Gostling, M.J., Savage, M.D. and Wilson, M.C.T. (2001) Flow in a doublefilmfed fluid bead between contrarotating rolls, Part 2: bead break and flooding. European Journal of Applied Mathematics, 12 (3). pp. 413431. ISSN 09567925

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Abstract
Twodimensional flow is considered in a fluid bead located in the gap between a pair of contrarotating 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 [zeta] of the outlet film on that roll. The flow is modelled via lubrication theory and for Ca [double lessthan sign] 1, where Ca represents the capillary number, boundary conditions are formally developed that account for S [not equal] 1 and the nonconstant 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 [zeta] [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 [zeta] [rightward arrow] 1 a transition from bead break to upstream flooding of the nip can occur and multiple twodimensionally stable solutions exist. For a varying input flux and fixed and ‘sufficiently large’ values of [zeta] there is a critical input flux &[lambda]macr;([zeta]) such that as S is increased from zero:
(i) bead break occurs for [lambda] < &[lambda]macr;;
(ii) upstream flooding occurs for [lambda] > &[lambda]macr;;
(iii) when [lambda] = &[lambda]macr; the flow becomes neutrally stable at a specific value of S beyond which there exist two steady solutions (twodimensionally stable) leading to bead break and upstream flooding, respectively.
Item Type:  Article 

Copyright, Publisher and Additional Information:  © Cambridge University Press, 2001 
Institution:  The University of Leeds 
Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Physics and Astronomy (Leeds) The University of Leeds > Faculty of Engineering (Leeds) > School of Mechanical Engineering (Leeds) > Institute of Engineering Thermofluids, Surfaces & Interfaces (iETSI) (Leeds) 
Depositing User:  Repository Assistant 
Date Deposited:  09 Jun 2006 
Last Modified:  06 Jun 2014 02:56 
Published Version:  http://dx.doi.org/10.1017/S0956792501004387 
Status:  Published 
Publisher:  Cambridge University Press 
Refereed:  Yes 
Identification Number:  10.1017/S0956792501004387 
URI:  http://eprints.whiterose.ac.uk/id/eprint/1269 