Yvonne Stokes

Unsteady stretching of a glass tube with internal channel pressurization

Publisher: AIP Publishing

Date: 08-2022

DOI: 10.1063/5.0096725

Abstract: Mathematical modeling is used to examine the unsteady problem of heating and pulling an axisymmetric cylindrical glass tube with an over-pressure applied within the tube to form tapers with a near uniform bore and small wall thickness at the tip. To allow for the dependence of viscosity on temperature, a prescribed axially varying viscosity is assumed. Our motivation is the manufacture of emitter tips for mass spectrometry which provide a continuous fluid flow and do not become blocked. We demonstrate, for the first time, the feasibility of producing such emitters by this process and examine the influence of the process parameters, in particular the pulling force and over-pressure, on the geometry. There is not a unique force and over-pressure combination to achieve the desired geometry at the tip but smaller over-pressure (hence force) yields a more uniform bore over the entire length of the emitter than does a larger over-pressure (and force). However, the sensitivity of the geometry to small fluctuations in the parameters increases as the over-pressure decreases. The best parameters depend on the accuracy of the puller used to manufacture the tapers and the permissible tolerances on the geometry. The model has wider application to the manufacture of other devices.

Thin-film flow in helically wound shallow channels of arbitrary cross-sectional shape

Publisher: AIP Publishing

Date: 2017

DOI: 10.1063/1.4973670

Abstract: We consider the steady, gravity-driven flow of a thin film of viscous fluid down a helically wound shallow channel of arbitrary cross-sectional shape with arbitrary torsion and curvature. This extends our previous work [D. J. Arnold et al., “Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015)] on channels of rectangular cross section. The Navier-Stokes equations are expressed in a novel, non-orthogonal coordinate system fitted to the channel bottom. By assuming that the channel depth is small compared to its width and that the fluid depth in the vertical direction is also small compared to its typical horizontal extent, we are able to solve for the velocity components and pressure analytically. Using these results, a differential equation for the free surface shape is obtained, which must in general be solved numerically. Motivated by the aim of understanding flows in static spiral particle separators used in mineral processing, we investigate the effect of cross-sectional shape on the secondary flow in the channel cross section. We show that the competition between gravity and inertia in non-rectangular channels is qualitatively similar to that in rectangular channels, but that the cross-sectional shape has a strong influence on the breakup of the secondary flow into multiple clockwise-rotating cells. This may be triggered by small changes to the channel geometry, such as one or more bumps in the channel bottom that are small relative to the fluid depth. In contrast to the secondary flow which is quite sensitive to small bumps in the channel bottom, the free-surface profile is relatively insensitive to these. The sensitivity of the flow to the channel geometry may have important implications for the design of efficient spiral particle separators.

Dynamics of Small Particle Inertial Migration in Curved Square Ducts

Publisher: Society for Industrial & Applied Mathematics (SIAM)

Date: 03-2022

DOI: 10.1137/21M1430935

Microstructured optical fibre drawing with active channel pressurisation

Publisher: Cambridge University Press (CUP)

Date: 13-10-2015

DOI: 10.1017/JFM.2015.570

Abstract: The use of channel pressurisation in drawing microstructured optical fibres (MOFs) potentially allows for fine control of the internal structure of the fibre. By applying extra pressure inside the channels it is possible to counteract the effect of surface tension which would otherwise act to close the channels in the fibre as it is drawn. This paper extends the modelling approach of Stokes et al. ( J. Fluid Mech. , vol. 755, 2014, pp. 176–203) to include channel pressurisation. This approach treats the problem as two submodels for the flow, one in the axial direction along the fibre and another in the plane perpendicular to that direction. In the absence of channel pressurisation these models decoupled and were solved independently we show that they become fully coupled when the internal channels are pressurised. The fundamental case of a fibre with an annular cross-section (containing one central channel) will be examined in detail. In doing this we consider both a forward problem to determine the shape of fibre from a known preform and an inverse problem to design a preform such that when drawn it will give a desired fibre geometry. Criteria on the pressure corresponding to fibre explosion and closure of the channel will be given that represent an improvement over similar criteria in the literature. A comparison between our model and a recent experiment is presented to demonstrate the effectiveness of the modelling approach. We make use of some recent work by Buchak et al. ( J. Fluid Mech. , vol. 778, 2015, pp. 5–38) to examine more complicated fibre geometries, where the cross-sectional shape of the internal channels is assumed to be elliptical and multiple channels are present. The ex les presented here demonstrate the versatility of our modelling approach, where the subtleties of the interaction between surface tension and pressurisation can be revealed even for complex patterns of cross-sectional channels.

Can We Fabricate That Fibre?

Publisher: Springer International Publishing

Date: 2019

DOI: 10.1007/978-3-030-13720-5_1

The evolution of a viscous thread pulled with a prescribed speed

Publisher: Cambridge University Press (CUP)

Date: 14-04-2016

DOI: 10.1017/JFM.2016.215

Abstract: We examine the extension of an axisymmetric viscous thread that is pulled at both ends with a prescribed speed such that the effects of inertia are initially small. After neglecting surface tension, we derive a particularly convenient form of the long-wavelength equations that describe long and thin threads. Two generic classes of initial thread shape are considered as well as the special case of a circular cylinder. In these cases, we determine explicit asymptotic solutions while the effects of inertia remain small. We further show that inertia will ultimately become important only if the long-time asymptotic form of the pulling speed is faster than a power law with a critical exponent. The critical exponent can take two possible values depending on whether or not the initial minimum of the thread radius is located at the pulled end. In addition, we obtain asymptotic expressions for the solution at large times in the case in which the critical exponent is exceeded and hence inertia becomes important. Despite the apparent simplicity of the problem, the solutions exhibit a surprisingly rich structure. In particular, in the case in which the initial minimum is not at the pulled end, we show that there are two very different types of solution that exhibit very different extension mechanics. Both the small-inertia solutions and the large-time asymptotic expressions compare well with numerical solutions.

Drawing of micro-structured fibres: circular and non-circular tubes

Publisher: Cambridge University Press (CUP)

Date: 14-08-2014

DOI: 10.1017/JFM.2014.408

Abstract: A general mathematical framework is presented for modelling the pulling of optical glass fibres in a draw tower. The only modelling assumption is that the fibres are slender cross-sections along the fibre can have general shape, including the possibility of multiple holes or channels. A key result is to demonstrate how a so-called reduced time variable $\\def \\xmlpi #1{}\\def \\mathsfbi #1{\\boldsymbol {\\mathsf {#1}}}\\let \\le =\\leqslant \\let \\leq =\\leqslant \\let \\ge =\\geqslant \\let \\geq =\\geqslant \\def \\Pr {\\mathit {Pr}}\\def \\Fr {\\mathit {Fr}}\\def \\Rey {\\mathit {Re}}\\tau $ serves as a natural parameter in describing how an axial-stretching problem interacts with the evolution of a general surface-tension-driven transverse flow via a single important function of $\\tau $ , herein denoted by $H(\\tau )$ , derived from the total rescaled cross-plane perimeter. For any given preform geometry, this function $H(\\tau )$ may be used to calculate the tension required to produce a given fibre geometry, assuming only that the surface tension is known. Of principal practical interest in applications is the ‘inverse problem’ of determining the initial cross-sectional geometry, and experimental draw parameters, necessary to draw a desired final cross-section. Two case studies involving annular tubes are presented in detail: one involves a cross-section comprising an annular concatenation of sintering near-circular discs, the cross-section of the other is a concentric annulus. These two ex les allow us to exemplify and explore two features of the general inverse problem. One is the question of the uniqueness of solutions for a given set of experimental parameters, the other concerns the inherent ill-posedness of the inverse problem. Based on these ex les we also give an experimental validation of the general model and discuss some experimental matters, such as buckling and stability. The ramifications for modelling the drawing of fibres with more complicated geometries, and multiple channels, are discussed.

A mathematical model for cell-induced gel contraction incorporating osmotic effects

Publisher: Cold Spring Harbor Laboratory

Date: 10-12-2021

DOI: 10.1101/2021.12.08.471846

Abstract: Biological tissues are composed of cells surrounded by the extracellular matrix (ECM). The ECM can be thought of as a fibrous polymer network, acting as a natural scaffolding to provide mechanical support to the cells. Reciprocal mechanical and chemical interactions between the cells and the ECM are crucial in regulating the development of tissues and maintaining their functionality. Hence, to maintain in vivo -like behaviour when cells are cultured in vitro , they are often seeded in a gel, which aims to mimic the ECM. In this paper, we present a mathematical model that incorporate cell-gel interactions together with osmotic pressure to study the mechanical behaviour of biological gels. In particular, we consider an experiment where cells are seeded within a gel, which gradually compacts due to forces exerted on it by the cells. Adopting a one-dimensional Cartesian geometry for simplicity, we use a combination of analytical techniques and numerical simulations to investigate how cell traction forces interact with osmotic effects (which can lead to either gel swelling or contraction depending on the gel’s composition). Our results show that a number of qualitatively different behaviours are possible, depending on the composition of the gel (i.e. the chemical potentials) and the strength of the cell traction forces. We observe an unusual case where the gel oscillates between swelling and contraction. We also consider on how physical parameters like drag and viscosity affect the manner in which the gel evolves.

Coupled fluid and energy flow in fabrication of microstructured optical fibres

Publisher: Cambridge University Press (CUP)

Date: 11-07-2019

DOI: 10.1017/JFM.2019.466

Abstract: We consider the role of heating and cooling in the steady drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape. The internal holes and the external boundary evolve as a result of the axial drawing and surface-tension effects. The heating and cooling affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We use asymptotic techniques to show that, under a suitable transformation, this complicated three-dimensional free boundary problem can be formulated in such a way that the transverse aspect of the flow can be reduced to the solution of a standard Stokes flow problem in the absence of axial stretching. The solution of this standard problem can then be substituted into a system of three ordinary differential equations that completely determine the flow. We use this approach to develop a very simple numerical method that can determine the way that thermal effects impact on the drawing of threads by devices that either specify the fibre tension or the draw ratio. We also develop a numerical method to solve the inverse problem of determining the initial cross-sectional geometry, draw tension and, importantly, heater temperature to obtain a desired cross-sectional shape and change in cross-sectional area at the device exit. This precisely allows one to determine the pattern of air holes in the preform that will achieve the desired hole pattern in the stretched fibre.

Utilizing bifurcations to separate particles in spiral inertial microfluidics

Publisher: AIP Publishing

Date: 2023

DOI: 10.1063/5.0132151

Abstract: Particles suspended in fluid flow through a closed duct can focus to specific stable locations in the duct cross section due to hydrodynamic forces arising from the inertia of the disturbed fluid. Such particle focusing is exploited in biomedical and industrial technologies to separate particles by size. In curved ducts, the particle focusing is a result of balance between two dominant forces on the particle: (i) inertial lift arising from small inertia of the fluid and (ii) drag arising from cross-sectional vortices induced by the centrifugal force on the fluid. Bifurcations of particle equilibria take place as the bend radius of the curved duct varies. By using the mathematical model of Harding et al. [J. Fluid Mech. 875, 1–43 (2019)], we illustrate via numerical simulations that these bifurcations can be leveraged in a spiral duct to achieve a large separation between different sized neutrally buoyant particles and identify a separation mechanism, not previously reported, which exploits the transient focusing of smaller particles near saddle points. We demonstrate this for similar sized particles, as well as particles that have a large difference in size, using spiral ducts with a square cross section. The novel formalism of using bifurcations to manipulate particle focusing can be applied more broadly to different geometries in inertial microfluidics, which may open new avenues in particle separation techniques.

Particle-laden thin-film flow in helical channels with arbitrary shallow cross-sectional shape

Publisher: AIP Publishing

Date: 07-2019

DOI: 10.1063/1.5092814

Abstract: Particle-laden flows in helical channels are of interest for their applications in spiral particle separators used in the mining and mineral processing industries. In this paper, we extend the previous work of Lee, Stokes, and Bertozzi [“Behaviour of a particle-laden flow in a spiral channel,” Phys. Fluids 26, 043302 (2014)] by studying thin-film flows of monodisperse particle-laden fluid in helically wound channels of arbitrary centerline curvature and torsion and arbitrary cross-sectional shape. In the case where the particles are uniformly distributed through the depth of the film, significant analytic progress can be made yielding insight into the influence of channel geometry on particle distribution across the channel cross section: the governing equations reduce to a single nonlinear ordinary differential equation, which is readily integrated numerically to obtain the solution subject to appropriate boundary conditions. Motivated by possible application to the design of spiral separators, we consider the effects of changing the channel centerline geometry, the cross-sectional shape and the particle density on the resulting flows, and the radial distribution of particles. Our results support the findings in the work of Arnold, Stokes, and Green [“Thin-film flow in helically wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015)] regarding the effect of channel centerline geometry and cross-sectional shape on flows in particle-free regions. In particle-rich regions, similar effects are seen although the primary velocity is lower due to increased effective mixture viscosity. Of key interest is the effect of channel geometry on the focusing of the particles for given fluxes of fluid and particles. We find that introducing a trench into the channel cross section, a feature often used in commercial spiral particle separators, leads to a smaller radial width of the particle-rich region, i.e., sharper focusing of the particles, which is consistent with experiments showing that channel geometry influences particle separation in a spiral separator.

Flowing windowpanes: fact or fiction?

Publisher: The Royal Society

Date: 08-07-1999

DOI: 10.1098/RSPA.1999.0425

Flowing windowpanes: a comparison of Newtonian and Maxwell fluid models

Publisher: The Royal Society

Date: 08-08-2000

DOI: 10.1098/RSPA.2000.0590

Bifurcations and Dynamics in Inertial Focusing of Particles in Curved Rectangular Ducts

Publisher: Society for Industrial & Applied Mathematics (SIAM)

Date: 02-12-2022

DOI: 10.1137/21M1451919

Computation of extensional fall of slender viscous drops by a one-dimensional eulerian method

Publisher: Society for Industrial & Applied Mathematics (SIAM)

Date: 2007

DOI: 10.1137/050646743

AN EXAMINATION OF THE “LANIER WING” DESIGN

Publisher: Cambridge University Press (CUP)

Date: 2023

DOI: 10.1017/S1446181123000068

Abstract: Six patents were secured by E. H. Lanier from 1930 to 1933 for aeroplane designs that were intended to be exceptionally stable. A feature of five of these was a flow-induced “vacuum chamber” which was thought to provide superior stability and increased lift compared to typical wing designs. Initially, this chamber was in the fuselage, but later designs placed it in the wing by replacing a section of the upper skin of the wing with a series of angled slats. We report upon an investigation of the Lanier wing design using inviscid aerodynamic theory and viscous numerical simulations. This took place at the 2005 Australia–New Zealand Mathematics-in-Industry Study Group. The evidence from this investigation does not support the claims but, rather, suggests that any improvement in lift and/or stability seen in the few prototypes that were built was, most probably, due to thicker airfoils than were typical at the time.

Stability of drawing of microstructured optical fibres

Publisher: Cambridge University Press (CUP)

Date: 27-04-2023

DOI: 10.1017/JFM.2023.267

Abstract: We consider the stability of the drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape that evolves due to axial drawing, inertia and surface tension effects. Despite the complicated geometry of the boundaries, we use asymptotic techniques to determine a particularly convenient formulation of the equations of motion that is well-suited to stability calculations. We will determine an explicit asymptotic solution for steady states with (a) large surface tension and negligible inertia, and (b) large inertia. In both cases, we will show that complicated boundary layer structures can occur. We will use linear stability analysis to show that the presence of an axisymmetric hole destabilises the flow for finite capillary number and which answers a question raised in the literature. However, our formulation allows us to go much further and consider arbitrary hole structures or non-axisymmetric shapes, and show that any structure with holes will be less stable than the case of a solid axisymmetric thread. For a solid axisymmetric thread, we will also determine a closed-form expression that delineates the unconditional instability boundary in which case the thread is unstable for all draw ratios. We will determine how the detailed effects of the microstructure affect the stability, and show that they manifest themselves only via a single function that occurs in the stability problem and hence have a surprisingly limited effect on the stability.

Mathematical modelling of oxygen concentration in bovine and murine cumulus-oocyte complexes.

Publisher: Bioscientifica

Date: 06-2006

DOI: 10.1530/REP.1.00974

Abstract: Immature oocytes benefit from nutrient modification of the follicular environment by the surrounding cumulus mass. However, the oxygen concentration that the oocyte may be exposed to could be lower than the antral follicular concentration due to the metabolism of surrounding cumulus cells. Using metabolic data previously determined, we have developed a mathematical model of O 2 diffusion across the bovine and murine cumulus–oocyte complex. From this we have determined that across a physiological range of external pO 2 , less than 0.25% and 0.5% O 2 is removed by cumulus cells within the bovine and murine cumulus–oocyte complex respectively. Our model differs from others as it: incorporates a term that allows for nonlinear variation of the oxygen consumption rate with oxygen concentration considers two regions (oocyte and cumulus) sharing a common boundary, both of which consume oxygen at different non linear rates. Cumulus cells therefore remove little O 2 , thus sparing this essential gas for the oocyte, which is dependent on ATP generation via oxidative phosphorylation.

Drawing tubular fibres: experiments versus mathematical modelling

Publisher: The Optical Society

Date: 15-12-2015

DOI: 10.1364/OME.6.000166

A mathematical model for cell-induced gel contraction incorporating osmotic effects

Publisher: Springer Science and Business Media LLC

Date: 16-03-2022

DOI: 10.1007/S00285-022-01730-6

Abstract: Biological tissues are composed of cells surrounded by the extracellular matrix (ECM). The ECM can be thought of as a fibrous polymer network, acting as a natural scaffolding to provide mechanical support to the cells. Reciprocal mechanical and chemical interactions between the cells and the ECM are crucial in regulating the development of tissues and maintaining their functionality. Hence, to maintain in vivo-like behaviour when cells are cultured in vitro, they are often seeded in a gel, which aims to mimic the ECM. In this paper, we present a mathematical model that incorporates cell-gel interactions together with osmotic pressure to study the mechanical behaviour of biological gels. In particular, we consider an experiment where cells are seeded within a gel, which gradually compacts due to forces exerted on it by the cells. Adopting a one-dimensional Cartesian geometry for simplicity, we use a combination of analytical techniques and numerical simulations to investigate how cell traction forces interact with osmotic effects (which can lead to either gel swelling or contraction depending on the gel’s composition). Our results show that a number of qualitatively different behaviours are possible, depending on the composition of the gel (i.e. its chemical potentials) and the strength of the cell traction forces. A novel prediction of our model is that there are cases where the gel oscillates between swelling and contraction to our knowledge, this behaviour has not been reported in experiments. We also consider how physical parameters like drag and viscosity affect the manner in which the gel evolves.

Mathematical modeling of glucose supply toward successful in vitro maturation of Mammalian oocytes.

Publisher: Mary Ann Liebert Inc

Date: 09-2008

DOI: 10.1089/TEN.TEA.2008.0036

Abstract: In vitro maturation-whereby an oocyte is harvested from an ovary just before full maturation, matured in the laboratory, fertilized, and then transplanted back to the uterus-has important benefits over, but is significantly less successful than, traditional in vitro fertilization. Inadequate in vitro nutrient environments are believed to be a prime reason for the low success, but understanding of the in vivo environment, which needs to be better replicated in the laboratory, is still lacking. We here consider mathematical modeling as an aid to increasing that understanding. A general mathematical model suitable for examining the in vivo concentrations of a nutrient in the cumulus-oocyte complex (COC) is presented. We then tailor the model to consider glucose concentration. Experimental data are used to obtain information on glucose uptake in the COC for use in the model. Finally, we solve the model to estimate glucose concentration in the COC. With the information currently available, the model indicates a significant reduction in glucose concentration from the follicular fluid across the cumulus matrix to the oocyte.

Thin-film flow in helically wound rectangular channels with small torsion

Publisher: AIP Publishing

Date: 08-2013

DOI: 10.1063/1.4818628

Abstract: Laminar gravity-driven thin-film flow down a helically wound channel of rectangular cross-section with small torsion in which the fluid depth is small is considered. Neglecting the entrance and exit regions we obtain the steady-state solution that is independent of position along the axis of the channel, so that the flow, which comprises a primary flow in the direction of the axis of the channel and a secondary flow in the cross-sectional plane, depends only on position in the two-dimensional cross-section of the channel. A thin-film approximation yields explicit expressions for the fluid velocity and pressure in terms of the free-surface shape, the latter satisfying a nonlinear ordinary differential equation that has a simple exact solution in the special case of a channel of rectangular cross-section. The predictions of the thin-film model are shown to be in good agreement with much more computationally intensive solutions of the small-helix-torsion Navier–Stokes equations. The present work has particular relevance to spiral particle separators used in the mineral-processing industry. The validity of an assumption commonly used in modelling flow in spiral separators, namely, that the flow in the outer region of the separator cross-section is described by a free vortex, is shown to depend on the problem parameters.

Pore Level Simulation of Miscible Injection with Gravity Domination

Publisher: Elsevier BV

Date: 2013

DOI: 10.1016/J.EGYPRO.2013.06.621

A two-dimensional asymptotic model for capillary collapse

Publisher: Cambridge University Press (CUP)

Date: 17-12-2020

DOI: 10.1017/JFM.2020.954

An inverse modelling technique for glass forming by gravity sagging

Publisher: Elsevier BV

Date: 05-2005

DOI: 10.1016/J.EUROMECHFLU.2004.10.002

Gravitational extension of a fluid cylinder with internal structure

Publisher: Cambridge University Press (CUP)

Date: 03-02-2016

DOI: 10.1017/JFM.2016.11

Abstract: Motivated by the fabrication of microstructured optical fibres, a model is presented for the extension under gravity of a slender fluid cylinder with internal structure. It is shown that the general problem decouples into a two-dimensional surface-tension-driven Stokes flow that governs the transverse shape and an axial problem that depends upon the transverse flow. The problem and its solution differ from those obtained for fibre drawing, because the problem is unsteady and the fibre tension depends on axial position. Solutions both with and without surface tension are developed and compared, which show that the relative importance of surface tension depends upon both the parameter values and the geometry under consideration. The model is compared with experimental data and is shown to be in good agreement. These results also show that surface-tension effects are essential to accurately describing the cross-sectional shape.

Prematuration with cyclic adenosine monophosphate modulators alters cumulus cell and oocyte metabolism and enhances developmental competence of in vitro- matured mouse oocytes

Publisher: Oxford University Press (OUP)

Date: 08-2014

DOI: 10.1095/BIOLREPROD.114.118471

Abstract: Oocyte in vitro maturation (IVM) is an important assisted reproductive technology and research tool. The adoption of IVM into routine clinical practice has been hindered by its significantly lower success rates compared to conventional in vitro fertilization. Cyclic AMP (cAMP) modulation and follicle-stimulating hormone (FSH), independently, have long been known to improve IVM oocyte developmental competence. This study comprehensively examined the effects of FSH and cAMP/cGMP modulation, alone and in combination, on IVM oocyte metabolism and developmental outcomes. Mouse cumulus-oocyte complexes (COCs) were subjected to a 1 h prematuration phase ± the cAMP modulator forskolin and cAMP/cGMP modulator 3-isobutyl-1-methylxanthine followed by IVM ± FSH. Prematuration with these cyclic nucleotide modulators or IVM with FSH significantly improved oocyte developmental competence and reduced spindle abnormalities compared to spontaneous IVM (no treatment) however, these two treatments in combination endowed even greater developmental competence (improved subsequent blastocyst rates and quality P < 0.05), albeit blastocyst yield and quality remained significantly lower than that of oocytes matured in vivo. A significant additive effect of combined IVM treatments was evident as increased COC lactate production and oxygen consumption and enhanced oocyte oxidative metabolism, ATP production, ATP:ADP ratio, and glutathione levels (P < 0.05). Nevertheless, IVM increased reactive oxygen species production, particularly as a consequence of FSH addition, relative to in vivo matured oocytes. In conclusion, improvements in the embryo yield following IVM is associated with increased COC oxygen consumption and oocyte oxidative metabolism, but these remain metabolically and developmentally less competent relative to in vivo derived oocytes.

Behavior of a particle-laden flow in a spiral channel

Publisher: AIP Publishing

Date: 04-2014

DOI: 10.1063/1.4872035

Abstract: Spiral gravity separators are devices used in mineral processing to separate particles based on their specific gravity or size. The spiral geometry allows for the simultaneous application of gravitational and centripetal forces on the particles, which leads to segregation of particles. However, this segregation mechanism is not fundamentally understood, and the spiral separator literature does not tell a cohesive story either experimentally or theoretically. While experimental results vary depending on the specific spiral separator used, present theoretical works neglect the significant coupling between the particle dynamics and the flow field. Using work on gravity-driven monodisperse slurries on an incline that empirically accounts for this coupling, we consider a monodisperse particle slurry of small depth flowing down a rectangular channel that is helically wound around a vertical axis. We use a thin-film approximation to derive an equilibrium profile for the particle concentration and fluid depth and find that, in the steady state limit, the particles concentrate towards the vertical axis of the helix, leaving a region of clear fluid.

Numerical design tools for thermal replication of optical-quality surfaces

Publisher: Elsevier BV

Date: 05-2000

DOI: 10.1016/S0045-7930(99)00030-4

Wet chemical etching of single-bore microstructured silicon dioxide fibers

Publisher: AIP Publishing

Date: 07-2020

DOI: 10.1063/5.0014335

Abstract: We model the process of wet chemical etching of the external surface of a single-bore microstructured silicon dioxide fiber in hydrofluoric acid (HFA) while water is pumped through the internal channel to prevent etching of it. The model uses the Stokes flow for the velocity throughout the system and the advection–diffusion equation for the concentration of HFA. We determine the etch rate as a function of HFA concentration using data from experiments designed for this purpose, from which we calculate the change in the fiber surface. We solve our equations using a time-stepping finite-element method and verify our model by comparing to results found experimentally. We investigate the effects of different water flow rates, diffusivity, buoyancy, and bore radius. We find the water being pumped through the bore does not fully protect it and there is some etching of the internal channel, which is difficult to see in experimental images. We also obtain an estimate of the diffusivity of high-concentration HFA in water.

Extensional flow at low Reynolds number with surface tension

Publisher: Springer Science and Business Media LLC

Date: 08-12-2011

DOI: 10.1007/S10665-010-9443-3

Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature

Publisher: Cambridge University Press (CUP)

Date: 23-12-2014

DOI: 10.1017/JFM.2014.703

Abstract: Laminar helically-symmetric gravity-driven thin-film flow down a helically-wound channel of rectangular cross-section is considered. We extend the work of Stokes et al. ( Phys. Fluids , vol. 25 (8), 2013, 083103) and Lee et al. ( Phys. Fluids , vol. 26 (4), 2014, 043302) to channels with arbitrary curvature and torsion or, equivalently, arbitrary curvature and slope. We use a non-orthogonal coordinate system and, remarkably, find an exact steady-state solution. We find that the free-surface shape and flow have a complicated dependence on the curvature, slope and flux down the channel. Moderate to large channel slope has a significant effect on the flow in the region of the channel near the inside wall, particularly when the curvature of the channel is large. This work has application to flow in static spiral particle separators used in mineral processing.

Determining rotational deformity in broken forearms

Publisher: Cambridge University Press (CUP)

Date: 04-2003

DOI: 10.1017/S144618110001292X

Abstract: To assess rotational deformity in a broken forearm, an orthopaedic surgeon needs to determine the amount of rotation of the radius from one or more two-dimensional x-rays of the fracture. This requires only simple first-year university mathematics — rotational transformations of ellipses plus a little differential calculus — which yields a general formula giving the rotation angle from information obtained from an x-ray. Preliminary comparisons with experimental results are excellent. This is a practical problem that may be useful to motivate the teaching of conic sections.

Simple Analysis of Line Packing, Attenuation, and Rarefaction Phenomena in Water Hammer

Publisher: American Society of Civil Engineers (ASCE)

Date: 10-2017

DOI: 10.1061/(ASCE)HY.1943-7900.0001367

Effect of inertial lift on a spherical particle suspended in flow through a curved duct

Publisher: Cambridge University Press (CUP)

Date: 18-07-2019

DOI: 10.1017/JFM.2019.323

Abstract: We develop a model of the forces on a spherical particle suspended in flow through a curved duct under the assumption that the particle Reynolds number is small. This extends an asymptotic model of inertial lift force previously developed to study inertial migration in straight ducts. Of particular interest is the existence and location of stable equilibria within the cross-sectional plane towards which particles migrate. The Navier–Stokes equations determine the hydrodynamic forces acting on a particle. A leading-order model of the forces within the cross-sectional plane is obtained through the use of a rotating coordinate system and a perturbation expansion in the particle Reynolds number of the disturbance flow. We predict the behaviour of neutrally buoyant particles at low flow rates and examine the variation in focusing position with respect to particle size and bend radius, independent of the flow rate. In this regime, the lateral focusing position of particles approximately collapses with respect to a dimensionless parameter dependent on three length scales: specifically, the particle radius, duct height and duct bend radius. Additionally, a trapezoidal-shaped cross-section is considered in order to demonstrate how changes in the cross-section design influence the dynamics of particles.

Flow in spiral channels of small curvature and torsion

Publisher: Springer Netherlands

Date: 2001

DOI: 10.1007/978-94-010-0796-2_35

The role of inertia in extensional fall of a viscous drop

Publisher: Cambridge University Press (CUP)

Date: 10-01-2004

DOI: 10.1017/S0022112003006682

A NOTE ON NAVIER-STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES

Publisher: Cambridge University Press (CUP)

Date: 2018

DOI: 10.1017/S144618111700058X

Abstract: There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

Estimation of glucose uptake by ovarian follicular cells.

Publisher: Springer Science and Business Media LLC

Date: 16-07-2011

DOI: 10.1007/S10439-011-0353-Y

Abstract: In vitro maturation (IVM) of mammalian oocytes provides an alternative to traditional in vitro fertilization techniques for clinical treatment of infertility or animal breeding. IVM involves the collection of oocytes from the ovary prior to ovulation, with maturation occurring in a laboratory environment. The success of IVM is highly sensitive to the in vitro nutrient environment. The nurse cells surrounding the oocyte, known as cumulus cells, regulate this environment and removal of these cells reduces the ability of the oocyte to develop following insemination. Determining the nature of the interaction between the oocyte and cumulus cells, collectively called the cumulus-oocyte complex (COC), is a difficult task experimentally. Here we use a combination of experimental and mathematical techniques to investigate glucose transport within bovine COCs and find quantitative estimates of the glucose uptake rates of the oocyte and cumulus cells. Surprisingly, our modeling shows the rate of uptake of glucose by the oocyte to increase and then decrease with concentration, a result that needs further experimental investigation but which supports the expectation that high and low glucose concentrations are detrimental to oocyte development. The methodology described is suitable for use across species and for investigating the transport of other important nutrients within the COC.

Inertial focusing of non-neutrally buoyant spherical particles in curved microfluidic ducts

Publisher: Cambridge University Press (CUP)

Date: 04-09-2020

DOI: 10.1017/JFM.2020.589

Pressure drop in pipelines due to pump trip event

Publisher: Australian Mathematical Publishing Association, Inc.

Date: 05-10-2016

DOI: 10.21914/ANZIAMJ.V57I0.10277

Fluid flow in a spiral microfluidic duct

Publisher: AIP Publishing

Date: 04-2018

DOI: 10.1063/1.5026334

Abstract: We consider the steady, pressure driven flow of a viscous fluid through a microfluidic device having the geometry of a planar spiral duct with a slowly varying curvature and height smaller than width. For this problem, it is convenient to express the Navier–Stokes equations in terms of a non-orthogonal coordinate system. Then, after applying appropriate scalings, the leading order equations admit a relatively simple solution in the central region of the duct cross section. First-order corrections with respect to the duct curvature and aspect ratio parameters are also obtained for this region. Additional correction terms are needed to ensure that no slip and no penetration conditions are satisfied on the side walls. Our solutions allow for a top wall shape that varies with respect to the radial coordinate which allows us to study the flow in a variety of cross-sectional shapes, including trapezoidal-shaped ducts that have been studied experimentally. At leading order, the flow is found to depend on the local height and slope of the top wall within the central region. The solutions are compared with numerical approximations of a classical Dean flow and are found to be in good agreement for a small duct aspect ratio and a slowly varying and small curvature. We conclude that the slowly varying curvature typical of spiral microfluidic devices has a negligible impact on the flow in the sense that locally the flow does not differ significantly from the classical Dean flow through a duct having the same curvature.

Lubrication analysis and numerical simulation of the viscous micropump with slip

Publisher: Elsevier BV

Date: 02-2012

DOI: 10.1016/J.IJHEATFLUIDFLOW.2011.09.009

Quantifying Oxygen Diffusion in Paraffin Oil Used in Oocyte and Embryo Culture

Publisher: Wiley

Date: 11-08-2009

DOI: 10.1002/MRD.21089

Abstract: Oxygen diffusion through oil is important in the culture of oocytes and embryos. A diffusion coefficient two orders of magnitude smaller than that of oxygen in water has been thought possible, and this has led to concerns of anoxia in cultures. Using an assay for determining the oxygen consumption rate of embryos and oocytes, along with a mathematical model, it is here shown that the oxygen diffusion rate in paraffin oil at 37 degrees C is about two-thirds of that in water at the same temperature. Although not previously recognised for the assay in question, the geometry is such that anoxia does occur for a period of time in excess of 1 hr and, by the completion of the assay, 30-40% of the medium is anoxic. Hence the quantity of oxygen consumed is less than would be consumed in conditions of plentiful oxygen supply. Nevertheless, using a model with a concentration dependent oxygen consumption rate, the oxygen consumption rate can be estimated.

Extrusion of fluid cylinders of arbitrary shape with surface tension and gravity

Publisher: Cambridge University Press (CUP)

Date: 24-11-2016

DOI: 10.1017/JFM.2016.729

Abstract: A model is developed for the extrusion in the direction of gravity of a slender fluid cylinder from a die of arbitrary shape. Both gravity and surface tension act to stretch and deform the geometry. The model allows for an arbitrary but prescribed viscosity profile, while the effects of extrudate swell are neglected. The solution is found efficiently through the use of a carefully selected axial Lagrangian coordinate and a transformation to a reduced-time variable. Comparisons between the model and extruded glass microstructured optical fibre preforms show that surface tension has a significant effect on the geometry but the model does not capture all of the behaviour observed in practice. Experimental observations are used in conjunction with the model to argue that some deformation, due neither to surface tension nor gravity, occurs in or near the die exit. Methods are considered to overcome deformation due to surface tension.

Asymptotic Modelling of a Six-Hole MOF

Publisher: Institute of Electrical and Electronics Engineers (IEEE)

Date: 15-12-2016

DOI: 10.1109/JLT.2016.2628438

Inertial focusing of spherical particles in curved microfluidic ducts at moderate Dean numbers

Publisher: Cambridge University Press (CUP)

Date: 20-02-2023

DOI: 10.1017/JFM.2023.43

Abstract: We examine the effect of Dean number on the inertial focusing of spherical particles suspended in flow through curved microfluidic ducts. Previous modelling of particle migration in curved ducts assumed the flow rate was small enough that a leading-order approximation of the background flow with respect to the Dean number produces a reasonable model. Herein, we extend our model to situations involving a moderate Dean number (in the microfluidics context) while the particle Reynolds number remains small. Variations in the Dean number cause a change in the axial velocity profile of the background flow which influences the inertial lift force on a particle. Simultaneously, changes in the cross-sectional velocity components of the background flow directly affect the secondary flow induced drag. In keeping the particle Reynolds number small, we continue to approximate the inertial lift force using a regular perturbation while capturing the subtle effects from the modified background flow. This approach pushes the limits at which a regular perturbation is applicable to provide some insights into how variations in the Dean number influence particle focusing. Our results illustrate that, as the extrema in the background flow move towards the outside of edge of the cross-section with increasing Dean number, we observe a similar shift in the stable equilibria of some, but not all, particle sizes. This might be exploited to enhance the lateral separation of particles by size in a number of practical scenarios.

Investigation of oversized channels in tubular fibre drawing

Publisher: Optica Publishing Group

Date: 26-02-2021

DOI: 10.1364/OME.419607

Abstract: In a previous study, we compared experiments on drawing of axisymmetric tubular optical fibres to a mathematical model of this process. The model and experiments generally agreed closely. However, for some preforms and operational conditions, the internal channel of the drawn fibre was larger than predicted by the model. We have further investigated this phenomenon of an oversized channel with to determine the mechanism behind the size discrepancy. In particular we have explored the possibility of channel expansion similar to ‘self-pressurisation’ in fibres drawn from preforms that have been first sealed to the atmosphere, as previously described by Voyce et al. [ J. Lightwave Technol. 27 , 871 ( 2009 ) 10.1109/JLT.2007.916489 ] . For this, two pieces from each of two preforms with different inner to outer diameter ratios were drawn to fibre, one open to the atmosphere and the other with a sealed end. In addition, we have sectioned a cooled neck-down region from a previous experiment, for which the fibre had an oversized channel compared to the model prediction, and measured the cross-sectional slices. We here compare this new experimental data with the predictions of the previously derived model for drawing of an unsealed preform and a new model, developed herein, for drawing of a sealed tube. We establish that the observed oversized channels are not consistent with the self-pressurisation model for the sealed tube.

On generalised penalty approaches for slip, free surface and related boundary conditions in viscous flow simulation

Publisher: Emerald

Date: 09-08-2011

DOI: 10.1108/09615531111148455

Abstract: The purpose of this paper is to extend the penalty concept to treat partial slip, free surface, contact and related boundary conditions in viscous flow simulation. The penalty partial‐slip formulation is analysed and related to the classical Navier slip condition. The same penalty scheme also allows partial penetration through a boundary, hence the implementation of porous wall boundaries. The finite element method is used for investigating and interpreting penalty approaches to boundary conditions. The generalised penalty approach is verified by means of a novel variant of the circular‐Couette flow problem, having partial slip on one of the cylindrical boundaries, for which an analytic solution is derived. Further verificationis provided by consideration of viscous flow over a sphere with partial slip on the surface, and comparison of numerical and classical solutions. Numerical studies illustrate the versatility of the approach. The penalty approach is applied to some different boundaries: partial slip and partial penetration with no/full slip enetration as limiting cases free surface space‐ and time‐varying boundary conditions which allow progressive contact over time. Application is made to curved and inclined boundaries. Sensitivity of flow to penalty parameters is an avenue for continued research, as is application of the penalty approach for non‐Newtonian flows. This is the first work to show the relation between penalty formulation of boundary conditions and physical boundary conditions. It provides a method that overcomes past difficulties in implementing partial slip on boundaries of general shape, and which handles progressive contact. It also provides useful benchmark problems for future studies.

Slow slumping of a very viscous liquid bridge

Publisher: Springer Science and Business Media LLC

Date: 1997

DOI: 10.1023/A:1004200926153

Pinch-off masses of very viscous fluids extruded from dies of arbitrary shape

Publisher: AIP Publishing

Date: 07-2018

DOI: 10.1063/1.5035090

Abstract: The extrusion of slender very viscous fluid cylinders from dies of arbitrary geometry is modeled to approximate the mass of the first drop to pinch off. The model neglects inertia, which, although important to the dynamics near pinch-off, does not have a significant impact on the drop mass. Extrudate swell is also assumed to be negligible. The model is able to compute the pinch-off mass for fluid cylinders with cross-sectional geometries of any connectivity. By way of illustration, pinch-off masses are computed for epicycloidal cross sections, two touching circular rods, and circular cylinders with a single circular hole placed both centrally and off-centre. It is shown that the drop mass may be controlled by altering either the extrusion conditions or the die geometry.

Follicle Structure Influences the Availability of Oxygen to the Oocyte in Antral Follicles

Publisher: Hindawi Limited

Date: 2011

DOI: 10.1155/2011/287186

Abstract: The ability of an oocyte to successfully mature is highly dependent on intrafollicular conditions, including the size and structure of the follicle. Here we present a mathematical model of oxygen transport in the antral follicle. We relate mean oxygen concentration in follicular fluid of bovine follicles to the concentration in the immediate vicinity of the cumulus-oocyte complex (COC). The model predicts that the oxygen levels within the antral follicle are dependent on the size and structure of the follicle and that the mean level of dissolved oxygen in follicular fluid does not necessarily correspond to that reaching the COC.

Very viscous flows driven by gravity with particular application to slumping of molten glass

Publisher: Cambridge University Press (CUP)

Date: 06-1999

DOI: 10.1017/S0004972700033220

ON THIN OR SLENDER BODIES

Publisher: Cambridge University Press (CUP)

Date: 2012

DOI: 10.1017/S1446181112000120

Abstract: This is a review of thin-body and slender-body theories, with indications of some new applications. Topics discussed include bodies with near-constant surface pressure, subsonic and supersonic aerodynamics, ship hydrodynamics, slender bodies in Stokes flow, slender footings in elastic media, and slender moonpools. Mathematical features of the thin- and slender-body approximations are also discussed, especially nonlocal convolution terms modelling three-dimensionality in the otherwise two-dimensional near field, end effects, and the role of the logarithm of the slenderness ratio. This review was presented by the first author as the IMA Lighthill Memorial Lecture at the British Applied Mathematics Colloquium (BAMC) 2004.

Pore Scale Visualization and Simulation of Miscible Displacement Process under Gravity Domination

Publisher: International Petroleum Technology Conference

Date: 2011

DOI: 10.2523/15310-MS

Elliptical pore regularisation of the inverse problem for microstructured optical fibre fabrication

Publisher: Cambridge University Press (CUP)

Date: 30-07-2015

DOI: 10.1017/JFM.2015.337

Abstract: A mathematical model is presented describing the deformation, under the combined effects of surface tension and draw tension, of an array of channels in the drawing of a broad class of slender viscous fibres. The process is relevant to the fabrication of microstructured optical fibres, also known as MOFs or holey fibres, where the pattern of channels in the fibre plays a crucial role in guiding light along it. Our model makes use of two asymptotic approximations, that the fibre is slender and that the cross-section of the fibre is a circular disc with well-separated elliptical channels that are not too close to the outer boundary. The latter assumption allows us to make use of a suitably generalised ‘elliptical pore model (EPM)’ introduced previously by one of the authors (Crowdy, J. Fluid Mech. , vol. 501, 2004, pp. 251–277) to quantify the axial variation of the geometry during a steady-state draw. The accuracy of the elliptical pore model as an approximation is tested by comparison with full numerical simulations. Our model provides a fast and accurate reduction of the full free-boundary problem to a coupled system of nonlinear ordinary differential equations. More significantly, it also allows a regularisation of an important ill-posed inverse problem in MOF fabrication: how to find the initial preform geometry and the experimental parameters required to draw MOFs with desired cross-plane geometries.

Extensional fall of a very viscous fluid drop

Publisher: Oxford University Press (OUP)

Date: 11-2000

DOI: 10.1093/QJMAM/53.4.565

University Of Adelaide

Location: Australia

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Mathematical Modelling Of Inertial Lift Driven Particle Migration In Complex Microfluidic Devices With Applications To Periodic Duct Geometries.

Start Date: 2024

End Date: 2027

Funder: Marsden Fund

ARC Future Fellowships - Grant ID: FT160100108

Start Date: 02-2017

End Date: 12-2023

Amount: $904,000.00

Funder: Australian Research Council

Linkage Projects - Grant ID: LP160100657

Start Date: 01-2017

End Date: 06-2021

Amount: $500,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP160102021

Start Date: 05-2016

End Date: 12-2019

Amount: $362,166.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP130101541

Start Date: 2013

End Date: 12-2016

Amount: $350,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP200100834

Start Date: 02-2021

End Date: 06-2024

Amount: $380,000.00

Funder: Australian Research Council

Special Research Initiatives - Grant ID: SR0354727

Start Date: 2003

End Date: 12-2003

Amount: $20,000.00

Funder: Australian Research Council

Linkage Projects - Grant ID: LP200100541

Start Date: 03-2021

End Date: 03-2024

Amount: $312,066.00

Funder: Australian Research Council

Linkage Projects - Grant ID: LP110100222

Start Date: 11-2011

End Date: 03-2016

Amount: $370,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP0450047

Start Date: 2004

End Date: 12-2012

Amount: $215,000.00

Funder: Australian Research Council