Coating flow on a rotating cylinder in the presence of an irrotational airflow with circulation

Abstract

A detailed analysis of steady coating flow of a thin film of a viscous fluid on the outside of a uniformly rotating horizontal circular cylinder in the absence of surface-tension effects but in the presence of a non-uniform pressure distribution due to an irrotational airflow with circulation shows that the presence of the airflow can result in qualitatively different behaviour of the fluid film from that in classical coating flow. Full-film solutions corresponding to a continuous film of fluid covering the entire cylinder are possible only when the flux and mass of fluid do not exceed critical values, which are determined in terms of the non-dimensional parameters $F$ and $K$ representing the speed of the far-field airflow and the circulation of the airflow, respectively. The qualitative changes in the behaviour of the film thickness as $F$ and $K$ are varied are described. In particular, the film thickness can have as many as four stationary points and, in general, has neither top-to-bottom nor right-to-left symmetry. In addition, when the circulation of the airflow is in the same direction as the rotation of the cylinder the maximum mass of fluid that can be supported on the cylinder is always less than that in classical coating flow, whereas when the circulation is in the opposite direction the maximum mass of fluid can be greater than that in classical coating flow.

Unsteady coating flow on a rotating cylinder in the presence of an irrotational airflow with circulation

“…( 1) and ( 2) reduce to Eqs. (2.1) and (3.3), respectively, of Mitchell et al 22 in the case of steady flow in the absence of capillarity, a ¼ 0.…”

“…(Newell and Viljoen 21 omitted the factor 2 from the definition of their parameter u, but this appears to be simply a typographical error.) As Mitchell et al 22 describe, Newell and Viljoen 21 (evidently following Hinch and Kelmanson 7 ) have the opposite signs on p in their Eqs. ( 5) and ( 6) from those in the present ( 2) and ( 1), but unfortunately, unlike for Hinch and Kelmanson, 7 these differences in sign do not cancel out in their evolution equation (7) [i.e., their version of the present Eq.…”

“…For the airflow around the cylinder to be even approximately irrotational, it is necessary that the Reynolds number based on the circumferential speed of the cylinder, q a a 2 X=l a ) 1, where l a is the viscosity of the air, is large, and that the boundary layer that forms in the air remains attached to the cylinder. As Mitchell et al 22 describe, the conclusions of a body of analytical and numerical studies of high-Reynolds-number flow around a rotating cylinder without a film of viscous fluid (notably the work by Glauert, 39 Moore, 40 Kang et al, 41 Stojkovic et al, 42 Mittal and Kumar, 43 and Aljure et al 44 ) are that the rotation of the cylinder tends to suppress the separation of the boundary layer, that when also the circumferential speed of the cylinder is large compared with the speed of the far-field airflow, aX ) U 1 , the boundary layer does indeed remain attached all the way around the cylinder, and that the circulation then takes the value j ¼ Referred to polar coordinates r-h with origin on the axis of the cylinder and with h measured from the horizontal, the pressure in the film, denoted by p ¼ pðh; tÞ, where t denotes time, is given by…”

“…Specifically, we investigate unsteady two-dimensional coating flow of a thin film of a viscous fluid on the outside of a uniformly rotating solid horizontal circular cylinder in the presence of a steady two-dimensional irrotational airflow with circulation. Three of the above-mentioned papers are particularly relevant to the present study, namely, those by Hinch and Kelmanson, 7 Newell and Viljoen, 21 and Mitchell et al 22 Very recently, Mitchell et al 22 investigated the steady version of the present problem in the absence of capillarity. They classified the possible steady solutions that can occur and proved (by a straightforward generalization of the argument of O'Brien 27 ) that subcritical solutions remain neutrally stable to small two-dimensional perturbations in the presence of the airflow.…”

“…Since the publication of the seminal papers by Moffatt 1 and Pukhnachev 2 in 1977, coating flow and rimming flow (i.e., flow of a film of fluid on the outside and inside, respectively, of a rotating solid horizontal cylinder) have come to be regarded as paradigm problems in the study of free-surface flows of viscous fluids. A rather large literature has grown up concerning these flows, including the study of discontinuous "shock" solutions in rimming flow by Johnson, 3 the pioneering numerical investigation of coating flow by Hansen and Kelmanson, 4 the study of "curtain" solutions, which are unbounded at the top and the bottom of the cylinder, by Duffy and Wilson, 5 the study of the effect of capillarity in rimming flow by Ashmore et al, 6 studies of the large-time dynamics of unsteady coating flow by Hinch and Kelmanson, 7 Hinch et al, 8 Kelmanson, 9 and Groh and Kelmanson, 10 the numerical investigations of two-and threedimensional coating flow by Evans et al, 11,12 the bifurcation analysis of coating flow by Lin et al, 13 and the discovery of new branches of steady solutions in coating and rimming flow by Lopes et al 14 In addition, there have been many extensions to the basic problems, including the studies of the effect of a uniform azimuthal shear stress on the free surface of the film by Black 15 and Villegas-D ıaz et al, 16 the studies of coating flow on elliptical cylinders by Hunt 17 and Li et al, 18 the study of thermoviscous effects in coating and rimming flow by Leslie et al, 19 the study of coating flow on patterned cylinders by Li et al, 20 the studies of unsteady and steady coating flow in the presence of an irrotational airflow with circulation by Newell and Viljoen 21 and Mitchell et al, 22 respectively, the study of a "thick-film" model for coating flow by Wray and Cimpeanu, 23 and the study of coating flow of a film laden with colloidal particles in the presence of solvent evaporation by Parrish and Kumar. 24 Studies of the stability of coating and/or rimming flow in various parameter regimes have been undertaken.…”