Rimming flows with an axially varying viscosity

Abstract

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.

Rheology of Concentrated Suspensions

“…This phenomenon, which has no counterpart in pure fluids, was recently explained by a novel stability analysis which yielded predictions in close agreement with the experimental results [17,18].…”

Rimming flows within a rotating horizontal cylinder: asymptotic analysis of the thin-film lubrication equations and stability of their solutions

“…As shown in Ref [7], Equation (14) has continuous, periodic solutions for all α when F ≤ 0·29. But, although, as was also shown in [7], the solutions of (14) are, numerically, very close to those of (11) for the same values of α and F ≤ 0·29, our discussion which follows will make use primarily of Equation (11) because of its slighter simplicity relative to (14).…”

“…Such a profile for β = 1·8 is shown in Figure 2(c). Moreover, θ * = −π/2 when β = β * * , and when β exceeds β * * no solution exists to Equation (5) subject to (7) which is everywhere real and positive. The discontinuity in the film-thickness profile is disquieting of course, since its existence is incompatible with the assumptions underlying the lubrication analysis leading to Equation (5).…”

“…Besides requiring that the liquid film be everywhere thin relative to R, the radius of the cylinder, Equation (5) presupposes steady-state (hence q is independent of θ ), plus, as was said earlier, negligible inertia and surface-tension effects, i.e., vanishingly small Reynolds number and inverse Bond number γ ≡ σ/ρgR 2 , where σ is the surface tension and ρ is the density of the fluid. Under these conditions, the remaining two independent dimensionless groups, F and α, can be combined into a single parameter β, the value of which implicitly determines q and therefore the solution of (5) subject to (7) plus the periodicity condition η(θ + 2π) = η(θ).…”

“…Thus, Equation (11) subject to (13) has a continuous periodic solution for all α with F fixed (below a critical value of F = 0·36 as shown in [7]), in contrast to (5) subject to (7) which does not have a solution if α < F/β * * .…”