# Simple Solutions of Boundary Layer Equations

Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

## Abstract

In this chapter two- and three-dimensional boundary layer flows in the vicinity of a stagnation point are studied as are flows around wedges and along wedge sidewalls. The flow, induced in the half plane above a rotating plane, is also analyzed. The technique of the boundary layer approach is commenced with the Blasius flow, but more importantly, the boundary layer solution technique for the Navier-Stokes equations is explained by use of the method of matched asymptotic expansions. Moreover, the global laws of the steady boundary layer theory are explained with the aid of the Holstein-Bohlen procedure. The chapter ends with a brief study of non-stationary boundary layers, in which e.g. an impulsive start from rest, flow in the vicinity of a pulsating body, oscillation induced drift current and non-stationary plate boundary layers are studied.

## Keywords

Stagnation point flow Flows around wedges and along wedge side walls Flow on top of a rotating plane Blasius flows Boundary layer solutions of the Navier-Stokes equations by matched asymptotic expansions Holstein-Bohlen procedure based on global laws Nonsteady boundary layers

## References

1. 1.
Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1974)Google Scholar
2. 2.
Belcher, R.J., Burggraf, O.R., Stewartson, K.: On the generalized-vortex boundary layers. J. Fluid Mech. 52, 753 (1972)
3. 3.
Benton, E.R.: Laminar boundary layer on an impulsively started rotating sphere. J. Fluid Mech. 23, 781–800 (1966)
4. 4.
Blasius, H.: Grenzschichten in Flüssigkeiten mit kleiner Reibung. Ph.D. thesis, University Göttingen (1908)Google Scholar
5. 5.
Blasius, H.: Funktionstheoretische methoden in der hydromechanik. Z. Math. Phys. 58, 90–110 (1910)Google Scholar
6. 6.
Blasius, H.: Wärmelehre—Physikalische Grundlagen vom technischen Standpunkt. Bosen and Maasch, Hamburg (1931). (3rd edn. 1949)Google Scholar
7. 7.
Cochran, W.G.: The flow due to a rotating disk. Proc. Camb. Phil. Soc. 30, 365–375 (1934)
8. 8.
Colins, W.M., Dennis, S.C.R.: The initial flow past an impulsively started circular cylinder. Q. Math. Appl. Math. 26, 53–75 (1973)
9. 9.
Coppel, W.A.: On a differential equation of boundary layer theory. Phil. Trans. Royal. Soc. London A253, 101–136 (1960)
10. 10.
Falkner, V.M., Skan, S. W.: Some approximate solutions of the boundary layer equations. Phil. Mag. Z. 865–896 (1931)Google Scholar
11. 11.
Goldstein, S., Rosenhead, L.: Boundary layer growth. Proc. Cambridge Philos. Soc. 32, 394–401 (1936)Google Scholar
12. 12.
Hager, W.H.: Blasius: a life in research and education. Exp. Fluids 34, 566–571 (2003)
13. 13.
Hager, W.H.: Karl Hiemenz, Hydraulicians in Europe 2: 1091. IAHR, Madrid (2009)Google Scholar
14. 14.
Hartree, D.R.: On an equation occurring in Falkner and Skan’s approximate treatment of the equation of the boundary layer. Proc. Camb. Phil. Soc. 33, 240–249 (1937)
15. 15.
Hiemenz, K.: Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers Polytech. J. 326, 321–410 (1911)Google Scholar
16. 16.
Homann, F.: Der Einfluss grosser Zähigkeit bei der Strömung um Zylinder und Kugel. Zeitschr. Angew. Math. Mech. (ZAMM) 16, 153–164 (1936)
17. 17.
Holstein, H., Bohlen, T.: Ein einafches Verfahren zur Berechnung laminarer Reibungsschichten die dem Nährungsverfahren von K. Pohlhausen genügen. Lilienthal-Bericht 10, 5–16 (1940)Google Scholar
18. 18.
Hussaini, M.Y., Lakin, W.D.: Existence and uniqueness of similarity solutions of a boundary layer problem. Q. J. Mech. Appl. Math. 39(1), 15–23 (1986)
19. 19.
Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling, p. 635. Springer, Berlin (2004)Google Scholar
20. 20.
Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics, p. 550. Springer, Berlin (1981)Google Scholar
21. 21.
Lagerstrom, P.A.: Asymptotic expansions—ideas and techniques. Applied Mathematical Sciences Nr 76. Springer, New York (1988)Google Scholar
22. 22.
Landau, L.: A new exact solution of Navier-Stokes equations. Akademija Nauk SSSR (Moskwa) 43, 286–288 (1944)Google Scholar
23. 23.
McLeod, J.B.: von Kármán’s swirling flow problem. Arch. Rational. Mech. Anal. 33, 91–102 (1969)
24. 24.
McLeod, J.B., Serrin, J.: The existence of similarity solutions for some boundary layer problems. Arch. Rational. Mech. Anal. 3, 288–303 (1968)Google Scholar
25. 25.
Miclavc̆ic̆, M., Wang. C.Y.: The flow due to a rough rotating disk. Z. Angew. Math. Phys. 54, 1–12 (2004)Google Scholar
26. 26.
Nayfeh, A.H.: Perturbation Methods. Wiley-VCH Verlag GmbH & Co.KGaA, Weinheim (2004)Google Scholar
27. 27.
Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley-VCH Verlag GmbH & Co.KGaA, Weinheim (2004)Google Scholar
28. 28.
Nigam, S.D.: Rotation of an infinite plane lamina: boundary layer growth. Motion started impulsively from rest. Quart. Appl. Math. 9, 89–91 (1951)Google Scholar
29. 29.
Pohlhausen, K.: Zur näherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht. ZAMM Z. Angew. Math. Mech. 1, 252–268 (1921)
30. 30.
Prandtl, L.: Flüssigkeitsbewegung bei sehr kleiner Reibung. Krazer, A. (ed.) Verhandlungen des dritten internationalen Mathematiker Kongresses in Heidelberg 1904. Teubner Verlag, Leipzig (1905)Google Scholar
31. 31.
Proudman, I., Johnson, K.: Boundary layer growth near a stagnation point. J. Fluid Mech. 12, 161–168 (1962)
32. 32.
Rayleigh, L.: On the motion of solid bodies through viscous liquids. Philos. Mag. 21, 697–711 (1911)
33. 33.
Riley, N.: Unsteady laminar boundary layers. SIAM Rev. 17, 274–297 (1975)
34. 34.
Robins, A.J., Howarth, J.A.: Boundary layer development at a two-dimensional rear stagnation point. J. Fluid Mech. 56, 161–171 (1972)
35. 35.
Rogers, M.H., Lance, G.S.: The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk. J. Fluid Mech. 7, 617–631 (1960)
36. 36.
Schlichting, H., Gersten, K.: Boundary Layer Theory, p. 799. Springer, Berlin (2000)Google Scholar
37. 37.
Sherman, F.S.: Viscous Flow. McGraw Hill Publ. Comp, New York (1990)Google Scholar
38. 38.
Squire, H.B.: The round laminar Jet. Quart. J. Mech. Appl. Math. 4, 321–329 (1951)
39. 39.
Stewardson, K.: The theory of unsteady laminar boundary layers. Adv. Appl. Mech. 6, 1–37 (1960)
40. 40.
Telionis, D.P.: Unsteady Viscous Flows. Springer, New York (1981)
41. 41.
Thirot, H.K.: Über die laminare Anlaufströmung einer Flüssigkeit mit einem rotierenden Boden bei plötzlicher Änderung des Drehungszustandes. Z. Angew. Math. Mech. (ZAMM) 20, 1–13 (1940)
42. 42.
van Dyke, M.: Perturbation Methods in Fluid Dynamics—Annotated Edition. Parabolic Press, p. 284 (1975)Google Scholar
43. 43.
von Kármán, Th: Über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1(2), 233–252 (1921)
44. 44.
Walz, A.: Ein neuer Ansatz für das Geschwindigkeitsprofil der laminaren Reibungsschicht. Lilienthal Ber. 141, 8–12 (1941)Google Scholar
45. 45.
Walz, A.: Strömungs- und Temperaturgrenzschichten. Braun-Verlag Karlsruhe (1966) English. Translation: boundary layers of flow and temperature. The M.I.T. Press, Cambridge, Mass (1969)Google Scholar
46. 46.
Weyl, H.: On the differential equations of the simplest boundary layer problems. Ann. Math. 43, 381–407 (1942)
47. 47.
Zandbergen, P.J., Dijkstra, D.: Non-unique solutions of the Navier-Stokes equations for the Karman swirling flow. J. Eng. Math. 11, 167–188 (1977)
48. 48.
Zandbergen, P.J., Dijkstra, D.: von Kármán swirling flows. Annu. Rev. Fluid Mech. 19, 465–491 (1987)