A mechanism for linear instability in two-dimensional rimming flow
Author:
S. B. G. O’Brien
Journal:
Quart. Appl. Math. 60 (2002), 283-299
MSC:
Primary 76D08; Secondary 76A20, 76D45, 76E17
DOI:
https://doi.org/10.1090/qam/1900494
MathSciNet review:
MR1900494
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Abstract: In rimming flow, a thin film of viscous liquid coats the inside of a cylinder whose axis is horizontal and which is rotating with constant angular velocity. It has been established experimentally that such flows are often unstable with a variety of secondary flow regimes having been observed experimentally [15]. We use a lubrication approximation extended to the first order in the dimensionless film thickness (including the small effects of the variation of the film pressure across its thickness and the surface tension) and study the stability of the steady solutions to two-dimensional disturbances. The modified evolution equation is found to have both asymptotically stable and unstable solutions arising from the pressure terms. Surface tension effects place a restriction on the critical wave number when instability occurs: in many cases, surface tension prevents instability.
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)
- S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961. MR 0128226
R. E. Johnson, Steady state coating flows inside a rotating horizontal cylinder, J. Fluid Mech. 190, 321–342 (1988)
M. J. Karweit and S. Corrsin, Observation of cellular patterns in a partly filled, horizontal, rotating cylinder, Phys. Fluids 18, 111–112 (1975)
J. P. Kovac and R. T. Balmer, Experimental studies of external hygrocysts, Journal of Fluids Engineering, 226–230 (June 1980)
Y. Kuramoto, Diffusion induced chaos in reaction systems, Progress of Theoretical Physics 64, 346–367 (1978)
J. Lammers, S. B. G. O’Brien, and M. N. M. Beerens, Proceedings of the first European symposium on The mechanics of thin film coatings, Leeds, 1995, pp. 397–403
H. K. Moffatt, Behaviour of a viscous film on the outer surface of a rotating cylinder, Journal de Mécanique 16, 651–674 (1977)
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, 649–670 (1993)
- S. B. G. O’Brien, Linear stability of rimming flow, Quart. Appl. Math. 60 (2002), no. 2, 201–211. MR 1900490, DOI https://doi.org/10.1090/qam/1900490
- L. W. Schwartz and D. E. Weidner, Modeling of coating flows on curved surfaces, J. Engrg. Math. 29 (1995), no. 1, 91–103. MR 1313419, DOI https://doi.org/10.1007/BF00046385
J. A. Moriarty, L. W. Schwartz, and E. O. Tuck, Unsteady spreading of thin liquid films with small surface tension, Phys. Fluids A, 3, 733–742 (1993)
J. L. Throne and J. Gianchandani, Reactive rotational molding, Polymer Engineering and Science 20, 899–919 (1980)
S. D. R. Wilson and J. Williams, The flow of a liquid film on the inside of a rotating cylinder, Phys. Fluids 4, 2184–2190 (1997)
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)
S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961
R. E. Johnson, Steady state coating flows inside a rotating horizontal cylinder, J. Fluid Mech. 190, 321–342 (1988)
M. J. Karweit and S. Corrsin, Observation of cellular patterns in a partly filled, horizontal, rotating cylinder, Phys. Fluids 18, 111–112 (1975)
J. P. Kovac and R. T. Balmer, Experimental studies of external hygrocysts, Journal of Fluids Engineering, 226–230 (June 1980)
Y. Kuramoto, Diffusion induced chaos in reaction systems, Progress of Theoretical Physics 64, 346–367 (1978)
J. Lammers, S. B. G. O’Brien, and M. N. M. Beerens, Proceedings of the first European symposium on The mechanics of thin film coatings, Leeds, 1995, pp. 397–403
H. K. Moffatt, Behaviour of a viscous film on the outer surface of a rotating cylinder, Journal de Mécanique 16, 651–674 (1977)
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, 649–670 (1993)
S. B. G. O’Brien, Linear stability of rimming flows, Quart. Appl. Math. LX, 201–211 (2002)
L. W. Schwartz and D. E. Weidner, Modelling of coating flows on curved surfaces, J. Engrg. Math. 29, 91–103 (1995)
J. A. Moriarty, L. W. Schwartz, and E. O. Tuck, Unsteady spreading of thin liquid films with small surface tension, Phys. Fluids A, 3, 733–742 (1993)
J. L. Throne and J. Gianchandani, Reactive rotational molding, Polymer Engineering and Science 20, 899–919 (1980)
S. D. R. Wilson and J. Williams, The flow of a liquid film on the inside of a rotating cylinder, Phys. Fluids 4, 2184–2190 (1997)
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Article copyright:
© Copyright 2002
American Mathematical Society
