Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T19:44:55.393Z Has data issue: false hasContentIssue false
  • English
  • Français

Lubricated pipelining: stability of core–annular flow. Part 4. Ginzburg–Landau equations

Published online by Cambridge University Press:  26 April 2006

Kangping Chen  and
Daniel D. Joseph
Kangping Chen
Affiliation:
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13676, USA
Daniel D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonlinear stability of core-annular flow near points of the neutral curves at which perfect core-annular flow loses stability is studied using Ginzburg-Landau equations. Most of the core-annular flows are always unstable. Therefore the set of core-annular flows having critical Reynolds numbers is small, so that the set of flows for which our analysis applies is small. An efficient and accurate algorithm for computing all the coefficients of the Ginzburg-Landau equation is implemented. The nonlinear flows seen in the experiments do not appear to be modulations of monochromatic waves, and we see no evidence for soliton-like structures. We explore the bifurcation structure of finite-amplitude monochromatic waves at criticality. The bifurcation theory is consistent with observations in some of the flow cases to which it applies and is not inconsistent in the other cases.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991 , pp. 587 - 615
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bai, R., Chen, K. & Joseph, D. D. 1991 Lubricated pipelining: stability of core—annular flow. Part 5. Experiments and comparison with theory. J. Fluid Mech. (submitted).Google Scholar
Bretherton, C. S. & Spiegel, E. A. 1983 Intermittency through modulational instability. Phys. Letts. 96 A (3), 152–156.Google Scholar
Charles, M. E., Govier, G. W. & Hodgson, G. W. 1961 The horizontal pipeline flow of equal density oil—water mixtures. Can. J. Chem. Engng 39, 17–36.Google Scholar
Chen, K. 1990 Lubricated pipelining: stability of core—annular flows. Ph.D thesis, University of Minnesota.
Chen, K., Bai, R. & Joseph, D. D. 1990 Lubricated pipelining. Part 3. Stability of core—annular flow in vertical pipes. J. Fluid Mech. 214, 251–286.Google Scholar
Chen, K. & Joseph, D. D. 1990 Application of the singular value decomposition to the numerical computation of the coefficients of amplitude equations and normal forms. Appl. Num. Math. 6, 425–430.Google Scholar
Craik, A. D. D. 1983 Wave Interactions and Fluid Flows. Cambridge University Press.
Davey, A., Hocking, L. M. & Stewartson, K. 1974 On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 63, 529–536.Google Scholar
Frenkel, A. L., Babchin, A. J., Levich, B. G., Shlang, T. & Sivashinsky, G. I. 1987 Annular flows can keep unstable films from breakup: Nonlinear saturation of capillary instability. J. Colloid Interface Sci. 115, 225–233.Google Scholar
Ginzburg, V. L. & Landau, L. D. 1950 Sov. Phys., J. Exp. Theor. Phys. 20, 1064.
Golub, G. H. & Van Loan, C. F. 1983 Matrix Computations. Baltimore: Johns Hopkins University Press.
Herbert, T. 1980 Nonlinear stability of parallel flows by higher order amplitude expansions. AIAA J. 18, 243–248.Google Scholar
Hocking, L. M. & Stewartson, K. 1972 On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance.. Proc. R. Soc. Lond. A 326, 289–313.Google Scholar
Hocking, L. M., Stewartson, K. & Stuart, J. T. 1972 A nonlinear instability burst in plane parallel flow. J. Fluid Mech. 51, 705–735.Google Scholar
Holmes, P. 1986 Spatial structure of time-periodic solutions of the Ginzburg–Landau equation. Physica 23 D, 84–90.Google Scholar
Hu, H. & Joseph, D. D. 1989 Lubricated pipelining: stability of core—annular flow. Part 2. J. Fluid Mech. 205, 359–396.Google Scholar
Hu, H., Lundgren, T. & Joseph, D. D. 1990 Stability of core—annular flow with a small viscosity ratio. Phys. Fluids A 2, 1945–1954.Google Scholar
Joseph, D. D., Renardy, Y. & Renardy, M. 1984 Instability of the flow of immiscible liquids with different viscosities in a pipe. J. Fluid Mech. 141, 319–345.Google Scholar
Joseph, D. D. & Sattinger, D. H. 1972 Bifurcating time periodic solutions and their stability. Arch. Rat. Mech. Anal. 45, 79–109.Google Scholar
Moon, H. T. 1982 Transition to chaos in the Ginzburg–Landau equation. Ph.D thesis, University of Southern California.
Moon, H. T., Huerre, P. & Redekopp, L. G. 1983 Transition to chaos in the Ginzburg–Landau equation. Physica 7 D, 135Google Scholar
Newell, A. C. 1974 Envelope equations In Lectures in Applied Mathematics, vol. 15, pp. 157–163.
Newell, A. C. 1985 Solitons in Mathematics and Physics (CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM publication).
Newell, A. C., Passot, T. & Souli, M. 1989 The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers I. Preprint.
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–304.Google Scholar
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core annular film flows.. Phys. Fluids A 2, 340–352.Google Scholar
Preziosi, L., Chen, K. & Joseph, D. D. 1989 Lubricated pipelining: stability of core—annular flow. J. Fluid Mech. 201, 323–356.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite amplitude instability of parallel shear flow. J. Fluid Mech. 27, 465–492.Google Scholar
Segel, L. A. 1969 Distance side-walls cause flow amplitude modulation of cellular convection. J. Fluid Mech. 38, 203–224.Google Scholar
Sen, P. K. & Venkateswarlu, D. 1983 On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients. J. Fluid Mech. 133, 179–206.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545.Google Scholar