Korea-Australia Rheology Journal

, Volume 27, Issue 3, pp 177–188 | Cite as

Efficient method to compute full eigenspectrum of incompressible viscous flows: Application on two-layer rectinear flow

  • Jaewook NamEmail author
  • Marcio S. Carvalho


An efficient algorithm based on the matrix transformation method (Valério et al., 2007) is presented for solving the generalized eigenvalue problem (GEVP) derived from linear stability analysis of incompressible viscous flow. The proposed method uses the formulation based on primitive variables, i.e. velocity and pressure, instead of streamfunction used by typical Orr-Sommerfeld equation. A series of matrix operations removes non-physical eigenvalues at infinity and leads to a non-singular smaller size eigenvalue problem (EVP), which contains full eigenspectrum, than the original GEVP. Two different solution strategies for the transformed EVP are proposed, and their accuracies are discussed. The proposed procedure is used to solve the stability of two layer rectilinear flow. The computed eigenspectrum are compared to previously reported values.


eigenvalues two-layer channel flow interlayer stability analysis interfacial mode finite element method 


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  1. Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenny, and D. Sorensen, 1999, LAPACK users guide Third Edition, SIAM.CrossRefGoogle Scholar
  2. Christodoulou, K.N. and L.E. Scriven, 1988, Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem, J. Sci. Comput. 3, 355–406.CrossRefGoogle Scholar
  3. Drazin, P.G. and W.H. Reid, 2004, Hydrodynamic stability, 2nd ed., Cambridge University Press, Cambridge, pp. 120–202.CrossRefGoogle Scholar
  4. Gary, J. and R. Helgason, 1970, A matrix method for ordinary differential eigenvalue problems, J. Comput. Phys. 5, 169–187.CrossRefGoogle Scholar
  5. Golub, G.H. and C.F. Van Loan, 1996, Matrix computation, The Johns Hopkins University Press, Baltimore.Google Scholar
  6. Goussis, D.A. and A.J. Perlstein, 1998, Removal of infinite eigenvalue in the generalized matrix problem, J. Comput. Phys. 84, 242–246.CrossRefGoogle Scholar
  7. Hu, H.H. and D.D. Joseph, 1989, Lubricated pipelining: stability of core-annular flow. Part 2, J. Fluid Mech. 205, 359–396.CrossRefGoogle Scholar
  8. Kaufman, L., 1975, The LZ algorithm to solve the generalized eigenvalue problem for complex matrices, ACM T. Math. Software 1, 271–281.CrossRefGoogle Scholar
  9. Lee, J.H., S.K. Han, J.S. Lee, H.W. Jung, and J.C. Hyun, 2010, Ribbing instability in rigid and deformable forward roll coating flows, Korea-Aust. Rheol. J. 22, 75–80.Google Scholar
  10. Li, Y.S. and S.C. Kot, 1981, One-dimensional finite element method in hydrodynamic stability, Int. J. Numer. Meth. Eng. 17, 853–870.CrossRefGoogle Scholar
  11. Nam, J. and M.S. Carvalho, 2010, Linear stability analysis of two-layer rectilinear flow in slot coating, AIChE J. 56, 2503–2512.CrossRefGoogle Scholar
  12. Ng, B.S. and W.H. Reid, 1979, A numerical method for linear two-point boundary-value problems using compound matrices, J. Comput. Phys. 33, 70–85.CrossRefGoogle Scholar
  13. Vinokur, M. On one-dimensional stretching functions for finitedifference calculations, J. Comput. Phys. 50, 215–234.Google Scholar
  14. Orszag, S., 1971, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech. 50, 689–703.CrossRefGoogle Scholar
  15. Saad, Y., 1984, Chebyshev accerlation techniques for solving nonsymmetric eigenvalue problems, Math. Comput. 42, 567–588.CrossRefGoogle Scholar
  16. Saad, Y., 1989, Numerical solution of large nonsymmetric eigenvalue problems, Comput. Phys. Commun. 54, 71–90.CrossRefGoogle Scholar
  17. Severtson, Y.C. and C.K. Aidun, 1996, Stability of two-layer stratified flow in inclined channels: applications to air entrainment in coating flow system, J. Fluid Mech. 312, 173–200.CrossRefGoogle Scholar
  18. Su, Y.Y. and B. Khomami, 1992, Numerical solution of eigenvalue problems using spectral techniques, J. Comput. Phys. 100, 297–305.CrossRefGoogle Scholar
  19. Valério, J., M.S. Carvalho, and C. Tomei, 2007, Filtering the eigenvalues at infinite from the linear stability analysis of incompressible flows, J. Comput. Phys. 227, 229–243.CrossRefGoogle Scholar
  20. Valério, J., M.S. Carvalho, and C. Tomei, 2009, Efficient computation of the spectrum of viscoelastic flows, J. Comput. Phys. 228, 1172–1187.CrossRefGoogle Scholar
  21. Yiantsios, S.G. and B.G. Higgins, 1986, Analysis of superposed fluids by the finite element method: Linear stability and flow development, Int. J. Numer. Meth. Eng. 7, 247–261.CrossRefGoogle Scholar
  22. Yiantsios, S.G. and B.G. Higgins, 1988a, Numerical solution of eigenvalue problem using the compound matrix method, J. Comput. Phys. 74, 25–40.CrossRefGoogle Scholar
  23. Yiantsios, S.G. and B.G. Higgins, 1988b, Linear stability of plane Poiseuille flow of two superimposed fluids, Phys. Fluids 31, 3225–3238.CrossRefGoogle Scholar
  24. Yih, C., 1967, Instability due to viscosity stratification, J. Fluid Mech. 27, 337–351.CrossRefGoogle Scholar

Copyright information

© Korean Society of Rheology (KSR) and the Australian Society of Rheology (ASR) and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Chemical EngineeringSungkyunkwan UniversitySuwon, Gyeonggi-doRepublic of Korea
  2. 2.Department of Mechanical EngineeringPontifícia Universidade Católica do Rio de JaneiroGávea, Rio de JanerioBrazil

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