We consider rimming flows in the presence of an axially varying viscosity but with inertia and surface tension effects being negligible. First, we find that a modified lubrication analysis ͑MLA͒ presented earlier ͓M. Tirumkudulu and A. Acrivos, Phys. Fluids 13, 14 ͑2001͔͒ can predict accurately the thickness of the film profile over the whole range of ⍀, the angular velocity of the rotating cylinder, even when the fill fraction F is as large as 0.36, where the film is far from thin. This is also the case with the analysis due to Benjamin et al. ͓T. B. Benjamin, W. G. Pritchard, and S. J. Tavener ͑preprint, 1993͔͒ except that, here, F cannot exceed 0.29. On the basis of this MLA, we propose a model to describe the three-dimensional free surface shape of rimming flows with an axially varying viscosity and show that the free surface profiles thereby obtained agree with those determined by solving numerically the three-dimensional Stokes equations. In the accompanying article, this model will be used as the basis of a stability analysis which will explain the origin of the observed particle band formation in rimming flows of suspensions containing neutrally buoyant particles ͓M. Tirumkudulu, A. Mileo, and A. Acrivos, Phys. Fluids 12, 1615 ͑2000͔͒.

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.

It has recently been reported that an initially uniform suspension of neutrally buoyant particles within a partially filled horizontal rotating cylinder can, under certain conditions, segregate into bands of particles separated by regions of low particle concentration or even particle-free liquid [M. Tirumkudulu, A. Mileo, and A. Acrivos, Phys. Fluids 12, 1615 (2000)]. An explanation for this phenomenon is proposed on the basis of a model of rimming flows with an axially varying viscosity plus the experimental observation that, when the liquid contains a recirculating region (puddle), the particles segregate radially by migrating out of the puddle into the unidirectional circumferential flow. A linear stability analysis for dilute suspensions shows that such a particle distribution is unstable to axial perturbations with the surface tension being responsible for the selection of the wavelength of the most rapidly amplified disturbance. The calculated and measured spacings between the bands are in good agreement. In addition, since, in the absence of a puddle, the particle concentration appears to remain uniform throughout the cross section of the film, no axial particle segregation is predicted to occur nor has it ever been seen experimentally, even when an axial viscosity variation is imposed on the flow by cooling a preselected portion of the cylinder.