European Journal of Applied Mathematics



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


M. J. GOSTLING a1, M. D. SAVAGE a1 and M. C. T. WILSON a2
a1 Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
a2 School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK

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 [double less-than sign] 1, where Ca represents the capillary number, boundary conditions are formally developed that account for S [not equal] 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 [rightward arrow] 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 [rightward arrow] 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 &λmacr;(ζ) such that as S is increased from zero:

(i) bead break occurs for λ < &λmacr;;

(ii) upstream flooding occurs for λ > &λmacr;;

(iii) when λ = &λmacr; 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.

(Received September 22 1999)
(Revised September 30 2000)