

  29. Thomas, J. M.: Systems of total differential equations, Annals of Mat. 28, 379–385, 1926.

    Google Scholar 

  30. Thomas, J. M.: Incomplete systems of partial differential equations, Proc. Nation. Acad. Sci. 14, 666–670, 1928.

    Article  ADS  MATH  Google Scholar 

  31. Thomas, J.M.: Riquier’s existence theorems, Annals of Math. 30, 285–310, 1928.

    Article  Google Scholar 

  32. Thomas, J. M.: Matrices of integers ordering derivatives, Trans. Am. Math. Soc. 33, 389–410, 1931.

    Article  Google Scholar 

  33. Thomas, J. M.: The condition for an ortho-nomic differential system, Trans. Am. Math. Soc. 34, 332–338, 1932.

    Article  Google Scholar 

  34. Thomas, J. M.: Regular differential systems of the first order, Proc. Nat. Acad. Sci. 19, 451–453, 1933.

    Article  ADS  Google Scholar 

  35. Thomas, J. M.: Riquier’s existence theorems, Annals of Math. 35, 306–311, 1934.

    Article  Google Scholar 

  36. Thomas, J. M.: Differential systems, Amer. Math. Soc. Collogquium Public, 21, New York, 1937.

    Google Scholar 

  37. Tifford, A. N.: On certain particular solutions of the laminar boundary layer equations. Proceed. First Midwestern Conf. Fluid Dynamics, Univ. Illinois 1950, publish. J.W. Edwards, Ann. Arbor, Michigan, 1951, p. 81–90.

    Google Scholar 

  38. Weyl, H.: Concerning the differential equations of some boundary layer equations. Proc. Nat. Acad. Sci. 27, 578–583, 1941.

    Article  ADS  MathSciNet  Google Scholar 

  39. Weyl, H.: II. Ibid, 28, 100–102, 1942.

    Google Scholar 

  40. Weyl, H.: On the differential equations of the simplest boundary layer problems. Annals of Math. 43, 2, 381–407, Apr. 1942.

    Article  MATH  MathSciNet  Google Scholar 

  41. Whyburn, W. M.: On the fundamental existence theorems for differential systems. Annals of Math. Second Ser. 30, 1, 31–38, Dec. 1928.

    Article  MATH  MathSciNet  Google Scholar 

  42. Witting, H.: Über zwei Differenzenverfahren der Grenzschichttheorie. Archiv der Mathem. 4, 3, 247–256, 1953.

    Article  MATH  MathSciNet  Google Scholar 

  43. Witting, H.: Verbesserung des Differenzenverfahrens von H. Görtier zur Berechnung laminarer Grenzschichten. ZAMP, 4, 5, 376–397, 1953.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  44. Görtier, H.: Ein Differenzenverfahren zur Berechnung laminarer Grenzschichten. Ing. Arch. 16, 173–187, 1948.

    Article  MathSciNet  Google Scholar 

  45. Ostrowski, M. A.: Sur le rayon de convergence de la série de Blasius. C. R. l’Acad. Sc. Paris, 227, 580–582, Sept. 1948.

    MATH  MathSciNet  Google Scholar 

  46. Oudart, A.: Les methodes scientifiques de la couche limite laminaire. Publ. Sc. et Tech. du Min. de l’Air, No. 213, 123–127, 1948.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1955 Springer Fachmedien Wiesbaden

About this chapter

Cite this chapter

v. Krzywoblocki, M.Z. (1955). On the boundary layer in liquids. In: Görtler, H., Tollmien, W. (eds) 50 Jahre Grenzschichtforschung. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-20219-6_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-663-20219-6_9

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-663-19879-6

  • Online ISBN: 978-3-663-20219-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics