Linear stability of rimming flow
Author:
S. B. G. O’Brien
Journal:
Quart. Appl. Math. 60 (2002), 201-211
MSC:
Primary 76D08; Secondary 76A20, 76E17
DOI:
https://doi.org/10.1090/qam/1900490
MathSciNet review:
MR1900490
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Abstract: We consider the case of rimming flow where a thin film of viscous liquid coats the walls of a cylinder whose axis is horizontal and which is rotating with constant angular velocity. It has been experimentally established that such flows are often unstable and that the liquid often segregates into “rings” along the length of the tube. Using a leading-order lubrication theory, we utilise recently established steady solutions [10], which in some instances contain shocks, to examine the linear stability of the flow when subjected to two-dimensional disturbances. All solutions are shown to be at least neutrally stable. We suggest that further investigations should include higher-order (small) effects and that the origin of the observed instabilities lies in these terms.
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F. Melo, Localised states in a film dragging experiment, Phys. Rev. E 48, 2704-2712 (1993)
F. Melo, From solitary waves to static patterns via spatio-temporal intermittency, Physical Review Letters 79, 3283 (1993)
S. B. G. O’Brien and E. G. Gath, The location of a shock in rimming flow, Phys. Fluids 10, 1-3 (1998)
S. B. G. O’Brien, Marangoni drying: non-linear waves in a thin film, J. Fluid Mech. 254, 649-671 (1993)
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Y. Kuramoto, Diffusion induced chaos in reaction systems, Progress of Theoretical Physics 64, 346-367 (1978)
R. T. Balmer and T. G. Wang, An experimental study of internal hygrocysts, Journal of Fluids Engineering, 688-693 (December 1976)
R. T. Balmer, The hygrocyst, a stability phenomenon in continuum mechanics, Nature 227, 600-601 (1970)
J. P. Kovac and R. T. Balmer, Experimental studies of external hygrocysts, Journal of Fluids Engineering, 226-230 (June 1980)
M. J. Karweit and S. Corrsin, Observation of cellular patterns in a partly filled, horizontal, rotating cylinder, Phys. Fluids 18, 111-112 (1975)
S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, International Series of Monographs on Physics, Clarendon Press, Oxford, 1961
H. K. Moffatt, Behaviour of a viscous film on the outer surface of a rotating cylinder, Journal de Mécanique 16, 651-674 (1977)
R. E. Johnson, Steady state coating flows inside a rotating horizontal cylinder, J. Fluid Mech. 190, 321-342 (1988)
F. Melo, Localised states in a film dragging experiment, Phys. Rev. E 48, 2704-2712 (1993)
F. Melo, From solitary waves to static patterns via spatio-temporal intermittency, Physical Review Letters 79, 3283 (1993)
S. B. G. O’Brien and E. G. Gath, The location of a shock in rimming flow, Phys. Fluids 10, 1-3 (1998)
S. B. G. O’Brien, Marangoni drying: non-linear waves in a thin film, J. Fluid Mech. 254, 649-671 (1993)
G. B. Whitham, Linear and non-linear waves, John Wiley and Sons, 1974
Y. Kuramoto, Diffusion induced chaos in reaction systems, Progress of Theoretical Physics 64, 346-367 (1978)
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© Copyright 2002
American Mathematical Society
