13 results
Homogenisation problems in reactive decontamination
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- E. LUCKINS, C. J. W. BREWARD, I. M. GRIFFITHS, Z. WILMOTT
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- Journal:
- European Journal of Applied Mathematics / Volume 31 / Issue 5 / October 2020
- Published online by Cambridge University Press:
- 30 September 2019, pp. 782-805
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The decontamination of hazardous chemical agents from porous media is an important and critical part of the clean-up operation following a chemical weapon attack. Decontamination is often achieved through the application of a cleanser, which reacts on contact with an agent to neutralise it. While it is relatively straightforward to write down a model that describes the interplay of the agent and cleanser on the scale of the pores in the porous medium, it is computationally expensive to solve such a model over realistic spill sizes.
In this paper, we consider the homogenisation of a pore-scale model for the interplay between agent and cleanser, with the aim of generating simplified models that can be solved more easily on the spill scale but accurately capture the microscale structure and chemical activity. We consider two situations: one in which the agent completely fills local porespaces and one in which it does not. In the case when the agent does not completely fill the porespace, we use established homogenisation techniques to systematically derive a reaction–diffusion model for the macroscale concentration of cleanser. However, in the case where the agent completely fills the porespace, the homogenisation procedure is more in-depth and involves a two-timescale approach coupled with a spatial boundary layer. The resulting homogenised model closely resembles the microscale model with the effect of the porous material being incorporated into the parameters. The two models cater for two different spill scenarios and provide the foundation for further study of reactive decontamination.
Out-of-plane buckling in two-dimensional glass drawing
- D. O’Kiely, C. J. W. Breward, I. M. Griffiths, P. D. Howell, U. Lange
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- Journal:
- Journal of Fluid Mechanics / Volume 869 / 25 June 2019
- Published online by Cambridge University Press:
- 29 April 2019, pp. 587-609
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We derive a mathematical model for the drawing of a two-dimensional thin sheet of viscous fluid in the direction of gravity. If the gravitational field is sufficiently strong, then a portion of the sheet experiences a compressive stress and is thus unstable to transverse buckling. We analyse the dependence of the instability and the subsequent evolution on the process parameters, and the mutual coupling between the weakly nonlinear buckling and the stress profile in the sheet. Over long time scales, the sheet centreline ultimately adopts a universal profile, with the bulk of the sheet under tension and a single large bulge caused by a small compressive region near the bottom, and we derive a canonical inner problem that describes this behaviour. The large-time analysis involves a logarithmic asymptotic expansion, and we devise a hybrid asymptotic–numerical scheme that effectively sums the logarithmic series.
Surface-tension- and injection-driven spreading of a thin viscous film
- K. B. Kiradjiev, C. J. W. Breward, I. M. Griffiths
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- Journal:
- Journal of Fluid Mechanics / Volume 861 / 25 February 2019
- Published online by Cambridge University Press:
- 28 December 2018, pp. 765-795
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We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form $t^{k}$, we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of $k$. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these.
The effect of ions on the motion of an oil slug through a charged capillary
- Z. M. Wilmott, C. J. W. Breward, S. J. Chapman
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- Journal:
- Journal of Fluid Mechanics / Volume 841 / 25 April 2018
- Published online by Cambridge University Press:
- 21 February 2018, pp. 310-350
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Numerous experimental studies have documented that injecting low-salinity water into an oil reservoir can increase the amount of oil recovered. However, owing to the complexity of the chemical interactions involved in this process, there has been much debate over the dominant mechanism causing this effect. In order to further understand one proposed mechanism, multicomponent ionic exchange, we study the motion of an oil slug through a clay pore throat filled with saline water. The pore throat is modelled as a capillary tube connecting two bulk regions of water. We assume that the surfaces of the oil and the capillary are negatively charged and that, due to repulsion between these surfaces, the oil slug is separated from the capillary surface by a thin film of water. Ion interactions at the oil–water and clay–water interfaces are modelled using the law of mass action. By using lubrication theory to describe the thin-film flow in the water layer separating the oil from the clay surface, and the macroscopic flow through the capillary, we derive expressions for the thickness of the wetting film, and the velocity of the oil slug, given a pressure difference across the ends of the capillary. Numerical results show that the thickness of the water layer and the velocity of the oil slug increase as the salinity of the water is reduced, suggesting that this mechanism contributes to the low-salinity effect. An analytical solution is presented in the limit in which the applied pressure is small.
Edge behaviour in the glass sheet redraw process
- D. O’Kiely, C. J. W. Breward, I. M. Griffiths, P. D. Howell, U. Lange
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- Journal:
- Journal of Fluid Mechanics / Volume 785 / 25 December 2015
- Published online by Cambridge University Press:
- 17 November 2015, pp. 248-269
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Thin glass sheets may be manufactured using a two-part process in which a sheet is first cast and then subsequently reheated and drawn to a required thickness. The latter redrawing process typically results in a sheet with non-uniform thickness and with smaller width than the cast glass block. Experiments suggest that the loss of width can be minimized and the non-uniformities can be essentially confined to thickening at the sheet edges if the heater zone through which the glass is drawn is made very short. We present a three-dimensional mathematical model for the redraw process and consider the limits in which (i) the heater zone is short compared with the sheet width, and (ii) the sheet thickness is small compared with both of these length scales. We show that, in the majority of the sheet, the properties vary only in the direction of drawing and the sheet motion is one-dimensional, with two-dimensional behaviour and the corresponding thick edges confined to boundary layers at the sheet extremities. We present numerical solutions to this boundary-layer problem and demonstrate good agreement with experiment, as well as with numerical solutions to the full three-dimensional problem. We show that the final thickness at the sheet edge scales with the inverse square root of the draw ratio, and explore the effect of tapering of the ends to identify a shape for the initial preform that results in a uniform rectangular final product.
Straining flow of a weakly interacting polymer–surfactant solution
- C. J. W. BREWARD, I. M. GRIFFITHS, P. D. HOWELL, C. E. MORGAN
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- European Journal of Applied Mathematics / Volume 26 / Issue 5 / October 2015
- Published online by Cambridge University Press:
- 16 July 2015, pp. 743-772
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In this paper, we consider the straining flow of a weakly interacting polymer–surfactant solution below a free surface, with the bulk surfactant concentration above the critical micelle concentration. We formulate a set of coupled differential equations describing the concentration of monomers, micelles, polymer, and polymer–micelle aggregates in the flow. We analyse the model in several asymptotic limits, and make predictions about the distribution of each of the species. In particular, in the large-reaction-rate limit we find that the model predicts a region near the free surface where no micelles or aggregates are present, and beneath this a region where the concentration of surfactant is constant, across which the concentration of aggregates increases until all the free polymer is consumed. For certain parameter regimes, a maximum in the concentration of the polymer–micelle complex occurs within the bulk fluid. In the finite-reaction-rate limit, micelles, and aggregates are present right up to the free surface, and the plateau in the concentration of surfactant in the bulk is no longer present. Results from the asymptotic theory compare favorably with full numerical solutions.
The influence of non-polar lipids on tear film dynamics
- M. Bruna, C. J. W. Breward
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- Journal of Fluid Mechanics / Volume 746 / 10 May 2014
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- 04 April 2014, pp. 565-605
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In this paper we examine the effect that physiological non-polar lipids, residing on the surface of an aqueous tear film, have on the film evolution. In our model we track the evolution of the thickness of the non-polar lipid layer, the thickness of the aqueous layer and the concentration of polar lipids which reside at the interface between the two. We also utilise a force balance in the non-polar lipid layer in order to determine its velocity. We show how to obtain previous models in the literature from our model by making particular choices of the parameters. We see the formation of boundary layers in some of these submodels, across which the concentration of polar lipid and the non-polar lipid velocity and film thickness vary. We solve our model numerically for physically realistic parameter values, and we find that the evolution of the aqueous layer and the polar lipid layer are similar to that described by previous authors. However, there are interesting dynamics for the non-polar lipid layer. The effects of altering the key parameters are highlighted and discussed. In particular, we see that the Marangoni number plays a key role in determining how far over the eye the non-polar lipid spreads.
Boundary conditions for free surface inlet and outlet problems
- M. Taroni, C. J. W. Breward, P. D. Howell, J. M. Oliver
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- Journal of Fluid Mechanics / Volume 708 / 10 October 2012
- Published online by Cambridge University Press:
- 10 August 2012, pp. 100-110
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We investigate and compare the boundary conditions that are to be applied to free-surface problems involving inlet and outlets of Newtonian fluid, typically found in coating processes. The flux of fluid is a priori known at an inlet, but unknown at an outlet, where it is governed by the local behaviour near the film-forming meniscus. In the limit of vanishing capillary number it is well known that the flux scales with , but this classical result is non-uniform as the contact angle approaches . By examining this limit we find a solution that is uniformly valid for all contact angles. Furthermore, by considering the far-field behaviour of the free surface we show that there exists a critical capillary number above which the problem at an inlet becomes over-determined. The implications of this result for the modelling of coating flows are discussed.
Contributors
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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Straining flow of a micellar surfactant solution
- C. J. W. BREWARD, P. D. HOWELL
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- European Journal of Applied Mathematics / Volume 15 / Issue 5 / October 2004
- Published online by Cambridge University Press:
- 04 March 2005, pp. 511-531
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We present a mathematical model describing the distribution of monomer and micellar surfactant in a steady straining flow beneath a fixed free surface. The model includes adsorption of monomer surfactant at the surface and a single-step reaction whereby $n$ monomer molecules combine to form each micelle. The equations are analysed asymptotically and numerically and the results are compared with experiments. Previous studies of such systems have often assumed equilibrium between the monomer and micellar phases, i.e. that the reaction rate is effectively infinite. Our analysis shows that such an approach inevitably fails under certain physical conditions and also cannot accurately match some experimental results. Our theory provides an improved fit with experiments and allows the reaction rates to be estimated.
Mathematical modelling of the overflowing cylinder experiment
- P. D. HOWELL, C. J. W. BREWARD
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- Journal:
- Journal of Fluid Mechanics / Volume 474 / 10 January 2003
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- 14 January 2003, pp. 275-298
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The overflowing cylinder (OFC) is an experimental apparatus designed to generate a controlled straining flow at a free surface, whose dynamic properties may then be investigated. Surfactant solution is pumped up slowly through a vertical cylinder. On reaching the top, the liquid forms a flat free surface which expands radially before over flowing down the side of the cylinder. The velocity, surface tension and surfactant concentration on the expanding free surface are measured using a variety of non-invasive techniques.
A mathematical model for the OFC has been previously derived by Breward et al. (2001) and shown to give satisfactory agreement with experimental results. However, a puzzling indeterminacy in the model renders it unable to predict one scalar parameter (e.g. the surfactant concentration at the centre of the cylinder), which must be therefore be taken from the experiments.
In this paper we analyse the OFC model asymptotically and numerically. We show that solutions typically develop one of two possible singularities. In the first, the surface concentration of surfactant reaches zero a finite distance from the cylinder axis, while the surface velocity tends to infinity there. In the second, the surfactant concentration is exponentially large and a stagnation point forms just inside the rim of the cylinder. We propose a criterion for selecting the free parameter, based on the elimination of both singularities, and show that it leads to good agreement with experimental results.
The drainage of a foam lamella
- C. J. W. BREWARD, P. D. HOWELL
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- Journal:
- Journal of Fluid Mechanics / Volume 458 / 10 May 2002
- Published online by Cambridge University Press:
- 23 May 2002, pp. 379-406
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We present a mathematical model for the drainage of a surfactant-stabilized foam lamella, including capillary, Marangoni and viscous effects and allowing for diffusion, advection and adsorption of the surfactant molecules. We use the slender geometry of a lamella to formulate the model in the thin-film limit and perform an asymptotic decomposition of the liquid domain into a capillary-static Plateau border, a time-dependent thin film and a transition region between the two. By solving a quasi-steady boundary-value problem in the transition region, we obtain the flux of liquid from the lamella into the Plateau border and thus are able to determine the rate at which the lamella drains.
Our method is illustrated initially in the surfactant-free case. Numerical results are presented for three particular parameter regimes of interest when surfactant is present. Both monotonic profiles and those exhibiting a dimple near the Plateau border are found, the latter having been previously observed in experiments. The velocity field may be uniform across the lamella or of parabolic Poiseuille type, with fluid either driven out along the centreline and back along the free surfaces or vice versa. We find that diffusion may be negligible for a typical real surfactant, although this does not lead to a reduction in order because of the inherently diffusive nature of the fluid–surfactant interaction. Finally, we obtain the surprising result that the flux of liquid from the lamella into the Plateau border increases as the lamella thins, approaching infinity at a finite lamella thickness.
Modelling the interactions between tumour cells and a blood vessel in a microenvironment within a vascular tumour
- C. J. W. BREWARD, H. M. BYRNE, C. E. LEWIS
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- Journal:
- European Journal of Applied Mathematics / Volume 12 / Issue 5 / October 2001
- Published online by Cambridge University Press:
- 28 November 2001, pp. 529-556
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In this paper, we develop a mathematical model to describe interactions between tumour cells and a compliant blood vessel that supplies oxygen to the region. We assume that, in addition to proliferating, the tumour cells die through apoptosis and necrosis. We also assume that pressure differences within the tumour mass, caused by spatial variations in proliferation and degradation, cause cell motion. We couple the behaviour of the blood vessel into the model for the oxygen tension. The model equations track the evolution of the densities of live and dead cells, the oxygen tension within the tumour, the live and dead cell speeds, the pressure and the width of the blood vessel. We present explicit solutions to the model for certain parameter regimes, and then solve the model numerically for more general parameter regimes. We show how the resulting steady-state behaviour varies as the key model parameters are changed. Finally, we discuss the biological implications of our work.