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

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 [zeta] 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 [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 two-dimensionally 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 (two-dimensionally stable) leading to bead break and upstream flooding, respectively

Stokes flow in a half-filled annulus between rotating coaxial cylinders

A model is presented for viscous flow in a cylindrical cavity (a half-filled annulus lying between horizontal, infinitely long concentric cylinders of radii R-i,R-0 rotating with peripheral speeds U-i,U-0). Stokes' approximation is used to formulate a boundary value problem which is solved for the streamfunction, phi, as a function of radius ratio (R) over bar = R-i/R-0 and speed ratio S = U-i/U-0. Results show that for S > 0 (S 1, a sequence of 'flow bifurcations' leads to a flow structure consisting of a set of nested separatrices, and provides the means by which the two-dimensional cavity flow approaches quasi-unidirectional flow in the small gap limit. Control-space diagrams reveal that speed ratio has little effect on the flow structure when S 0 and aspect ratios are small (except near S = 1). For S > 0 and moderate to large aspect ratios the bifurcation characteristics of the two large eddies are quite different and depend on both (R) over bar and S

Nested separatrices in simple shear flows: the effect of localized disturbances on stagnation lines

The effects of localized two-dimensional disturbances on the structure of shear flows featuring a stagnation line are investigated. A simple superposition of a planar Couette flow and Moffatt's [J. Fluid Mech. 18, 1--18 (1964)] streamfunction for the decay of a disturbance between infinite stationary parallel plates shows that in general the stagnation line is replaced by a chain of alternating elliptic and hyperbolic stagnation points with a separation equal to 2.78 times the half-gap between the plates. The flow structure associated with each saddle point consists of a homoclinic separatrix and two other separatrices which locally diverge but become parallel far from the disturbance. This basic structure repeats to give a sequence of nested separatrices permitting the streamfunction to approach that of simple shear flow far from the disturbance. Using the finite element method, the specific disturbance caused by a stationary cylinder placed on the stagnation line is considered, and results confirm the existence of the stagnation point chain, with computed separations and velocity damping ratios in very good agreement with those obtained from the Couette-Moffatt superposition. Numerical solutions also illustrate that while Reynolds number greatly affects the stagnation point separation and velocity damping ratio, these two quantities are the same for any pair of adjacent stagnation points in a given chain. Insight gained from the analysis of planar shear flows is applied to the flow in a half-filled horizontal annulus between rotating coaxial cylinders, and is used to explain why only certain flow patterns from the range of mathematically possible structures arise in previous numerical solutions. By way of contrast, the concentric annulus solution is then perturbed to allow for a small eccentricity. The non-uniformity of the inter-cylinder gap is shown to destroy the chain of stagnation points, but also to unfold additional flow structures not realizable when the gap is uniform

An experimental investigation of meniscus roll coating

A two-roll apparatus is used to explore experimentally the detailed fluid mechanics of meniscus roll coating in which inlets are starved and flow rates are small. Both forward and reverse modes of operation (with contra- and co-rotating rolls) are investigated using optical sectioning combined with dye injection and particle imaging techniques. That part of parameter space where meniscus coating occurs is identified by varying the roll separation and roll speeds and hence flow rate and capillary number. Key features of the flow structures identified in the forward mode include two large eddies (each with saddle point, separatrix and sub-eddies), a primary fluid transfer jet and the existence of two critical flow rates associated with the switching-on of a second fluid transfer jet and the switching-off of the primary transfer jet followed by a change in the flow structure. In the reverse mode, the key features are a single large eddy consisting of two sub-eddies, a saddle point and separatrix, a primary fluid transfer jet and once again two critical flow rates. These correspond to (i) the switching-on of a secondary transfer jet and (ii) the disappearance of a saddle point at the nip resulting in the merger of the primary and secondary transfer jets. Measurements of film thickness and meniscus location made over a range of speed ratios and capillary numbers are compared with theoretical predictions. A plate-roll apparatus is used to confirm the presence, for very small flow rates, of a sub-ambient, almost linear, pressure profile across the bead. Investigated also is the transition from inlet-starved to fully flooded roll coating as flow rate is increased and the changes in flow structure and pressure profile are observed

Flow in a double-film-fed fluid bead between contra-rotating rolls, Part 1: equilibrium flow structure

In multiple-roll coaters thin liquid films are transferred from roll to roll by means of liquid ‘beads’ which occupy the small gaps between adjacent rolls. Double-Film-Fed (DFF) beads are those which feature two ingoing films instead of the usual one, and arise in the intermediate stages of certain types of roll coater. One of the ingoing films, h1, is supplied from the previous inter-roll gap while the other, h2, ‘returns’ from the subsequent gap. Such a flow is investigated here under the conditions of low flow rate, small capillary number and negligible gravity and inertia, using lubrication theory and finite element analysis. The thickness of film h1 is fixed independently, while that of h2 is specified as a fraction, [zeta], of the film output on the same roll. This simple approach allows a degree of feedback between the output and input of the bead, and enables one to simulate different conditions in the subsequent gap. Predictions of outgoing film thicknesses made using the two models agree extremely well and show that, for each value of [zeta] < 1, one outgoing film thickness decreases monotonically with speed ratio, S, while the other features a maximum. Good agreement is also seen in the pressure profiles, which are entirely sub-ambient in keeping with the small capillary number conditions. The finite element solutions reveal that in the ‘zero-flux’ case (when [zeta] = 1) the flow structures are very similar to those seen in an idealized cavity problem. In the more general ([zeta] < 1) situation, as in single-film-fed meniscus roll coating, several liquid transfer-jets occur by which liquid is conveyed through the bead from one roll to the other. The lubrication model is used to calculate several critical flow rates at which the flow is transformed, and it is shown that when the total dimensionless flow rate through the bead exceeds 1/3, the downstream flow structure is independent of the relative sizes of the ingoing films

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

The steady flow is considered of a Newtonian fluid, of viscosity mu, between contra-rotating 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 = mu(Ulf U-2)/H-2(A). Several additional results then follow. (i) The effect of non-dimensional flow rate, lambda: A(2) = 2RH(0)(3 lambda - 1) and so the stagnation-saddle points are absent for lambda 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 = i). The effect of unequal cylinder speeds is to introduce a displacement that increases to a maximum of O(RH0)(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 not equal 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, where H-1/H-2 = S(S + 3)/3S + 1, provided the flux in the transfer jet is assumed to be negligible. (iii) The effect of capillary number, Ca: as Ca 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

A model for deformable roll coating with negative gaps and incompressible compliant layers

A soft elastohydrodynamic lubrication model is formulated for deformable roll coating involving two contra-rotating rolls, one rigid and the other covered with a compliant layer. Included is a finite-strip model (FSM) for the deformation of the layer and a lubrication model with suitable boundary conditions for the motion of the fluid. The scope of the analysis is restricted to Newtonian fluids, linear elasticity/viscoelasticity and equal roll speeds, with application to the industrially relevant highly loaded or 'negative gap' regime. Predictions are presented for coated film thickness, interroll thickness, meniscus location, pressure and layer deformation as the control parameters - load (gap), elasticity, layer thickness and capillary number, Ca - are varied. There are four main results: \ud (i) Hookean spring models are shown to be unable to model effectively the deformation of a compliant layer when Poisson's ratio nu --> 0.5. In particular, they fall to predict the swelling of the layer at the edge of the contact region which increases as v - 0.5; they also fail to locate accurately the position of the meniscus, X-M, and to identify the presence, close to the meniscus, of a 'nib' (constriction in gap thickness) and associated magnification of the sub-ambient pressure loop. (ii) Scaling arguments suggest that layer thickness and elasticity may have similar effects on the field variables. It is shown that for positive gaps this is true, whereas for negative gaps they have similar effects on the pressure profile and flow rate yet quite different effects on layer swelling (deformation at the edge of the contact region) and different effects on X-M. (iii) For negative gaps and Ca similar to O(1), the effect of varying either viscosity or speed and hence Ca is to significantly alter both the coating thickness and X-M. This is contrary to the case of fixed-gap rigid roll coating. (iv) Comparison between theoretical predictions and experimental data shows quantitive agreement in the case of X-M and qualitive agreement for flow rate. It is shown that this difference in the latter case may be due to viscoelastic effects in the compliant layer

Silicon isotopes reveal recycled altered oceanic crust in the mantle sources of ocean island basalts

EP thanks the Chateaubriand STEM fellowship program for funding. FM thanks the European Research Council under the European Community’s H2020 framework program/ERC grant agreement #637503 (Pristine) and the Agence Nationale de la Recherche for a chaire d’Excellence Sorbonne Paris Cité (IDEX13C445) and for the UnivEarthS Labex program (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02). PS thanks the support of the Marie Curie FP7-IOF fellowship “Isovolc”.The study of silicon (Si) isotopes in Ocean Island Basalts (OIB) has the potential to discern between different models for the origins of geochemical heterogeneities in the mantle. Relatively large (∼several per mil per atomic mass unit) Si isotope fractionation occurs in low-temperature environments during biochemical and geochemical precipitation of dissolved Si, where the precipitate is preferentially enriched in the lighter isotopes relative to the dissolved Si. In contrast, only a limited range (∼tenths of a per mil) of Si isotope fractionation has been observed from high-temperature igneous processes. Therefore, Si isotopes may be useful as tracers for the presence of crustal material within OIB mantle source regions that experienced relatively low-temperature surface processes in a manner similar to other stable isotope systems, such as oxygen. Characterizing the isotopic composition of the mantle is also of central importance to the use of the Si isotope system as a basis for comparisons with other planetary bodies (e.g., Moon, Mars, asteroids). Here we present the first comprehensive suite of high-precision Si isotope data obtained by MC-ICP-MS for a diverse suite of OIB. Samples originate from ocean islands in the Pacific, Atlantic, and Indian Ocean basins and include representative end-members for the EM-1, EM-2, and HIMU mantle components. On average, δ30Si values for OIB (−0.32 ± 0.09‰, 2 sd) are in general agreement with previous estimates for the δ30Si value of Bulk Silicate Earth (−0.29 ± 0.07‰, 2 sd; Savage et al., 2014). Nonetheless, some small systematic variations are present; specifically, most HIMU-type (Mangaia; Cape Verde; La Palma, Canary Islands) and Iceland OIB are enriched in the lighter isotopes of Si (δ30Si values lower than MORB), consistent with recycled altered oceanic crust and lithospheric mantle in their mantle sources.PostprintPeer reviewe

Bose-Einstein Condensates in Optical Lattices: Band-Gap Structure and Solitons

We analyze the existence and stability of spatially extended (Bloch-type) and localized states of a Bose-Einstein condensate loaded into an optical lattice. In the framework of the Gross-Pitaevskii equation with a periodic potential, we study the band-gap structure of the matter-wave spectrum in both the linear and nonlinear regimes. We demonstrate the existence of families of spatially localized matter-wave gap solitons, and analyze their stability in different band gaps, for both repulsive and attractive atomic interactions

A Conflict Model for Strategists and Managers

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/67210/2/10.1177_000276427201500604.pd