Hostname: page-component-6bf8c574d5-h6jzd Total loading time: 0.001 Render date: 2025-02-23T21:08:49.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Measurements of the primary instabilities of film flows

Published online by Cambridge University Press:  26 April 2006

Jun Liu ,
Jonathan D. Paul  and
J. P. Gollub
Jun Liu
Affiliation:
Department of Physics, Haverford College, Haverford, PA 19041, USAand Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA
Jonathan D. Paul
Affiliation:
Department of Physics, Haverford College, Haverford, PA 19041, USAand Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA
J. P. Gollub
Affiliation:
Department of Physics, Haverford College, Haverford, PA 19041, USAand Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We present novel measurements of the primary instabilities of thin liquid films flowing down an incline. A fluorescence imaging method allows accurate measurements of film thickness h(x, y, t) in real time with a sensitivity of several microns, and laser beam deflection yields local measurements with a sensitivity of less than one micron. We locate the instability with good accuracy despite the fact that it occurs (asymptotically) at zero wavenumber, and determine the critical Reynolds number Rc for the onset of waves as a function of angle β. The measurements of Rc(β) are found to be in good agreement with calculations, as are the growth rates and wave velocities. We show experimentally that the initial instability is convective and that the waves are noisesustained. This means that the waveform and its amplitude are strongly affected by external noise at the source. We investigate the role of noise by varying the level of periodic external forcing. The nonlinear evolution of the waves depends strongly on the initial wavenumber (or the frequency f). A new phase boundary f*s(R) is measured, which separates the regimes of saturated finite amplitude waves (at high f) from multipeaked solitary waves (at low f). This boundary probably corresponds approximately to the sign reversal of the third Landau coefficient in weakly nonlinear theory. Finally, we show that periodic waves are unstable over a wide frequency band with respect to a convective subharmonic instability. This instability leads to disordered two-dimensional waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 250 , May 1993 , pp. 69 - 101
Copyright
© 1993 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

Agrawal, S. K. & Lin, S. P. 1975 Nonlinear spatial instability of a film coating on a plate. Trans. ASME E: J. Appl. Mech. 42, 580–583.Google Scholar
Alekseenko, S. V., Nakoryakov, V. Y. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460.Google Scholar
Anshus, B. E. & Goren, S. L. 1966 A method of getting approximate solutions to the Orr–Sommerfeld equation for flow on a vertical wall. AIChE J. 12, 1004–1008.Google Scholar
Babcock, K. L., Ahlers, G. & Cannell, D. S. 1991 Noise-sustained structure in Taylor–Couette flow with through-flow. Phys. Rev. Lett. 67, 3388–3391.Google Scholar
Babcock, K. L., Cannell, D. S. & Ahlers, G. 1992 Stability and noise in Taylor–Couette flow with through-flow. Preprint.
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.Google Scholar
Benjamin, T. B. 1961 The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane. J. Fluid Mech. 10, 401–419.Google Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150–155.Google Scholar
Bertschy, J. R., Chin, R. W. & Abernathy, F. H. 1983 High-strain-rate free-surface boundary-layer flows. J. Fluid Mech. 126, 443–461.Google Scholar
Brauner, N. & Maron, D. M. 1982 Characteristics of inclined thin films, waviness and the associated mass transfer. Intl J. Heat Mass Transfer 25, 99–110.Google Scholar
Chang, H.-C. 1989 Onset of nonlinear waves on falling films. Phys. Fluids A 1, 1314–1327.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1993 Nonlinear evolution of waves on a falling film. J. Fluid Mech. (in press).Google Scholar
Cheng, M. & Chang, H.-C. 1992 Subharmonic instabilities of finite-amplitude monochromatic waves. Phys. Fluids A 4, 505–523.Google Scholar
Chin, R. W., Aberhathy, F. F. & Bertschy, J. R. 1986 Gravity and shear wave stability of free surface flows. Part 1. Numerical calculations. J. Fluid Mech. 168, 501–513.Google Scholar
Chu, K. J. & Dukler, A. E. 1974 Statistical characteristics of thin wavy films, Part II: Studies of the substrate and its wave structure. AIChE J. 20, 695–706.Google Scholar
Chu, K. J. & Dukler, A. E. 1975 Statistical characteristics of thin wavy films, Part III: Structure of the large waves and their resistance to gas flow. AIChE J. 21, 583–595.Google Scholar
Deissler, R. J. 1987a Spatially growing waves, intermittency, and convective chaos in an open flow system. Physica D 25, 233–260.Google Scholar
Deissler, R. J. 1987b The convective nature of instability in plane Poiseuille flow. Phys. Fluids 30, 2303–2305.Google Scholar
Deissler, R. J. 1989 External noise and the origin and dynamics of structure in convective unstable systems. J. Statist. Phys. 54, 1459–1488.Google Scholar
Deissler, R. J., Oron, A. & Lee, Y. C. 1991 Evolution of two-dimensional waves in externally perturbed flow on a vertical cylinder. Phys. Rev. A 43, 4558–4561.Google Scholar
Deissler, R. J. & To, W.-M. 1992 Noise-sustained structure in the Navier–Stokes equations: Taylor–Couette flow with through-flow. Preprint.
Dukler, A. E. 1972 Characterization, effects and modeling of the wavy gas–liquid interface. In Progress in Heat and Mass Transfer (ed. G. Hetsroni, S. Sideman & J. P. Hartnet), vol. 6, pp. 207–234. Pergamon.
Fulford, G. D. 1964 The flow of liquids in thin films. Adv. Chem. Engng 5, 151–236.Google Scholar
Gjevik, B. 1970 Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1918–1925.Google Scholar
Gjevik, B. 1971 Spatially varying finite-amplitude wave trains on falling liquid films. Acta Polytech. Scand. Me 61, 1–16.Google Scholar
Goussis, D. A. & Kelly, R. E. 1991 Surface wave and thermocapillary instabilities in a liquid film flow. J. Fluid Mech. 223, 25–45.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473–537.Google Scholar
Joo, S. W. & Davis, S. H. 1992a Instabilities of three-dimensional viscous falling films. J. Fluid Mech. 242, 529–547.Google Scholar
Joo, S. W. & Davis, S. H. 1992b Irregular waves on viscous falling films. Chem. Engng Commun. 118, 111–123.Google Scholar
Joo, S. W., Davis, S. H. & Bankoff, S. G. 1991 Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers. J. Fluid Mech. 230, 117–146.Google Scholar
Kapitza, P. L. 1948 Wave flow of thin layers of a viscous fluid: I. The free flow. Zh. Exp. Teor. Fiz. 18, 3. Also in Collected Papers of P. L. Kapitza (ed. D. Ter Haar), vol. 2, pp. 662–679. Pergamon, 1965.Google Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin layers of a viscous fluid: III. Experimental study of undulatory flow conditions. Zh. Exp. Teor. Fiz. 19, 105. Also in Collected Papers of P. L. Kapitza (ed. D. Ter Haar), vol. 2, pp. 690–709. Pergamon, 1965.Google Scholar
Kelly, R. E., Goussis, D. A., Lin, S. P. & Hsu, F. K. 1989 The mechanism for surface wave instability in film flow down an inclined plane. Phys. Fluids A 1, 819–828.Google Scholar
Koehler, R. 1968 Dissertation, Georg-August-Universität, Gottingen.
Krantz, W. B. & Goren, S. L. 1970 Finite-amplitude, long waves on liquid films flowing down a plane. Ind. Engng Chem. Fundam. 9, 107–113.Google Scholar
Krantz, W. B. & Goren, S. L. 1971a Stability of thin liquid films flowing down a plane. Ind. Engng Chem. Fundam. 10, 91–101.Google Scholar
Krantz, W. B. & Goren, S. L. 1971b Bimodal wave formation on thin liquid films flowing down a plane. AIChE J. 17, 494–496.Google Scholar
Krantz, W. B. & Owens, W. B. 1973 Spatial formulation of the Orr–Sommerfeld equation for thin liquid films flowing down a plane. AIChE J. 19, 1163–1169.Google Scholar
Lacy, C. E., Sheintuch, M. & Dukler, A. E. 1991 Methods of deterministic chaos applied to the flow of thin wavy films. AIChE J. 37, 481–489.Google Scholar
Lee, J. 1969 Kapitza's method of film flow description. Chem. Engng Sci. 24, 1309–1320.Google Scholar
Lin, S. P. 1969 Finite-amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113–126.Google Scholar
Lin, S. P. 1974 Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417–429.Google Scholar
Lin, S. P. & Wang, C. Y. 1985 Modeling wavy film flows. In Encyclopedia of Fluid Mechanics (ed. N. P. Cheremisinoff), vol. 1, pp. 931–951. .
Liu, J. & Gollub, J. P. 1993 Onset of spatially chaotic waves on flowing films. Phys. Rev. Lett. (submitted.)Google Scholar
Liu, J., Paul, J. D., Banilower, E. & Gollub, J. P. 1992 Film flow instabilities and spatiotemporal dynamics. In Proc. First Experimental Chaos Conf. (ed. S. Vohra, M. Spano, M. Shlesinger, L. M. Pecora & W. Ditto), pp. 225–239. World Scientific.
Nakaya, C. 1975 Long waves on a thin fluid layer flowing down an inclined plane. Phys. Fluids 18, 1407–1420.Google Scholar
Nakaya, C. 1989 Waves on a viscous fluid film down a vertical wall. Phys. Fluids A 1, 1143–1154.Google Scholar
Pierson, F. W. & Whitaker, S. 1977 Some theoretical and experimental observations of the wave structure of falling liquid films. Ind. Engng Chem. Fundam. 16, 401–408.Google Scholar
Prokopiou, T., Cheng, M. & Chang, H.-C. 1991 Long waves on inclined films at high Reynolds number. J. Fluid Mech. 222, 665–691.Google Scholar
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.Google Scholar
Roskes, G. J. 1970 Three dimensional long waves on a liquid film. Phys. Fluids 13, 1440–1445.Google Scholar
Schatz, M. F., Tagg, R. P. & Swinney, H. L. 1991 Supercritical transition in plane channel flow with spatially periodic perturbations. Phys. Rev. Lett. 66, 1579–1582.Google Scholar
Shkadov, V. Y. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Izv. Akad. Nauk. SSSR, Mekh. Zhid. Gaza, No. 1, 43–51. (English translation: Fluid Dyn. 2, 29–34.)Google Scholar
Sivashinsky, G. I. & Michelson, D. M. 1980 On irregular wavy flow of a liquid film down a vertical plane. Prog. Theor. Phys. 63, 2112–2114.Google Scholar
Steinberg, V. & Tsameret, A. 1991 Noise-modulated propagating pattern in a convectively unstable system. Phys. Rev. Lett. 67, 3392–3395.Google Scholar
Tailby, S. R. & Portalski, S. 1962 The determination of the wavelength on a vertical film of liquid flowing down a hydrodynamically smooth plate. Trans. Inst. Chem. Engrs 40, 114–122.Google Scholar
Trifonov, Y. Y. 1992 Steady-state traveling waves on the surface of a viscous liquid film falling down on vertical wires and tubes. AIChE J. 38, 821–834.Google Scholar
Trifonov, Y. Y. & Tsvelodub, O. Y. 1991 Nonlinear waves on the surface of a falling liquid film. Part 1. Waves of the first family and their stability. J. Fluid Mech. 229, 531–554.Google Scholar
Tsvelodub, O. Y. 1980 Stationary travelling waves on a film falling down an inclined plane. Izv. Akad. Nauk. SSSR, Mekh. Zhid. Gaza, No. 4, 142–146.(English translation: Fluid Dyn. 15, 591–594.)Google Scholar
Whitaker, S. 1964 Effect of surface active agents on stability of falling liquid films. Ind. Engng Chem. Fundam. 3, 132–142.Google Scholar
Yih, C. S. 1955 Stability of parallel laminar flow with a free surface. In Proc. 2nd US Congr. on Applied Mechanics, pp. 623–628. ASME.
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–330.Google Scholar