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Instabilities of longitudinal rolls in the presence of Poiseuille flow

Published online by Cambridge University Press:  26 April 2006

R. M. Clever  and
F. H. Busse
R. M. Clever
Affiliation:
Institute of Geophysics and Planetary Physics, University of California at Los Angeles, CA 90024, USAand Institute of Physics, University of Bayreuth, 858 Bayreuth, Germany
F. H. Busse
Affiliation:
Institute of Geophysics and Planetary Physics, University of California at Los Angeles, CA 90024, USAand Institute of Physics, University of Bayreuth, 858 Bayreuth, Germany
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Abstract

Longitudinal rolls represent the preferred form of convection in a horizontal fluid layer heated from below in the presence of parallel shear flows for sufficiently low Reynolds numbers and for a finite range of the Rayleigh number above the critical value Rac. In this paper properties of the longitudinal rolls and their stability with respect to three-dimensional disturbances are investigated in the case of Poiseuille flow. While the convective heat transport is independent of the Reynolds number, the mass flux through the channel at a given Reynolds number decreases with increasing Rayleigh number. A wavy instability is found to set in at a finite Reynolds number and relatively low Rayleigh numbers, depending on the Prandtl number P. In particular, the stability region for longitudinal rolls is analysed for P = 0.025, 0.1, 0.71, 2.5, and 7. For sufficiently small Reynolds number the oscillatory, the skewed varicose or the knot instability can precede the wavy instability. For P = 7 the wavy instability is preceded by a modified knot instability throughout the Reynolds-number range that has been investigated. In spite of the difference hi symmetry, the results for Poiseuille flow resemble those obtained earlier hi the case of plane Couette flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 517 - 529
Copyright
© 1991 Cambridge University Press

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References

Alekseenko, S. V., Nakoryakov, V. Ye. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460.Google Scholar
Bach, P. & Villadsen, J. 1984 Simulation of the vertical flow of a thin wavy film using a finite-elements method. Intl J. Heat Mass Transfer 27, 815–827.Google Scholar
Benjamin, G. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.Google Scholar
Benney, D. J. 1966 Long waves in liquid films. J. Math. Phys. 45, 150–155.Google Scholar
Berbente, C. P. & Ruckenstein, E. 1968 Hydrodynamics of wave flow. AIChE J. 14, 772–782.Google Scholar
Bunov, A. V., Demekhin, Ye. A. & Shkadov, V. Ya. 1984 Nonuniqueness of wave solutions in viscous layer. Prikl. Mekh. i Mat. 48, 691–696.Google Scholar
Chang, H.-C. 1987 Evolution of nonlinear waves on vertically falling films — A normal form analysis. Chem. Engng Sci. 42, 515–533.Google Scholar
Chang, H.-C. 1989 Onset of nonlinear waves on falling films.. Phys. Fluids A 1, 1314–1327.Google Scholar
Demekhin, Ye. A., Demekhin, I. A. & Shkadov, V. Ya. 1983 Solitons in flowing layer of a viscous fluid. Izv. Akad. Nauk. SSSR, Mekh. Zhid. i Gasa 4, 9–16.Google Scholar
Geshev, P. I. & Ezdin, B. S. 1985 Calculation of the velocity profile and that of the wave shape of flowing liquid film. In Hydrodynamics and Heat-Mass Transfer of the Liquid Flow with Free Surface (ed. I. R. Shreyber), pp. 49–58. Inst. Thermophysics, Novosibirsk.
Gjevik, B. 1970 Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1918–1925.Google Scholar
Kapitza, P. L. 1948 Wave flow of thin viscous liquid films. Zh. Teor. Fiz. 18, 3–28.Google Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin viscous liquid films. Zh. Teor. Fiz. 19, 105–120.Google Scholar
Krantz, W. B. & Goren, S. L. 1971 Stability of thin liquid films flowing down a plane. Indust. Engng Chem. Fundam. 10, 91–101.Google Scholar
Levich, V. G. 1959 Physics-Chemical Hydrodynamics. Moscow, 699 pp.
Nakoryakov, V. E., et al. 1977 Instantaneous velocity profiles in a wave liquid film. Inzh. Fiz. J., SSSR 33, 399–405.Google Scholar
Nakoryakov, V. E., Pokusaev, B. G. & Alekseenko, S. V. 1981 Desorbtion of weakly-solution gas from flowing liquid films. In Calculation of Heat and Mass Transfer in Energy-Chemical Processes (ed. A. P. Burdukov), pp. 23–36. Institute of Thermophysics, Novosibirsk.
Nepomnyashchy, A. A. 1974 Wave regime stability in a film falling down inclined surfaces. Izv. Akad. Nauk. SSSR, Mekh. Zhid. i Gasa 3, 28–34.Google Scholar
Nusselt, W. 1916 Die oberflachenkondesation des Wasserdampfes. Teil I, II. Z.VDI. 27, 541; 28, 569.Google Scholar
Shkadov, V. Ya. 1967 Wave modes in the gravity flow of a thin layer of a viscious fluid. Izv. Akad. Nauk. SSSR, Mekh. Zhid. i Gasa 1, 43–51.Google Scholar
Shkadov, V. Ya. 1968 On the theory of wave flows of a thin layer of a viscous fluid. Izv. Akad. Nauk. SSSR, Mekh. Zhid. i Gasa 3, 20–25.Google Scholar
Shkadov, V. Ya. 1977 Single waves in the thin layer of a viscous fluid. Izv. Akad. Nauk. SSSR, Mekh. Zhid. i Gasa 1, 63–66.Google Scholar
Sivashinsky, G. I. & Michelson, O. M. 1980 On irregular wavy flow of a liquid film down a vertical plate. Prog. Theor. Phys. 63, 2112–2114.Google Scholar
Tsvelodub, O. Yu. 1980 Stationary running waves on a film falling down inclined surfaces. Izv. Akad. Nauk. SSSR, Mekh. Zhid. i Gasa 4, 142–146.Google Scholar
Tsvelodub, O. Yu. & Trifonov, Yu. Ya. 1989 On steady-state travelling solutions of an evolution equation describing the behaviour of disturbances in active dissipative media. Physica D 39, 336–351.Google Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–335.Google Scholar