Flow in a double-film-fed fluid bead between contra-rotating rolls Part 2: bead break and flooding

Abstract

Two-dimensional flow is considered in a fluid bead located in the gap between a pair of contra-rotating cylinders and bounded by two curved menisci. The stability of such bead flows with two inlet films, and hence no contact line, are analysed as the roll speed ratio S is increased. One of the inlet films can be regarded as an 'input flux' whilst the other is a 'returning film' whose thickness is specified as a fraction ζ of the outlet film on that roll. The flow is modelled via lubrication theory and for Ca ≪ 1, where Ca represents the capillary number, boundary conditions are formally developed that account for S 1 and the non-constant gap. It is shown that there is a qualitative difference in the results between the single and double inlet film models unless small correction terms to the pressure drops at the interfaces are taken into account. Futhermore, it is shown that the inclusion of these small terms produces an O(1) effect on the prediction of the critical value of S at which bead break occurs. When the limits of the returning film fraction are examined it is found that as ζ → 0 results are in good agreement with those for the single inlet film. Further it is shown for a fixed input flux that as ζ → 1 a transition from bead break to upstream flooding of the nip can occur and multiple two-dimensionally stable solutions exist. For a varying input flux and fixed and 'sufficiently large' values of ζ there is a critical input flux λ(ζ) such that as S is increased from zero: (i) bead break occurs for λ<λ;(ii) upstream flooding occurs for λ>λ;(iii) when λ = λ the flow becomes neutrally stable at a specific value of S beyond which there exist two steady solutions (two-dimensionally stable) leading to bead break and upstream flooding, respectively.

Topology of two-dimensional flow associated with degenerate dividing streamline on a free surface

“…The flow pattern (9) in Figure 4 deserves further consideration since the interaction of two vortices with opposite rotation occurs only near the non-simple degenerate critical point. Two topologically different flow structures (8) and (10) appear in the same area, but this is possible only in the event of global bifurcation (9) which divides the region into subregions (8) and (10). Examining the changes in the structure near the free surface, we encounter a vortex ring bifurcation (11) which attaches to the free surface at a singular point.…”

“…The topological fluid dynamics of flows with interfaces and free surfaces is a topic of interest, with a wide variety of scientific and engineering applications (see [3,10,14,17,20,21]).…”

Effects of shear thinning on forward roll coating

“…The model is divided into two components illustrated in figure 2; (i) the lubrication model of the flow between the two rolls together with an upstream boundary condition p ∞ = 0, consistent with a fully flooded inlet [12], (ii) the film splitting model that describes the downstream meniscus. …”

“…From an initial flux the pressure at the downstream meniscus is calculated using the equations describing the film splitting. Equa-13 tions (19), (20), (23), (24), (26), (27) and (28) (11) and (12). The search for the flux through the coating bead was then repeated until the downstream pressures were entirely consistent (within an error of ∆p < 1 × 10 −6 ).…”

“…When the thickness of the liquid film entering the coating bead is of the same order or greater than the gap between the rolls a rolling bank of fluid may form and be returned to the coating pool [6]. When the liquid film is significantly thinner than the roll separation, the upstream meniscus plays an important part in determining the pressures throughout the coating pool and on the entire coating process in general [5,11,12,10]. In the current work the flooded forward roll coating case is examined, a problem encountered in a range of coating processes, for example the transfer of fluid from a metering to applicator roll.…”