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Acta Mechanica

, Volume 209, Issue 3–4, pp 187–199 | Cite as

Stability of two-layered fluid flows in an inclined channel

  • Bhadraiah VempatiEmail author
  • Alparslan Oztekin
  • Sudhakar Neti
Article

Abstract

A linear stability analysis of two-layer fluid flows in an inclined channel geometry has been carried out. The onset of flow transitions and the spatio-temporal characteristics of secondary flows produced by the flow instabilities have been examined. The effects of density and viscosity stratifications and surface tension on flow structures also have been investigated at various values of Froude numbers (channel inclinations). Multi-domain Chebyshev–Tau spectral methods along with MATLAB QZ eigenvalue solver are used to determine the whole spectrum of the eigenvalues and associated eigenfunctions. The neutral stability diagrams and stability boundaries are constructed for various values of flow parameters. The onset of flow transitions and flow structures predicted by linear stability analysis are compared against experimental results and they agree reasonably well. The results presented in the present paper imply that the shear mode of flow transitions is the one likely to be identified in experiments.

Keywords

Froude Number Viscosity Ratio Linear Stability Analysis Incline Plane Shear Mode 
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

Matrix of size N × N in D 2 method or 2N × 2N in D 4 method

B

Matrix of size N × N in D 2 method or 2N × 2N in D 4 method

c

Complex wave speed (= c r  + ic i )

D

Derivate with respect to \({z, \frac{{d}}{{d}z}}\)

d

Height of the channel (m)

G

Froude number, \({\frac{gd^{3}}{\upsilon_1^2}}\)

g

Acceleration due to gravity (ms−2)

h

Dimensionless fully developed interface height

M

Number of Chebyshev Polynomials

N

Size of the matrix

Q

Dimensionless flow rate through the channel

q

Flow rate through the channel (m2/s)

S

Surface tension (N/m)

t

Time (s)

X

Vector matrix N × 1 in D 2 method or 2N × 1 in D 4 method

x

Axial co-ordinate direction

z

Vertical co-ordinate direction

H(x, t)

Fully developed interface height w.r.t x and t

\({\begin{array}{l} a_{11},\,a_{12},\\ a_{21},\,a_{22} \end{array}}\)

Constants

aj,k

Chebyshev coefficient of arbitrary variables j and k

ci

Amplitude of the disturbance

cr

Frequency of the disturbance

D2

Derivate with respect to \({z,\, \frac{{d}^{2}}{{d}z^{2}}}\)

D3

Derivate with respect to \({z,\, \frac{{d}^{3}}{{d}z^{3}}}\)

D4

Derivate with respect to \({z,\, \frac{{d}^{4}}{{d}z^{4}}}\)

f1

Constant

f2

Constant

Qc

Critical dimensionless flow rate through the channel

U1

Fully developed velocity of the bottom fluid

U2

Fully developed velocity of the top fluid

μ

Ratio of top layer and bottom layer fluid viscosities, \({\frac{\mu_2}{\mu_1}}\)

ρ

Ratio of top layer and bottom layer fluid densities, \({\frac{\rho_2}{\rho_1}}\)

υ

Ratio of top layer and bottom layer fluid kinematic viscosities, \({\frac{\upsilon_2}{\upsilon_1}}\)

β

Angle of the inclined channel with axial direction, degrees [radians in Eqs. (2.1, z)]

σ

Dimensionless surface tension (or Inverse Weber number), \({\frac{Sd}{\rho_1\upsilon_1^2}}\)

κ

Curvature (m−1)

α

Wave number

αc

Critical wavenumber

μj

Viscosity of fluid layer j (Pa s); j = 1, 2 for bottom and top layer fluids, respectively

ρj

Density of fluid layer j (kg/m3); j = 1, 2 for bottom and top layer fluids, respectively

υj

Kinematic viscosity of fluid layer j (m2/s); j = 1, 2 for bottom and top layer fluids, respectively

\({\phi_j}\)

Stream function of the fluid layer’s disturbance; j = 1, 2 for bottom and top layer fluids, respectively

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Bhadraiah Vempati
    • 1
    Email author
  • Alparslan Oztekin
    • 1
  • Sudhakar Neti
    • 1
  1. 1.Lehigh UniversityBethlehemUSA

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