An axisymmetric boundary layer on a needle

Bryuno, A.; Shadrina, T.

doi:10.1090/S0077-1554-07-00165-3

An axisymmetric boundary layer on a needle
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by A. D. Bryuno and T. V. Shadrina
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2007, 201-259

DOI: https://doi.org/10.1090/S0077-1554-07-00165-3

Published electronically: November 15, 2007

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Abstract:

Methods of power geometry are used to study the boundary layer on a semi-infinite needle, due to a steady flow of a viscous fluid or gas parallel to the needle. The purpose is to find the asymptotics of the flow in the boundary layer at infinity along the needle. Two variants of the flow are considered: (a) an incompressible non-heat-conducting fluid, and (b) a compressible heat-conducting gas. It is shown that variant (a) has no asymptotics for solutions satisfying all the boundary conditions, whereas variant (b) has several families of asymptotics for solutions that satisfy all the boundary conditions. These asymptotic expansions have power or logarithmic singularities near the needle.

References

L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Verhandlungen des III. Internat. Math.-Kongr., Heidelberg, 1904, Teubner, Leipzig, 1905, S. 484–491.
H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys. 56 (1908), 1–37.
S. Goldstein, On the two-dimensional steady flow of a viscous fluid behind a solid body. I, II, Proc. Royal Soc. London A 142 (1933), 545–573.
K. Stewartson, On asymptotic expansion in the theory of boundary layers, J. Math. and Phys. 36 (1957), 173–191. MR 93232, DOI 10.1002/sapm1957361173
A. I. van de Vooren and D. Dijkstra, The Navier–Stokes solution for laminar flow past a semi-infinite flat plate, J. Engineer. Math. 4 (1970), 9–27.
R. I. MacLachlan, The boundary layer on a finite flat plate, Phys. Fluids A 3 (1991), 341–348.
R. A. Seban and R. Bond, Skin-friction and heat-transfer characteristics of a laminar boundary layer on a cylinder in axial incompressible flow, J. Aeronaut. Sci. 18 (1951), 671–675.
H. R. Kelly, A note on the laminar boundary layer on a circular cylinder in axial incompressible flow, J. Aeronaut. Sci. 21 (1954), 634.
Lord Rayleigh, On the motion of solid bodies through viscous liquid, Phil. Mag. (6) 21 (1911), 697–711.
K. Pohlhausen, Zur näherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht, Z. Angew. Math. Mech. 1 (1921), 252–268.
M. B. Glauert and M. J. Lighthill, The axisymmetric boundary layer on a long thin cylinder, Proc. Roy. Soc. London Ser. A 230 (1955), 188–203. MR 70373, DOI 10.1098/rspa.1955.0121
K. Stewartson, The asymptotic boundary layer on a circular cylinder in axial incompressible flow, Quart. Appl. Math. 13 (1955), 113–122. MR 68954, DOI 10.1090/S0033-569X-1955-68954-X
A. D. Bryuno, Stepennaya geometriya v algebraicheskikh i differentsial′nykh uravneniyakh, Fizmatlit “Nauka”, Moscow, 1998 (Russian, with Russian summary). MR 1734769
A. D. Bryuno and T. V. Shadrina, On the axisymmetric flow of a viscous incompressible fluid around a needle, Dokl. Akad. Nauk 387 (2002), no. 5, 589–595 (Russian). MR 2004520
T. V. Shadrina, The axially symmetric boundary layer around a needle, Proc. BAIL 2002 (Eds. S. Wang and N. Fowkes), Univ. of Western Australia, Perth, 2002, pp. 213–220.
A. D. Bryuno and T. V. Shadrina, An axisymmetric boundary layer on a needle, Dokl. Akad. Nauk 394 (2004), no. 3, 298–304 (Russian). MR 2088482
T. V. Shadrina, Boundary layer for axisymmetric flow past a needle, Differential Equations 38 (2002), 853–854. (Russian)
T. V. Shadrina, On axisymmetric flow of a viscous incompressible fluid past a needle, XXVIII Gagarin lectures, Absracts, vol. 2, MATI, Moscow, 2002, 98–99. (Russian)
A. D. Bryuno and T. V. Shadrina, The axially symmetric boundary layer around a needle, Intern. Conf. on Differential and Functional Differential Equations, MAI, Moscow, 2002, pp. 18–19.
A. D. Bryuno and T. V. Shadrina, The axially symmetric boundary layer around a needle, Intern. Conf. “Navier–Stokes Equations and Related Topics (NSEC8)”, Abstracts, Euler Inst., S. Petersburg, 2002, pp. 18–19.
T. V. Shadrina, Axisymmetric boundary layer on a needle, XIV All-Russia Conf. “Theoretical Foundations and Construction of Algorithms for Solving Problems of Mathematical Physics”, Abstracts, Abrau-Dyurso, 2002, pp. 167–168. (Russian)
A. D. Bryuno and T. V. Shadrina, The axially symmetric boundary layer on a needle, Intern. Conf. on Boundary and Interior Layers, ONERA, Toulouse, 2004, July 5th, 11.20, pp. 1–10.
A. D. Bryuno and T. V. Shadrina, Boundary layer on a needle, II All-Russia Conf. “Topical Problems of Applied Mathematics and Mechanics”, Abrau-Dyurso, 2004, pp. 30–32. (Russian)
T. V. Shadrina, On axisymmetric flow of a viscous incompressible fluid past a needle, Preprint no. 36, Keldysh Inst. Appl. Math., Moscow, 2002. (Russian)
A. D. Bryuno and T. V. Shadrina, Axisymmetric boundary layer on a needle, Preprint no. 64, Keldysh Inst. Appl. Math., Moscow, 2003. (Russian)
A. D. Bryuno and T. V. Shadrina, Methods of investigation of a boundary layer on a needle, Preprint no. 35, Keldysh Inst. Appl. Math., Moscow, 2004. (Russian)
A. D. Bryuno and T. V. Shadrina, On incompressible boundary layer on a needle, Preprint no. 36, Keldysh Inst. Appl. Math., Moscow, 2004. (Russian)
A. D. Bryuno and T. V. Shadrina, Compressible heat-conducting boundary layer on a needle, Preprint no. 37, Keldysh Inst. Appl. Math., Moscow, 2004. (Russian)
A. D. Bryuno, Self-similar solutions and power geometry, Uspekhi Mat. Nauk 55 (2000), no. 1(331), 3–44 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 1, 1–42. MR 1751817, DOI 10.1070/rm2000v055n01ABEH000248
A. D. Bryuno, Power expansions of solutions of a system of algebraic and differential equations, Dokl. Akad. Nauk 380 (2001), no. 3, 298–304 (Russian). MR 1868120
A. D. Bryuno, Power asymptotics of solutions of an ordinary differential equation, Dokl. Akad. Nauk 392 (2003), no. 3, 295–300 (Russian). MR 2088468
A. D. Bryuno, Asymptotics and expansions of solutions of an ordinary differential equation, Preprint no. 9, Keldysh Inst. Appl. Math., Moscow, 2003. (Russian)
A. D. Bryuno, Asymptotic behavior and expansions of solutions of an ordinary differential equation, Uspekhi Mat. Nauk 59 (2004), no. 3(357), 31–80 (Russian, with Russian summary); English transl., Russian Math. Surveys 59 (2004), no. 3, 429–480. MR 2116535, DOI 10.1070/RM2004v059n03ABEH000736
A. D. Bryuno, Power expansions of solutions of an algebraic or a differential equation, Dokl. Akad. Nauk 380 (2001), no. 2, 155–159 (Russian). MR 1873273
A. D. Bryuno, Power-logarithmic expansions of solutions of an ordinary differential equation, Dokl. Akad. Nauk 392 (2003), no. 4, 439–444 (Russian). MR 2081499
A. D. Bryuno, Nonpower asymptotics of solutions of an ordinary differential equation, Dokl. Akad. Nauk 392 (2003), no. 5, 586–591 (Russian). MR 2082848
L. G. Loitsyanskii, Mechanics of liquids and gases, Pergamon Press, Oxford-New York-Toronto, 1966. Translated from the Russian by J. Berry and H. K. Zienkiewicz; Edited by K. Stewartson. MR 0197015
G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
L. G. Loĭtsyanskiĭ, Laminar boundary layer, Nauka, Fizmatlit, Moscow, 1962.
Erich Kamke, Differentialgleichungen, B. G. Teubner, Stuttgart, 1977 (German). Lösungsmethoden und Lösungen. I: Gewöhnliche Differentialgleichungen; Neunte Auflage; Mit einem Vorwort von Detlef Kamke. MR 0466672
M. M. Vasil’ev, On axisymmetric flows of a viscous heat-conducting gas, Preprint no. 11, Keldysh Inst. Appl. Math., Moscow, 2001. (Russian)
M. M. Vasiliev, Obtaining the self-similar asymptotics of solutions to the Navier-Stokes equations by power geometry, Progress in analysis, Vol. I, II (Berlin, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 93–101. MR 2032673
A. D. Bryuno, Complicated expansions of solutions of an ordinary differential equation, Preprint no. 36, Keldysh Inst. Appl. Math., Moscow, 2005. (Russian)