S. J. S. Morris

The motion of long bubbles in polygonal capillaries. Part 1. Thin films

Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure–velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The ‘piston’ is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments. Part 1 of this work studies the fluid films deposited on capillary walls in the limit Ca → 0 ( Ca ≡ μ U /σ, where μ is the fluid viscosity, U the bubble velocity, and σ the surface tension). Part 2 (Wong et al. 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x . For 1 [Lt ] x [Lt ] Ca −1 , the film profile is frozen with a thickness of order Ca 2/3 at the centre and order Ca at the sides. For x ∼ Ca −1 , surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca 4/3 . For x [Gt ] Ca −1 , the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ∼ Ca −5/3 , the height of the parabola is order Ca 2/3 . Finally, for x [Gt ] Ca −5/3 , the height decreases as Ca 1/4 x −1/4 .

The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow

This work determines the pressure–velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ca 2/3 in the limit Ca → 0 ( Ca μ U /σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca 2/3 . The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary. The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure–velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σ a 2 /μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Q c , whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Q c [Lt ] 10 −6 . For Q c [Lt ] Q [Lt ] 1, the plug flow dominates, and the gradient in liquid pressure varies with Q 2/3 . For Q [Lt ] Q c , the corner flow dominates, and the pressure gradient varies linearly with Q . A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.

A boundary-layer analysis of Benard convection in a fluid of strongly temperature-dependent viscosity

We give an asymptotic analysis of 2-dimensional Benard convection in a fluid of infinite Prandtl number and strongly temperature-dependent viscosity ν(T)=c e−γT. All surfaces of the box containing the fluid are traction-free. We assume that the Nusselt number N ⪢ 1. The only temperature differences in the cell then occur in a cold horizontal thermal layer at the top of the cell, a hot horizontal layer at the base of the cell and two plumes on the sidewalls. We assume further that the viscosity ratio is so large that the upper part of the cold, horizontal thermal-layer is stagnant. We show that the essential viscosity variations occur in the sublayer at the base of the stagnant region. The thickness l of the sublayer is small compared to the total thickness Λ of the stagnant layer. This separation of scales allows a simple solution for the sublayer. At the top edge of this sublayer we find the viscosity increases exponentially with height, and the velocity vanishes exponentially. We find that the temperature drop across the sublayer is ∼ 3γ−1. We show that the viscosity variations in the underlying flow play no dynamical role. The viscosity ratio across the entire convecting region is independent of the viscosity ratio eγΔT for the whole cell and depends only weakly on the Rayleigh number; for typical Rayleigh numbers achieved in experiments, the viscosity ratio is ∼ 55 when θ≡γΔT → ∞. We find the heat-transfer law for the cell to have the form θNF(Rγ) The Rayleigh number if Rγαgγ−1d3ν0κ is based on the rheological temperature scale γ−1 and the viscosity ν0 at the hot surface. We give F as a function of Rγ, the aspect ratio of A of the cell, and the fraction f of the total heat flow carried by the hot plume. The results depend weakly on ƒ. In particular we find that if ƒ12, A1 and 3 × 104 ⩽ Rγ ⩽ × 107 θN=1.61Rγ0.20 We compare our predictions with observations by Richter et al. and Lux and Sacks. The theory predicts the observations with an error of ∼50%. This error is comparable to that in boundary layer theories for isoviscous flows. We discuss its probable causes. We show that the success of empirical correlations due to Booker is a natural consequence of the boundary-layer structure of the cell.

Three-dimensional Menisci in Polygonal Capillaries

Abstract The shapes of gravity-free, three-dimensional menisci are computed from the augmented Young-Laplace equation. Incorporation of disjoining thin-film forces in the Young-Laplace relation eliminates the contact line, thereby eliminating the free boundary from the problem. To calculate a meniscus with finite contact angles, the conjoining/disjoining pressure isotherm must also contain an attractive, sharply varying, spike function. The width of this function, w, reflects the range of the thin-film forces. In the limit of w approaching zero, a solution of the Young-Laplace equation is recovered. The proposed calculation method is demonstrated for menisci in two different types of capillaries. In the first case, the capillary is regular-polygonal in cross section with either 3, 4, or 6 sides and with contact angles Φ ranging from 0 to 45°. In the second case, the capillary is rectangular in section with aspect ratios ranging from 1.2 to 5 and with Φ = 0°, 15°, or 30°. Gas-liquid menisci inside a square glass capillary of 0.5 mm inscribed radius are measured optically for air bubbles immersed in a solution of di-n-butyl phthalate and mineral oil. This liquid mixture exhibits a zero contact angle with the wall and matches the refractive index of the glass capillary, permitting precise visual location of the interface. Excellent agreement is found with the numerical results which further demonstrates that the limiting process of the proposed method is valid. Because it avoids the issue of locating the contact line, solution of the augmented Young-Laplace equation is a simple and powerful method for the calculation of three-dimensional menisci.

The effects of a strongly temperature-dependent viscosity on slow flow past a hot sphere

This work determines analytically the drag on, and heat flux out of, a hot sphere that translates steadily in a fluid of strongly temperature-dependent viscosity. There is no dissipative heating. The essentials are illustrated by an exact solution for the flow induced by slowly squeezing two parallel planes together. The lower plane is hot and stationary; the upper is cold and advances in a direction normal to itself at uniform speed U . The gap is completely filled by a fluid of strongly temperature-dependent viscosity. We find the temperature and velocity profiles, and determine the Nusselt number N and Peklet number P as functions of the normal force D on the lower plane. The large viscosity variation tries to concentrate the flow into a relatively thin softened layer in which the viscosity is of order its value μ 0 at the hot plane. In the limit of infinite viscosity ratio (fixed P ), it succeeds (lubrication limit): if P [Lt ] 1, the width of the softened layer is determined by conduction and D ∝ μ 0 U ; but D ∝ μ 0 U 4 when forced convection is important. If P → ∞ (fixed viscosity ratio), the softened layer is so thin that it chokes, and all the deformation occurs outside the thermal layer in the fluid of uniform viscosity μ ∞ (Stokes limit); then D ∝ μ ∞ U . These mechanisms appear as three distinct legs in our plot of log P against log D . There are similar transitions in the plot of log N against log D . The solution gives an estimate of the drag on a sphere. We test this estimate against an analytical solution for the sphere in the lubrication limit. Then we extend the solution to cover power-law fluids, and apply it to a model (by Marsh) of magma transport beneath island-arc volcanoes. The results suggest that the magma covers the first 50 km of its ascent by an isoviscous mechanism, with the lubrication mechanism operating in the remaining 50 km. To open a fresh pathway from the source to the surface takes about 10 6 years and uses about 10 27 erg.

Femtoliter-Scale Patterning by High-Speed, Highly Scaled Inverse Gravure Printing

Pattern printing techniques have advanced rapidly in the past decade, driven by their potential applications in printed electronics. Several printing techniques have realized printed features of 10 μm or smaller, but unfortunately, they suffer from disadvantages that prevent their deployment in real applications; in particular, process throughput is a significant concern. Direct gravure printing is promising in this regard. Gravure printing delivers high throughput and has a proven history of being manufacturing worthy. Unfortunately, it suffers from scalability challenges because of limitations in roll manufacturing and limited understanding of the relevant printing mechanisms. Gravure printing involves interactions between the ink, the patterned cylinder master, the doctor blade that wipes excess ink, and the substrate to which the pattern is transferred. As gravure-printed features are scaled, the associated complexities are increased, and a detailed study of the various processes involved is lacking. In this work, we report on various gravure-related fluidic mechanisms using a novel highly scaled inverse direct gravure printer. The printer allows the overall pattern formation process to be studied in detail by separating the entire printing process into three sequential steps: filling, wiping, and transferring. We found that pattern formation by highly scaled gravure printing is governed by the wettability of the ink to the printing plate, doctor blade, and substrate. These individual functions are linked by the apparent capillary number (Ca); the printed volume fraction (φ(p)) of a feature can be constructed by incorporating these basis functions. By relating Ca and φ(p), an optimized operating point can be specified, and the associated limiting phenomena can be identified. We used this relationship to find the optimized ink viscosity and printing speed to achieve printed polymer lines and line spacings as small as 2 μm at printing speeds as high as ∼1 m/s.

Cost-Utility Analysis of Mechanical Thrombectomy Using Stent Retrievers in Acute Ischemic Stroke

Background and Purpose— Recently, 5 randomized controlled trials demonstrated the benefit of endovascular therapy compared with intravenous tissue-type plasminogen activator in acute stroke. Economic evidence evaluating stent retrievers is limited. We compared the cost-effectiveness of intravenous tissue-type plasminogen activator alone versus mechanical thrombectomy and intravenous tissue-type plasminogen activator as a bridging therapy in eligible patients in the UK National Health Service. Methods— A model-based cost-utility analysis was performed using a lifetime horizon. A Markov model was constructed and populated with probabilities, outcomes, and cost data from published sources, including 1-way and probabilistic sensitivity analysis. Results— Mechanical thrombectomy was more expensive than intravenous tissue-type plasminogen activator, but it improved quality-adjusted life expectancy. The incremental cost per (quality-adjusted life year) gained of mechanical thrombectomy over a 20 year period was

Buoyant Instability of a Viscous Film Over a Passive Fluid

11 651 (£7061). The probabilistic sensitivity analysis demonstrated that thrombectomy had a 100% probability of being cost-effective at the minimum willingness to pay for a quality-adjusted life year commonly used in United Kingdom. Conclusions— Although the upfront costs of thrombectomy are high, the potential quality-adjusted life year gains mean this intervention is cost-effective. This is an important factor for consideration in deciding whether to commission this intervention.

Methodology for Inkjet Printing of Partially Wetting Films

The article of record as published may be located at http://dx.doi.org/10.1017/S0022112093002514

The evaporating meniscus in a channel

Inkjet printing of precisely defined structures is critical for the realization of a range of printed electronics applications. We develop and demonstrate a methodology to optimize the inkjet printing of two-dimensional, partially wetting films. When printed inks have a positive retreating contact angle, we show that any fixed spacing is ineffective for printing two-dimensional features. With fixed spacing, the bead contact angle begins large, leading to a bulging overflow of its intended footprint. Each additional line reduces the bead contact angle, eventually leading to separation of the bead. We propose a printing scheme that adjusts the line-to-line spacing to maintain a beads contact angle between its advancing and retreating values as it is printed. Implementing this approach requires an understanding of the two-dimensional bead surface and compensation for evaporation during the print. We derive an analytic equation for the beads surface with pinned contact lines and use an empirical fit for mass loss due to evaporation. Finally, we demonstrate that enhanced contact angle hysteresis, achieved by preprinting a features border, leads to better corner definition.