Roll coating of purely viscous liquids

Benkreira, Hadj; Edwards, M.F.; Wilkinson, W.L.

doi:10.1016/0009-2509(81)85025-7

Cited by 77 publications

(48 citation statements)

References 5 publications

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“…Figure 18 compares analytical and numerical predictions for H 1 /H 2 with recent experimental data collected from a high-precision, laboratory-scale apparatus having horizontally-aligned cylinders, Gaskell et al (1998). Film thicknesses were measured with a maximum error of 5% and like those of Benkreira et al (1981), where cylinders were aligned vertically, are in close accord with the theoretical predictions.…”

Section: Displacement Of Lsupporting

confidence: 74%

Stagnation–saddle points and flow patterns in Stokes flow between contra-rotating cylinders

1998

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The steady flow is considered of a Newtonian fluid, of viscosity µ, between contrarotating cylinders with peripheral speeds U 1 and U 2 . The two-dimensional velocity field is determined correct to O(H 0 /2R) 1/2 , where 2H 0 is the minimum separation of the cylinders and R an 'averaged' cylinder radius. For flooded/moderately starved inlets there are two stagnation-saddle points, located symmetrically about the nip, and separated by quasi-unidirectional flow. These stagnation-saddle points are shown to divide the gap in the ratio U 1 : U 2 and arise at |X| = A where the semi-gap thickness is H(A) and the streamwise pressure gradient is given by dP /dX = µ(U 1 + U 2 )/H 2 (A). Several additional results then follow.(i) The effect of non-dimensional flow rate, λ: A 2 =2 RH 0 (3λ − 1) and so the stagnation-saddle points are absent for λ<1/3, coincident for λ =1/3 and separated for λ>1/3.(ii) The effect of speed ratio, S = U 1 /U 2 : stagnation-saddle points are located on the boundary of recirculating flow and are coincident with its leading edge only for symmetric flows (S = 1). The effect of unequal cylinder speeds is to introduce a displacement that increases to a maximum of O(RH 0 ) 1/2 as S → 0. Five distinct flow patterns are identified between the nip and the downstream meniscus. Three are asymmetric flows with a transfer jet conveying fluid across the recirculation region and arising due to unequal cylinder speeds, unequal cylinder radii, gravity or a combination of these. Two others exhibit no transfer jet and correspond to symmetric (S = 1) or asymmetric (S = 1) flow with two asymmetric effects in balance. Film splitting at the downstream stagnation-saddle point produces uniform films, attached to the cylinders, of thickness H 1 and H 2 , whereprovided the flux in the transfer jet is assumed to be negligible.(iii) The effect of capillary number, Ca:asCa is increased the downstream meniscus advances towards the nip and the stagnation-saddle point either attaches itself to the meniscus or disappears via a saddle-node annihilation according to the flow topology.Theoretical predictions are supported by experimental data and finite element computations.

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Section: Displacement Of Lsupporting

confidence: 74%

Stagnation–saddle points and flow patterns in Stokes flow between contra-rotating cylinders

1998

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“…Tracing the x u solution curve from S =0 in Figures 5(e) and (f) only temporarily results in bead contraction before it begins to expand again, and then at some value of S, S f say, the upstream meniscus moves rapidly outwards -a feature which is identified as 'upstream flooding'. Despite the fact that three-dimensional instabilities cannot be ruled out in Figures 5(e) and (f) the plots do mimic the experimental results of Benkreira [2] and Benjamin [1], who investigated the flux through a DFF bead. Benjamin reported that for a DFF bead with Ca =0 .83 and S = 1, when the flux q equalled 1.315 the upstream interface flooded and the flow became three-dimensional.…”

Section: Effect On Meniscus Locations and S Cmentioning

confidence: 59%

“…The flux through a moderately starved/flooded inlet has been studied experimentally [2] and the flooding of a DFF bead has also been investigated [1].…”

Section: J Gostling Et Almentioning

confidence: 99%

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

Gostling¹,

Savage²,

Wilson³

2001

Eur. J. Appl. Math

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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.

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“…Coyle found an accurate flow distribution at a film splitting point to solve a full Navier-Stokes equation using a finite element method (14) . Before Coyle's study, Benkreira (15), (16) experimentally obtained a simple power-law equation to predict the accurate film splitting ratio for Newtonian and non-Newtonian fluids, and Savage (17) theoretically obtained an algebraic-type equation to predict the accurate film splitting ratio. The third level for investigating a multiple coating system is by considering circulation flows between the coating rolls.…”

Section: Journal Of Fluidmentioning

confidence: 99%

Numerical Study of a Four-Roll Coating System

Tsuda

2010

JFST

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The characteristics of a four-roll coating system were numerically investigated and compared with experimental data to validate the theoretical models used in this study. In the theoretical models, a film splitting model using a power-law-type equation, a roll-gap model based on elastohydrodynamics, and a flow model from a rotating-cylinder system were applied. The parametric computations for each operational condition revealed the steady and dynamic behaviors of a coating film and liquid films on the coating rolls. The results of the frequency response to the speed disturbances of the coating rolls indicated that the sensitivity of the lowest coating roll to the disturbance was half that of the others; this implies that the requirement for the accuracy of a driving system of the coating roll is not as severe as compared with others. The experimental data and the numerical results at steady state agreed well. Therefore, the theoretical models used in this research were found to be appropriate.

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Roll coating of purely viscous liquids

Cited by 77 publications

References 5 publications

Stagnation–saddle points and flow patterns in Stokes flow between contra-rotating cylinders

Stagnation–saddle points and flow patterns in Stokes flow between contra-rotating cylinders

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

Numerical Study of a Four-Roll Coating System

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