Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-20T05:43:12.605Z Has data issue: false hasContentIssue false
  • English
  • Français

An asymptotic theory for the linear stability of a core–annular flow in the thin annular limit

Published online by Cambridge University Press:  26 April 2006

E. Georgiou ,
C. Maldarelli ,
D. T. Papageorgiou  and
D. S. Rumschitzki
E. Georgiou
Affiliation:
Levich Institute and Chemical Engineering Department, City College of New York, New York, NY 10031, USA
C. Maldarelli
Affiliation:
Levich Institute and Chemical Engineering Department, City College of New York, New York, NY 10031, USA
D. T. Papageorgiou
Affiliation:
Department of Mathematics, New Jersey Institute of Technology, Newark, NJ 07102, USA
D. S. Rumschitzki
Affiliation:
Levich Institute and Chemical Engineering Department, City College of New York, New York, NY 10031, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the linear stability of a vertical, perfectly concentric, core–annular flow in the limit in which the gap is much thinner than the core radius. The analysis includes the effects of viscosity and density stratification, interfacial tension, gravity and pressure-driven forcing. In the limit of small annular thicknesses, several terms of the expression for the growth rate are found in order to identify and characterize the competing effects of the various physical mechanisms present. For the sets of parameters describing physical situations they allow immediate determination of which mechanisms dominate the stability. Comparisons between the asymptotic formula and available full numerical computations show excellent agreement for non-dimensional ratio of undisturbed annular thickness to core radius as large as 0.2.

The expansion leads to new linear stability results (an expression for the growth rate in powers of the capillary number to the $\frac{1}{3}$ power) for wetting layers in low-capillary-number liquid–liquid displacements. The expression includes both capillarity and viscosity stratification and agrees well with the experimental results of Aul & Olbrich (1990).

Finally, we derive Kuramoto–Sivashinsky-type integro-differential equation for the later nonlinear stages of the interfacial dynamics, and discuss their solutions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 243 , October 1992 , pp. 653 - 677
Copyright
© 1992 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

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.
Aul, R. W. & Olbricht, W. L. 1990 Stability of a thin annular film in pressure-driven, low-Reynolds-number flow through a capillary. J. Fluid Mech. 215, 585–599.Google Scholar
Bai, R., Chen, K. & Joseph, D. D. 1990 Lubricated pipelining: stability of core—annular flow. Part 5. Experiments and comparison with theory. J. Fluid Mech. 240, 97–132.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Chandrasekhar, S. 1968 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Chen, K. P. 1992 Short wave instabilities of core annular flow.. Phys. Fluids A 4, 186–188.Google Scholar
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 (referred to herein as CBJ).Google Scholar
Chen, K. & Joseph, D. D. 1991 Long waves and lubrication theories for core annular flow.. Phys. Fluids A 3, 2672–2679.Google Scholar
Gauglitz, P. A. & Radke, C. J. 1990 The dynamics of liquid film breakup in constricted cylindrical capillaries. J. Colloid Interface Sci. 134, 14–40.Google Scholar
Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 27, 309–319.Google Scholar
Hickox, C. E. 1971 Instability due to viscosity and density stratification in axisymmetric pipe flow., Phys. Fluids 14, 251–262.Google Scholar
Hu, H. H. & Joseph, D. D. 1989 Lubricated pipelines: 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, 309–317.Google Scholar
Newhouse, L. A. & Pozrikidis, C. 1992 The capillary instability of annular layers and liquid threads. J. Fluid Mech. 242, 193–209.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
Papageorgiou, D. T. & Smith, F. T. 1988 Nonlinear instability of the wake behind a flat plate placed parallel to a uniform stream.. Proc. R. Soc. Lond. A 419, 1–28.Google Scholar
Park, C. W. & Homsy, G. M. 1984 Two-phase displacement in Hele—Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Pekeris, C. L. 1948 Stability of the laminar flow through a straight pipe of circular cross-section to infinitesimal disturbances which are symmetrical about the axis of the pipe. Proc. Natl Acad. Sci. USA 34, 285–295.Google Scholar
Preziosi, K., Chen, K. & Joseph, D. D. 1989 Lubricated pipelines: Stability of core—annular flow. J. Fluid Mech. 201, 323–356.Google Scholar
Ratolowski, J. & Change, H. C. 1989 In Snap-off at strong constrictions: effect of pore geometry. In Surface-Based Mobility Control: Progress in Miscible Flood Enhanced Oil Recovery (ed. D. H. Smith). ACS Symposium Series, vol. 33, pp. 282–314. Hemisphere.
Smith, M. K. 1989 The axisymmetric long wave instability of a concentric two-phase pipe flow.. Phys. Fluids A 1, 494–506.Google Scholar
Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous liquid.. Proc. R. Soc. Lond. A 150, 322–337.Google Scholar
Yih, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.Google Scholar