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The nonlinear growth of surface-tension-driven instabilities of a thin annular film

Published online by Cambridge University Press:  26 April 2006

Mark Johnson ,
Roger D. Kamm ,
Lee Wing Ho ,
Ascher Shapiro  and
T. J. Pedley
Mark Johnson
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Roger D. Kamm
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Lee Wing Ho
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Ascher Shapiro
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
T. J. Pedley
Affiliation:
Department of Applied Mathematical Studies, The University of Leeds, Leeds LS2 9JT, UK
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Abstract

The stability and initial growth rate of disturbances on an annular film lining a cylindrical tube have been the focus of several previous works. The further development of thsse disturbances as they grow to form stable unduloids or liquid bridges is investigated by means of a thin-film integral model. The model is compared both with perturbation theories for early times, and a numerical solution of the exact equations (NEKTON) for later times. The thin-film model gave results that were in good agreement with solutions of the exact equations. The results show that linear perturbation theory can be used to give good estimates of the times for unduloid and liquid bridge formation. The success of the model derives from the dominant influence of narrow draining regions that feed into the growing unduloid, and these regions remain essentially one-dimensional throughout the growth of the instability.

The model is used to analyse the evolution of the liquid layer lining the small airways of the lung during a single breath. The timescales for formation of unduloids and liquid bridges are found to be short enough for the liquid layer to be in a virtually quasi-equilibrium state throughout the breathing cycle. This conclusion is only tentative, however, because the model assumes that the surface tension of the airway liquid lining does not change with changes in interfacial area despite the known presence of pulmonary surfactant.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 233 , December 1991 , pp. 141 - 156
Copyright
© 1991 Cambridge University Press

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References

Bogy, D. B. 1979 Drop formation in a circular liquid jet. Ann. Rev. Fluid Mech. 11, 207.Google Scholar
Carey, G. F. & Oden, J. T. 1986 Finite Elements: Fluid Mechanics, vol. vi. Prentice-Hall.
Everett, D. H. & Haynes, J. M. 1972 Model studies of capillary condensation: I. Cylindrical pore model with zero contact angle. J. Colloid Interface Sci. 38, 125.Google Scholar
Frazer, D. G. & Khoshnood, B. 1979 A model of the gas trapping mechanism in excised lungs. Proc. 7th New England Bioengng Conf., vol. 9, pp. 482.
Gauglitz, P. A. & Radke, C. J. 1988 An extended evolution equation for liquid film breakup in cylindrical capillaries. Chem. Engng Sci. 43, 1457.Google Scholar
Girault, V. & Raviart, P. A. 1986 Finite Element Approximation of the Navier—Stokes Equations. Springer.
Goren, S. L. 1962 The stability of an annular thread of fluid. J. Fluid Mech. 12, 309.Google Scholar
Goren, S. L. 1964 The shape of a thread of liquid undergoing break-up. J. Colloid Sci. 19, 81.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Methods of Spectral Methods: Theory and Applications. SIAM.
Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363.Google Scholar
Ho, L. W. 1989 A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows. Ph.D. thesis, MIT.
Ho, L. W. & Patera, A. T. 1990a A Legendre spectral element method for stimulation of unsteady incompressible viscous free-surface flows. Comput. Meth. Appl. Mech. Engng 80, 355–366.Google Scholar
Ho, L. W. & Patera, A. T. 1990b Variational formulation of three-dimensional viscous free-surface flows: natural imposition of surface tension boundary conditions. Intl J. Numer. Meth. Fluids (to appear).Google Scholar
Kamm, R. D. & Schroter, R. C. 1989 Is airway closure caused by a liquid film instability? Respir. Physiol. 75, 141.Google Scholar
Kheshgi, H. S. 1989 Profile equations for film flows at moderate Reynolds numbers. AIChE J. 35, 1719.Google Scholar
Macklem, P. T. 1971 Airway obstruction and collateral ventilation. Physiol. Rev. 51, 368–385.Google Scholar
Maday, Y. & Patera, A. T. 1989 Spectral element methods for the Navier—Stokes equations. In State-of-the-art Surveys on Computational Mechanics (ed. J. T. Oden & A. K. Norr). ASME.
Patera, A. T. 1984 A spectral element method for fluid dynamics; laminar flow in a channel expansion. J. Comput. Phys. 54, 468.Google Scholar
Rayleigh, Lord 1879 On the capillary phenomena of jets. Appendix I.. Proc. R. Soc. Lond. A 29, 71.Google Scholar
Rayleigh, Lord 1902 On the instability of cylindrical fluid surfaces. Scientific Papers, vol. 3. pp. 594–596. Cambridge University Press.
Rønquist, E. M. 1988 Optimal spectral element methods for the unsteady three-dimensional incompressible Navier—Stokes equations, Ph.D. thesis, MIT.
Ruschak, K. J. 1978 Flow of a falling film into a pool. AIChE J. 24, 705.Google Scholar
Weibel, E. R. 1963 Morphometry of the Human Lung. Springer.