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Forschung im Ingenieurwesen

, Volume 64, Issue 4–5, pp 111–120 | Cite as

Stability analysis of plane laminar Poiseuille flow with constant heat flux across the wall: asymptotic methods

  • H. Herwig
  • X. You
Originalarbeiten

Abstract

Classical linear stability theory is extended to include the effects of temperature and pressure dependent fluid properties. These effects are studied asymptotically by using Taylor series expansions for all properties with respect to temperature and pressure. In this asymptotic approach all effects are well separated from each other, and only the Prandtl number remains as a parameter. In their general form the asymptotic solutions hold for all Newtonian fluids. Two different methods are introduced, a straightforward expansion method and a so called combined method. In the combined method results are found by combining the knowledge of the asymptotic form of the final solution with particular numerical results of the unexpanded basic equations. From the asymptotic results general conclusions can be drawn on how the critical Reynolds number is affected when variable property effects are taken into account.

Keywords

Expansion Method Taylor Series Expansion Combine Method Poiseuille Flow Critical Reynolds Number 
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.

List of symbols

a, b

general quantity

b

shape function of b

ci

amplification rate

cr

phase velocity

cp

specific heat

H

half channel height

k

thermal conductivity

Kap, KaT

nondimensional coefficients in Taylor expansion, Eq. (1)

p

pressure

Pr

Prandtl number μ*R c* p K*

qw

wall heat flux density

Re

Reynolds number ρ* U*R H* / μ*R

t

time

T

temperature

TB

bulk temperature

u

streamwise velocity

UR

reference velocity

v

velocity normal to the wall

x,y

Cartesian coordinates

Greek symbols

α

wave length parameter

εp, εT

perturbation parameter, Eq. (3)

μ

viscosity

ψ

stream function

ρ

density

Superscripts

*

dimensional quanity

mean value

disturbance quantity

Λ

complex quantity

Subscripts

i

imaginary part

r

real part

R

reference state

w

wall

0

zero order

μT

viscosity effect

ρP, ρT

density effect

Die Stabilitätsanalyse der ebenen laminaren Poiseuille Strömung mit konstantem Wandwärmestrom: Asymptotische Methoden

Zusammenfassung

In der vorliegenden Studie wird die klassische lineare Stabilitätstheorie um die Effekte temperatur-und druckabhängiger Stoffwerte erweiter. Diese Effekte werden asymptotisch durch Verwendung von Taylor-Reihenentwicklungen aller Stoffwerte erfaßt. Dabei sind alle Effekte klar voneinander getrennt. Ledighlich die Pradtl-Zahl verbleibt als Parameter.

In ihrer allegemeinen Form gelten die asymptotischen Ergebnisse für alle Newtonschen Fluide. Zwei Methoden finden Verwendung: eine klassische Reihenentwicklung und die sog. “kombinierte Method”. In dieser Method folgen die Ergebnisse aus einer Kombination der allgemeinen asymptotischen Form der Ergebnisse und konkreten Zahlenwerten aus einigen wenigen Fällen der nicht entwickelten Gleichungen. Aus den asymptotischen Ergebnissen können generelle Schlüsse über die Wirkung der Wärmeübertragung auf die Strömungsstabilität gezogen werden.

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Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • H. Herwig
    • 1
  • X. You
    • 1
  1. 1.Technische ThermodynamikTU ChemnitzChemnitzGermany

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