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On absolute linear instability analysis of plane Poiseuille flow by a semi-analytical treatment

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An Erratum to this article was published on 27 June 2017

Abstract

The absolute linear hydrodynamic instability of the plane Poiseuille flow is investigated by solving the Orr–Sommerfeld equation using the semi-analytical treatment of the Adomian decomposition method (ADM). In order to use the ADM, a new zero-order ADM approximation is defined. The results for the spectrum of eigenvalues are obtained using various orders of the ADM approximations and discussed. A comparative study of the results for the first, second and third eigenvalues with the ones from a previously published work is also presented. A monotonic trend of approach of decreasing relative error with the increase of the orders of ADM approximation is indicated. The results for the first, second and third eigenvalues show that they are in good agreement within 1.5 % error with the ones obtained by a previously published work using the Chebyshev spectral method. The results also show that the first eigenvalue is positioned in the unstable zone of the spectrum, while the second and third eigenvalues are located in the stable zone.

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Abbreviations

a :

Initial condition in Eq. (16) (value of second derivative of φ with respect to z at z = −1)

b :

Initial condition in Eq. (16) (value of third derivative of φ with respect to z at z = −1)

c 1 :

Lower boundary limit

c 2 :

Higher boundary limit

D :

Differential operator in Eq. (3)

e i :

Coefficients of φ 1(z) in Eq. (21)

f i :

Coefficients of φ 2(z) in Eq. (21)

g :

Zero order approximation

h :

Source term in Eq. (3)

i :

Index for imaginary; indicator of imaginary part

L :

Linear invertible part of differential operator D

L −1 :

Inverse of operator L

m :

Index of φ in Eq. (17)

N :

Nonlinear part of differential operator D

r :

Index for real

R :

Reynolds number

S :

Rest linear part of differential operator D

u :

Dependent variable in Eq. (3)

\(U\) :

Velocity of mean (base) flow 

z :

Transversal coordinate in Eq. (1), independent variable in Eq. (3)

α :

Axial wave-umber

α r :

Real axial wave-umber

λ :

Frequency

λ i :

Imaginary part of frequency λ

λ r :

Real part of frequency λ

φ :

Amplitude of velocity disturbance

Λ :

Coefficient in Eq. (14)

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Authors and Affiliations

  1. Department of Mechanical Engineering, Amirkabir University of Technology, 158754413, Tehran, Iran

    Nemat Dalir & S. Salman Nourazar

Authors
  1. Nemat Dalir
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  2. S. Salman Nourazar
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Correspondence to S. Salman Nourazar.

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Technical Editor: Francisco Ricardo Cunha.

An erratum to this article is available at http://dx.doi.org/10.1007/s40430-017-0838-1.

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Dalir, N., Nourazar, S.S. On absolute linear instability analysis of plane Poiseuille flow by a semi-analytical treatment. J Braz. Soc. Mech. Sci. Eng. 37, 495–505 (2015). https://doi.org/10.1007/s40430-014-0187-2

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