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 [Lt ] 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
&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.