# Stability of two-layered fluid flows in an inclined channel

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## 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## List of symbols

*A*Matrix of size

*N*×*N*in*D*^{2}method or 2*N*× 2*N*in*D*^{4}method*B*Matrix of size

*N*×*N*in*D*^{2}method or 2*N*× 2*N*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 (m

^{2}/s)*S*Surface tension (N/m)

*t*Time (s)

*X*Vector matrix

*N*× 1 in*D*^{2}method or 2*N*× 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

*a*_{j,k}Chebyshev coefficient of arbitrary variables

*j*and*k**c*_{i}Amplitude of the disturbance

*c*_{r}Frequency of the disturbance

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

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

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

*f*_{1}Constant

*f*_{2}Constant

- Q
_{c} Critical dimensionless flow rate through the channel

*U*_{1}Fully developed velocity of the bottom fluid

*U*_{2}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

*α*_{c}Critical wavenumber

*μ*_{j}Viscosity of fluid layer

*j*(Pa s);*j*= 1, 2 for bottom and top layer fluids, respectively*ρ*_{j}Density of fluid layer

*j*(kg/m^{3});*j*= 1, 2 for bottom and top layer fluids, respectively*υ*_{j}Kinematic viscosity of fluid layer

*j*(m^{2}/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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