The present study is related to the rimming flow of non-Newtonian fluid on the inner surface of a horizontal rotating cylinder. Using a scale analysis, the main characteristic scales and nondimensional parameters, which describe the principal features of the process, are found. Exploiting the fact that one of the parameters is very small, an approximate asymptotic mathematical model of the process is developed and justified. For a wide range of fluids, a general constitutive law can be presented by a single function relating shear stress and shear rate that corresponds to a generalized Newtonian model. For this case, the run-off condition for rimming flow is derived. Provided the run-off condition is satisfied, the existence of a steady-state solution is proved. Within the bounds stipulated by this condition, film thickness admits a continuous solution, which corresponds to subcritical and critical flow regimes. It is proved that for the critical regime the solution has a corner on the rising wall of the cylinder. In the supercritical flow regime, a discontinuous solution is possible and a hydraulic jump may occur. It is shown that straightforward leading order steady-state theory can work well to study the shock location and height. For the particular case of a power-law model, the analytical solution of a steady-state equation for the fluid film thickness is found in explicit form. More complex rheological models, which show linear Newtonian behavior at low shear rates with transition to power law at moderate shear rates, are also considered. In particular, numerical computations were carried out for the Ellis model. For this model, some analytical asymptotic solutions have also been obtained in explicit form and compared with the results of numerical computations. Based on these solutions, the optimal values of parameters, which should be used in the Ellis equation for the correct simulation of the coating flows, are determined; the criteria that guarantee the steady-state continuous solutions are defined; and the size and location of the stationary hydraulic jumps, which form when the flow is in the supercritical state, are obtained for the different flow parameters.

1.
Ruschak
,
K. J.
, and
Scriven
,
L. E.
, 1976, “
Rimming Flow of Liquid in a Rotating Horizontal Cylinder
,”
J. Fluid Mech.
0022-1120,
76
, pp.
113
125
.
2.
Moffatt
,
H. K.
, 1977, “
Behaviour of Viscous Film on the Surface of a Rotating Cylinder
,”
J. Mec.
0021-7832,
16
, pp.
651
673
.
3.
Preziosi
,
L.
, and
Joseph
,
D. D.
, 1988, “
The Run-Off Condition for Coating and Rimming Flows
,”
J. Fluid Mech.
0022-1120,
187
, pp.
99
113
.
4.
Melo
,
F.
, 1993, “
Localized States in Film-Dragging Experiments
,”
Phys. Rev. E
1063-651X,
48
, pp.
2704
2712
.
5.
O’Brien
,
S. B. G.
, and
Gath
,
E. G.
, 1998, “
Location of a Shock in Rimming Flow
,”
Phys. Fluids
1070-6631,
10
, pp.
1040
1042
.
6.
Harkin-Jones
,
E.
, and
Crawford
,
R. J.
, 1996, “
Rotational Moulding of Liquid Polymers
,” in
Rotational Moulding of Plastics
,
R. J.
Crawford
(Ed.),
John Wiley & Sons
,
London
, pp.
243
255
.
7.
Fomin
,
S.
,
Watterson
,
J.
,
Raghunathan
,
S.
, and
Harkin-Jones
,
E.
, 2002, “
Steady-State Rimming Flow of the Generalized Newtonian Fluid
,”
Phys. Fluids
1070-6631,
14
(
9
), pp.
3350
3353
.
8.
Jenekhe
,
S. A.
, 1983, “
The Rheology and Spin Coating of Polyimide Solutions
,”
Polym. Eng. Sci.
0032-3888,
23
, pp.
830
834
.
9.
Jenekhe
,
S. A.
, and
Schuldt
,
S. B.
, 1984, “
Coating of Non-Newtonian Fluids on a Flat Rotating Disk
,”
Ind. Eng. Chem. Fundam.
0196-4313,
23
, pp.
432
436
.
10.
Lawrence
,
C. J.
, and
Zhou
,
W.
, 1991, “
Spin Coating of Non-Newtonian Fluids
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
39
, pp.
137
187
.
11.
Bird
,
R. B.
,
Armstrong
,
R. C.
, and
Hassager
,
O.
, 1977,
Dynamics of Polymeric Liquids, Vol. 1, Fluid Dynamics
,
Wiley
,
New York
.