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Wärme - und Stoffübertragung

, Volume 20, Issue 1, pp 47–57 | Cite as

The effect of variable properties on laminar boundary layer flow

  • H. Herwig
  • G. Wickern
Article

Abstract

The influence of temperature dependent fluid properties on laminar boundary layer flows is examined for wedge flows (Falkner-Skan). An asymptotic expansion for small heat transfer rates is applied under the boundary conditionsTw= const andqw=const. Linear deviations from the zero order solution (constant properties) can be given in a form known as the property ratio method. As a consequence this method is no longer an empirical one.

Keywords

Heat Transfer Transfer Rate Asymptotic Expansion Heat Transfer Rate Variable Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

cf

skin friction coefficient

cp

specific heat at constant pressure

e

exponent, Eq. (19)

f(ηs)

dimensionless stream function

0

f for zero order solution

fli

f for first order solutions

j,jcf j,Nu

reference temperature factor

ki

exponents

Kα

dimensionless property, Eq. (3)

Kα

dimensionless property, Eq. (4)

Kα

dimensionless property, Eq. (4a)

L

characteristic length

m

exponent in velocity, Eq. (11)

Ma

Mach number

m, mi

exponents

Nu

Nusselt number

ni, ni

exponents

p

pressure

Pr

Prandtl number

qw

heat flux across the wall

Re

Reynolds number

T

temperature

u, v

velocity components

U

outer velocity

x, y

cartesian coordinates

α

physical property

β

wedge angle

γ

ratio of heat capacities

δe

unit length, Eq. (13)

ɛ

perturbation parameter, Eq. (20)

η

viscosity

ηs

similarity variable, Eq. (12)

λ.

thermal conductivity

ϱ

density

Θ (ηs)

dimensionless temperature

Θ0

Θ for zero order solution

Θ1i

Θ for first order solutions

τw

wall shear stress

ψ

stream function

Subscripts

c.p

constant property

L

at x*=L*

s

separation

W

wall

infinity

Der Einfluß temperaturabhängiger Stoffwerte auf laminare Grenzschichtströmung

Zusammenfassung

Es wird der Einfluß temperaturabhängiger Stoffwerte auf laminare Grenzschichtströmungen am Beispiel der Keilströmungen (Falkner-Skan) betrachtet. Für die RandbedingungenTw= konst und qw= konst wird eine asymptotische Entwicklung für kleine Wärmeübergänge vorgenommen. Lineare Abweichungen von der Grundlösung für konstante Stoffwerte können in Form einer Korrektur geschrieben werden, wie sie in der Literatur als Stoffwertverhältnis-Ansatz bekannt ist. Diese bisher empirische Methode ist damit zu einer theoretischen Methode geworden.

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References

  1. 1.
    Wilson, A. H.: Thermodynamics and statistical mechanics, Cambridge University Press, 1966Google Scholar
  2. 2.
    Gersten, K.; Herwig, H.: Impuls- und Wärmeübertragung bei variablen Stoffwerten für die. laminare Plattenströmung. Wärme-Stoffübertrag. 18 (1984) 25–35Google Scholar
  3. 3.
    Gersten, K.: Single phase fluid flow. In: Heat Exchanger Design Handbook, Vol. 2. Ed.: E. U. Schlünder, Hemisphere Publ. Corp. 1983Google Scholar
  4. 4.
    White, F. M.: Viscous Fluid Flow. New York: McGraw-Hill 1974Google Scholar
  5. 5.
    Harpole, G. M.; Berger, S. A.; Aroesty, J.: Approximate methods for calculating heated water laminar boundary-layer properties. J. of Appl. Mech. 46 (1979) 9–14Google Scholar
  6. 6.
    Stewartson, K.: Further solutions of the Falkner-Skan equation. Proc. Cambr. Phil. Soc. 50 (1954) 454–465Google Scholar
  7. 7.
    Brown, S. N.; Stewartson, K.: Laminar separation. Ann. Rev. Fluid Mech. 1 (1969) 45–72Google Scholar
  8. 8.
    Evans, H. L.: Laminar boundary-layer theory. Massachusetts: Addison-Wesley Publ. Comp. 1968Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • H. Herwig
    • 1
  • G. Wickern
    • 1
  1. 1.Institut für Thermo- und FluiddynamikRuhr-Universität BochumBochum-Querenburg

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