Abstract
It is well-known that a standard lubrication analysis of the equations of motion in thin liquid films coating the inside surface of a rotating horizontal cylinder leads, under creeping-flow conditions, to a cubic equation for the film thickness profile which, depending on the fluid properties of the liquid, the speed of rotation and the fill fraction F, has either (a) a continuous, symmetric (homogeneous) solution; (b) a solution containing a shock; or (c) no solution below a certain speed. By means of an asymptotic analysis of the recently proposed “modified lubrication equation” (MLE) [M. Tirumkudulu and A. Acrivos, Phys. Fluid 13 (2000) 14–19], it is shown that the solutions of the cubic equation referred to above correctly describe the film-thickness profiles although, when shocks are involved, under exceedingly restrictive conditions, typically F~ 10−3 or less. In addition, using the MLE, the linear stability of these film profiles is investigated and it is shown that: the “homogeneous” profiles are neutrally stable if surface-tension effects are neglected but, if the latter are retained, the films are asymptotically stable to two-dimensional disturbances and unstable to axial disturbances; on the other hand, the non-homogeneous profiles are always asymptotically stable, thus confirming results given earlier [T.B. Benjamin, W.G. Pritchard, and S.J. Tavener (preprint, 1993)] on the basis of the standard lubrication analysis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H.K. Moffatt (1977) ArticleTitleBehaviour of a viscous film on the outer surface of a rotating cylinder J. de Mecanique 16 651–673 Occurrence Handle1977JMec...16..651M
R.E. Johnson. (1988) ArticleTitleSteady-state coating flows inside a rotating horizontal cylinder J. Fluid Mech. 190 321–342 Occurrence Handle1988JFM...190..321J Occurrence Handle0642.76042
Benjamin T.B., Pritchard W.G., Tavener S.J. Steady and unsteady flow of a highly viscous fluid inside of a rotating horizontal cylinder. preprint (1993)
S.D.R. Wilson J. Williams (1997) ArticleTitleThe flow of a liquid film on the inside of a rotating cylinder, and some related problems Phys. Fluids 9 2184–2190 Occurrence Handle1997PhFl....9.2184W
S.B.G. O’Brien E.G. Gath (1998) ArticleTitleThe location of a shock in rimming flow Phys. Fluids 10 1040–1042 Occurrence Handle1998PhFl...10.1040O
M. Tirumkudulu A. Acrivos (2001) ArticleTitleCoating flows within a rotating horizontal cylinder: lubrication analysis, numerical computations and experimental measurements Phys. Fluids 13 14–19 Occurrence Handle10.1063/1.1329909 Occurrence Handle2001PhFl...13...14T
B. Jin A. Acrivos (2004) ArticleTitleRimming flows with an axial varying viscosity Phys. Fluids 16 633–640 Occurrence Handle2004PhFl...16..633J Occurrence Handle2036025
A.E. Hosoi L. Mahadevan (1999) ArticleTitleAxial instability of a free-surface front in a partially filled horizontal rotating cylinder Phys. Fluids 11 97–106 Occurrence Handle10.1063/1.869905 Occurrence Handle1999PhFl...11...97H Occurrence Handle1665036 Occurrence Handle1147.76416
J. Ashmore A.E. Hosoi H.A. Stone (2003) ArticleTitleThe effect of surface tension on rimming flows in a partially filled rotating cylinder J. Fluid Mech. 479 65–98 Occurrence Handle10.1017/S0022112002003312 Occurrence Handle2003JFM...479...65A Occurrence Handle02065441
S.B.G. O’Brien (2002) ArticleTitleLinear stability of rimming flow Quart. Appl. Math. 60 201–211 Occurrence Handle1900490 Occurrence Handle1026.76018
S.B.G. O’Brien (2002) ArticleTitleA mechanism for linear instability in two-dimensional rimming flow Quart. Appl. Math. 60 283–299 Occurrence Handle1900494 Occurrence Handle1075.76029
E.S. Benilov S.B.G. O’Brien I.A. Sazonov (2003) ArticleTitleA new type of instability: explosive disturbances in a liquid film inside a rotating horizontal cylinder J. Fluid Mech. 497 201–224 Occurrence Handle10.1017/S0022112003006633 Occurrence Handle2003JFM...497..201B Occurrence Handle2033849 Occurrence Handle1072.76027
E.J. Hinch M.A. Kelmanson (2003) ArticleTitleOn the decay and drift of free-surface perturbations in viscous thin-film flow exterior to a rotating cylinder Proc. R. Soc. London A 459 1193–1213 Occurrence Handle2003RSPSA.459.1193H Occurrence Handle1997097 Occurrence Handle10.1098/rspa.2002.1069 Occurrence Handle1092.76503
E.S. Benilov (2004) ArticleTitleExplosive instability in a linear system with neutrally stable eigenmodes. Part 2. Multi-dimensional disturbances J. Fluid Mech. 501 105–124 Occurrence Handle10.1017/S0022112003007237 Occurrence Handle2004JFM...501..105B Occurrence Handle1134.76344 Occurrence Handle2258206
B. Jin A. Acrivos (2004) ArticleTitleTheory of particle segregation in rimming flows of suspensions containing neutrally buoyant particles Phys. Fluids 16 641–651 Occurrence Handle2004PhFl...16..641J Occurrence Handle2036026
F. Melo (1993) ArticleTitleLocalized states in a film-dragging experiment Phys. Rev. E 48 2704–2712 Occurrence Handle10.1103/PhysRevE.48.2704 Occurrence Handle1993PhRvE..48.2704M
Jin B. Particle segregation in a sheared suspension with a free surface. Ph.D. thesis, The City University of New York (2004).
M. Tirumkudulu A. Mileo A. Acrivos (2000) ArticleTitleParticle segregation in monodisperse sheared suspensions in a partially filled rotating horizontal cylinder Phys. Fluids 12 1615–1618 Occurrence Handle10.1063/1.870411 Occurrence Handle2000PhFl...12.1615T Occurrence Handle1149.76569
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Acrivos, A., Jin, B. Rimming flows within a rotating horizontal cylinder: asymptotic analysis of the thin-film lubrication equations and stability of their solutions. J Eng Math 50, 99–121 (2004). https://doi.org/10.1007/s10665-004-1772-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10665-004-1772-7