m.sc. in mathematical modelling and scienti c computing

Dissertation Projects
December 2013
1 Projects with the Industrial Sponsors of the M.Sc. 3
1.1 Sharp — Flow and Solidification in Confined Geometries with Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Numerical Analysis Projects 4
2.1 Chebfun Dissertation Topics . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Multi-Structures and Computing in Mixed Dimensions . . . . . . . . . . . 5
2.3 Parallel Computing for ODEs/PDEs with Constraints . . . . . . . . . . . 6
2.4 Segmentation and Registration of Lung Images . . . . . . . . . . . . . . . 7
2.5 Repairing Damaged Volumetric Data using Fast 3D Inpainting . . . . . . 8
2.6 Constraints and Variational Problems in the Closest Point Method . . . . 9
2.7 Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8 Optimisation of Tidal Turbines for Renewable Energy . . . . . . . . . . . 11
2.9 Uncertainty Quantification in Glaciological Inverse Problems . . . . . . . 13
2.10 Edge Source Modelling for Diffraction by Impedance Wedges . . . . . . . 14
2.11 What to do with DLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.12 Random Plane Wave and Percolation . . . . . . . . . . . . . . . . . . . . . 16
2.13 Numerical Solution of Equations in Biochemistry . . . . . . . . . . . . . . 18
2.14 Numerical Solution of the Rotating Disc Electrode Problem . . . . . . . . 19
3 Biological and Medical Application Projects 21
3.1 Circadian Rhythms and their Robustness to Noise . . . . . . . . . . . . . 21
3.2 The Analysis of Low Dimensional Plankton Models . . . . . . . . . . . . . 22
3.3 Individual and Population-Level Models for Cell Biology Processes . . . . 23
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3.4 A New Model for the Establishment of Morphogen Gradients . . . . . . . 24
3.5 Modelling the Regrowth and Homoeostasis of Skin . . . . . . . . . . . . . 26
3.6 Modelling the Growth of Tumour Spheroids . . . . . . . . . . . . . . . . . 27
3.7 Discrete/Hybrid Modelling of Lymphangiogenesis . . . . . . . . . . . . . . 28
3.8 Mathematical Modelling of the Negative Selection of T Cells in the Thymus 29
3.9 The Dynamics and Mechanics of The Eukaryotic Axoneme . . . . . . . . 30
4 Physical Application Projects 32
4.1 Swarm Robotics: From Experiments to Mathematical Models . . . . . . . 32
4.2 A Simple Model for Dansgaard-Oeschger Events . . . . . . . . . . . . . . 33
4.3 Modelling Snow and Ice Melt . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 A Network-Based Computational Approach to Erosion Modelling . . . . . 35
4.5 Retracting Rims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 Mathematical Modelling of Membrane Fouling for Water Filtration . . . . 37
4.8 Flow-Induced “Snap-Through” . . . . . . . . . . . . . . . . . . . . . . . . 38
4.9 Plumes with Buoyancy Reversal . . . . . . . . . . . . . . . . . . . . . . . . 39
4.10 Dislocation Structures in Microcantilevers . . . . . . . . . . . . . . . . . . 40
4.11 Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.12 Finger Rafting: The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 Networks 44
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1 Projects with the Industrial Sponsors of the M.Sc.
1.1 Sharp — Flow and Solidification in Confined Geometries with In- dustrial Applications
Supervisor: Prof. John Wettlaufer Industrial Collaborator: Philip Roberts Contact: [email protected]
In a recent Applied Mathematics Industrial Workshop Philip Roberts from Sharp presented a class of problems motivated by a device to be used in water purification. The device consists of a series of channels in a metallic mass through which water is flowing. The problem is that the water can freeze too quickly and stop subsequent flow. This has a profound influence on the efficacy and long-term design issues for the company.
The questions involve basic aspects of moving boundaries, fluid flow and solidification and geometry. There is a class of questions that can be addressed using the theoretical edifice of these topics suitably modified for the relevant geometry. The project will involve development of a mathematical model that addresses the role of time dependence in the thermal boundary conditions for the inflow, the role of interfacial kinetics and the importance of crystallinity in controlling the purification properties. The mathematical methodology of moving phase boundaries will be modified to consider the specificity of the design problem and they will be tested with experimental measurements in the Mathematical Observatory.
An expected outcome includes a new working model to provide the basis for further collaboration with Sharp.
References
[1] J. A. Neufeld and J.S. Wettlaufer. Shear flow, phase change and matched asymptotic expansions: pattern formation in mushy layers, Physica D 240, 140, 2011.
[2] M. G. Worster. Solidification of fluids, in: Perspectives in Fluid Dynamics: A Collec- tive Introduction to Current Research, Cambridge University Press, 2000, pp. 393–446.
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2 Numerical Analysis Projects
2.1 Chebfun Dissertation Topics
Supervisor: Prof. Nick Trefethen in collaboration with other members of the Chebfun team Contact: [email protected]
Chebfun is an algorithms and software project based on the idea of overloading Mat- lab’s vectors and matrices to functions and operators. We like to think that “Cheb- fun can do almost anything in 1D” (integration, optimization, rootfinding, differential equations,...) and recently a good deal of it has been extended to 2D too. See http:///www.maths.ox.ac.uk/chebfun, especially the Guide and Examples.
A number of M.Sc. dissertations related to Chebfun have been written in recent years. There are many possibilities and we can tailor the project to the student’s interests and expertise.
Here are three specific possibilities with the flavours of 2D computing, quadrature, and classic approximation theory.
1. Numerical vector calculus: The Chebfun2 extension to 2D has made it possible for us to compute numerically with “div, grad, curl and all that”. These operations can even be then mapped to 2D surfaces in 3D (see http://www.maths.ox.ac.uk/chebfun/examples/geom/html/VolumeOfHeart.shtml). Next to nothing has been done to utilize these capabilities so far, and there are many possibilities to explore.
2. Computation in inner product spaces: Chebfun computes inner products in the vanilla-flavoured way, with (f, g) defined as the integral of f(x)g(x) over their interval of definition. Yet it has the quadrature capabilities to handle other weight functions such as Chebyshev, Gauss-Jacobi, or Gegenbauer weights. In fact, Chebfun even includes delta functions, making computation of Stieltjes integrals possible. It would be very interesting to explore building these notions into a Chebfun “domain” class, so that machine-precision computation in nonstandard inner products could be automated and exploited.
3. The Remez algorithm for rational best approximation: Chebfun’s existing REMEZ command works well for polynomial approximations, but for rational approximations it is very fragile. Can it be improved?
The Chebfun team consists of about 8-10 people, and an M.Sc. student doing a project in this area would be welcome to participate in our weekly team meetings. Ideally, a student wishing to do a Chebfun-related thesis should have taken the Approximation Theory course in Michaelmas term.
2.2 Multi-Structures and Computing in Mixed Dimensions
Supervisor: Dr Colin Macdonald (OCCAM) Contact: [email protected]
Figure 1: Heat equation in mixed dimensions.
The Closest Point Method is a recently developed simple technique for computing the numerical solution of PDEs on general surfaces [2,4]. It is so general that it can compute on surfaces where I don’t understand the results. For example, it can compute on problems with variable dimension just as easily as a simple sphere. In Figure 1, the pig and sphere are connected with a one dimensional filament and heat flow is solved over the composite domain. But what does such a calculation mean? What is the correct solution to such a problem?
Figure 2: A multi- structure from [3].
A presentation by Prof. Vladimir Maz’ya (Liverpool) introduced me to multi-structures [1,3]. An example of a multi-structure problem would to be determine the eigenvalues of a bridge consisting of solid structures coupled to thin cables. The aim of this project is to learn about multi-structure problems and do some calculations (e.g, heat equation or Laplace–Beltrami eigenvalues) using the Closest Point Method. There are also asymptotic techniques that can be applied here, letting ε represent the “radius” of the one-dimensional parts (e.g., [3]).
A reasonable achievement would be showing that the Closest Point Method computes a solution which is consistent with an asymptotic analysis for some mixed-dimension multi-structures. Or maybe it is not consistent: that would be equally interesting!
References
[1] V. Kozlov, V. G. Maz’ya, and A. B. Movchan. Asymptotic analysis of fields in multi-structures. Oxford University Press, 1999.
[2] C. B. Macdonald and S. J. Ruuth. The implicit Closest Point Method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput., 31(6):4330–4350, 2009.
[3] A. B. Movchan. Multi-structures: asymptotic analysis and singular perturbation problems. European Journal of Mechanics/A Solids, 25(4):677–694, 2006.
[4] S. J. Ruuth and B. Merriman. A simple embedding method for solving partial differential equations on surfaces. J. Comput. Phys., 227(3):1943–1961, 2008.
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Supervisor: Dr Colin Macdonald (OCCAM) Collaborators: Prof. Raymond Spiteri (Saskatchewan), Prof. Ping Lin (Dundee) Contact: [email protected]
b b b b
b b b b
b b b b
tm−3 tm−2 tm−1 tm tm+1. . . . . .
Figure 3: Each row of computations happens in parallel.
This project proposes a parallel time-stepping routine for differential equation with constraints. The incompressible Navier–Stokes are an example of constrained PDEs where the divergence free condition (the constraint) is enforced by the pressure [3]. A multicore idea for ODEs and time- dependent PDEs was developed in [2] where parallelism was exploited to obtain higher-order ac- curacy. Here we propose to exploit parallelism to impose the constraint, in an iterative fashion where all iterations happen in parallel.
The project would begin with a brief review of differential-algebraic equations (DAEs) which are a framework for dealing with differential equations with constraints. The proposed numerical algorithm is the Sequential Regularization Method [1]. An implementation, in OpenMP, Python (using the multiprocessing module), or perhaps Matlab would be programmed. Applications would include multi-body systems (e.g., the pendu- lum and slider-crank mechanisms) and incompressible Navier–Stokes, and these would form the test cases.
The anticipated achievements include improving a DAE solver, a projection-free incompressible fluid solver and experience in multicore and parallel computing.
References
[1] U. Ascher and P. Lin. Sequential regularization methods for nonlinear higher-index DAEs. SIAM J. Sci. Comput., 18(1):160–181, 1997.
[2] A. Christlieb, C. B. Macdonald, and B. Ong. Parallel high-order integrators. SIAM J. Sci. Comput., 32(2):818–835, 2010.
[3] P. Lin. A sequential regularization method for time-dependent incompressible Navier– Stokes equations. SIAM J. Numer. Anal., 34(3):1051–1071, 1997.
[4] C. B. Macdonald and R. J. Spiteri. The predicted sequential regularization method for differential-algebraic equations. In C. D’Attellis, V. Kluev, and N. Mastorakis, editors, Mathematics and Simulation with Biological, Economic, and Musicoacoustical Applica- tions, pages 107–112. WSES Press, 2001.
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2.4 Segmentation and Registration of Lung Images
Supervisors: Dr Colin Macdonald and Dr Julia Schnabel (Oxford Biomedical Image Analysis) Contact: [email protected]
http://www.ibme.ox.ac.uk/research/biomedia
Figure 4: (a) lung scan; (b) magnitude of displacement w/o slip; (c) with slip; (e)– (f) zoom near sliding boundary.
Computer tomography (CT), magnetic resonance imaging (MRI), and positron emis- sion tomography (PET) result in 3D volume data containing representations of tissues and organs such as the lungs. Mathemat- ical image processing techniques are crucial to acquiring and analysing this data. Com- monly, the 3D data must be aligned via some function which maps it onto another data set—this is known as registration. Ex- amples include multimodal imaging (where both CT and less harmful but less accurate PET data are recorded simultaneously). Comparing one patient to a normal healthy sample or tracking change over time also require registration. During each inhale/exhale cycle, the lungs experience significant translation/slip relative to the torso. Until recently, registration techniques did not explicitly account for this motion and subsequently gave poor results.
This project would begin with a review of image processing and specifically level set techniques [2]. We would try to construct a mathematical model for the registration problem that treats the surface of the lungs as well as the 3D voxel data. The registration problem should not penalize for motion parallel to this surface [3, 4]. Numerically, the surface would be represented implicitly using level-set or closest-point based techniques [1]. We would implement our algorithm (in Matlab or Python) and perform experiments using both simulated and real data.
As part of a brand-new collaboration, this project is very much “blue skies”. We hope to show feasibility of incorporating more “prior knowledge” of the particular problem of lung image segmentation. This could lead to improved registration. As the Biomedical Image Analysis lab is motivated by clinical diagnosis, therapy planning, and image-based treatment guidance, the long term goal would be to improve the tools used in practice.
References
[1] C. B. Macdonald and S. J. Ruuth. The implicit Closest Point Method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput., 31(6):4330–4350, 2009.
[2] S. Osher and R. Fedkiw. Level set methods and dynamic implicit surfaces. Springer- Verlag, 2003.
[3] B. Papiez, M. Heinrich, L. Risser and J. A. Schnabel. Complex lung motion estimation via adaptive bilateral filtering of the deformation field. Proceedings for the Medical Image Computing and Computer Assisted Intervention (MICCAI), 2013.
[4] L. Risser, F.-X. Vialard, H. Y. Baluwala, and J. A. Schnabel. Piecewise-diffeomorphic image registration: Application to the motion correction of 3d CT lung images using sliding conditions. Medical Image Analysis, 2013.
2.5 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors: Dr Tom Marz and Dr Colin Macdonald Contact: [email protected] and [email protected]
Figure 5: Before (top) and after (bottom) inpainting.
Digital inpainting fills in missing pixels in a damaged image, such as a creased photograph. It can also be used to manipu- late images as in Figure 5. This is an inverse problem and is usually regularized in some way so that the resulting image is pleasing in the “eye-ball norm”. Defining the latter is where the mathematics gets interesting! There are analogous problems in three-dimensions, for example, removing watermarks or subtitles from video (2D + time). There are also likely applications in medical imaging. This project would investigate the 3D image inpainting problem on voxel data (see for example [1, 7]).
The Bornemann–Marz inpainting algorithm is a recent and fast image inpainting technique [4, 5]. This project would begin by reviewing the mathematics and implementation of this algorithm (we have existing software in Matlab and the GIMP to experiment with). We would then extend the approach to three dimensions. The techniques include finite difference methods for PDEs, fast marching methods, structure tensors and some numerical linear algebra. There are plenty of theoretical mathematical issues too depending on interest.
A minimum goal would be to understand the mathematics and extend our software (github.com/maerztom/inpaintBCT) to 3D. We could then look at applications such as video inpainting. Depending on interest, we could also look at inpainting of colour data on triangulated surfaces (using the Closest Point Method [3, 2, 6]) or investigate applications in medical imaging.
References
[1] M. Bertalmo, A. L. Bertozzi, and G. Sapiro. Navier–Stokes, fluid dynamics, and image and video inpainting. In Proc. of IEEE International Conference on Computer Vision and Pattern Recognition, 2001.
[2] H. Biddle. Nonlinear diffusion filtering on surfaces. M.Sc. dissertation, Oxford Uni-
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versity, 2011.
[3] H. Biddle, I. von Glehn, C. B. Macdonald, and T. Marz. A volume-based method for denoising on curved surfaces, 2013. To appear in Proc. ICIP13, 20th IEEE International Conference on Image Processing.
[4] F. Bornemann and T. Marz. Fast image inpainting based on coherence transport. Journal of Mathematical Imaging and Vision, 28(3):259–278, 2007.
[5] T. Marz. Image inpainting based on coherence transport with adapted distance functions. SIAM Journal on Imaging Sciences, 4(4):981–1000, 2011.
[6] E. Naden. Fully anisotropic diffusion on surfaces and applications in image processing. M.Sc. dissertation, Oxford University, 2013.
[7] K. A. Patwardhan, G. Sapiro, and M. Bertalmo. Video inpainting of occluding and occluded objects. In Proc. of IEEE International Conference on Image Processing, 2005.
2.6 Constraints and Variational Problems in the Closest Point Method
Supervisor: Dr Colin Macdonald Contact: [email protected]
Figure 6: Reaction- diffusion equations on a red blood cell surface [5].
The Closest Point Method is a recently developed simple technique for computing the numerical solution of PDEs on general surfaces [4]. The method works by embedding the surface in three-dimensions an imposing a constraint to keep the solution constant in the normal direction. Despite the work [5] and von Glehn’s thesis, we still have many basic questions. Here are a couple which could make for good M.Sc. projects:
(a) Can we interpret the constraint as a differential-algebraic equation (DAE)? See also my other project on DAEs, which this could easily tie into. If the constrained problem is indeed a DAE, what is its index? If its not (technically) a DAE, can we still use DAE techniques to solve it? How would these compare to [5]?
(b) How do we deal with variational approaches to surface problems in this constrained closest-point framework. This ties into image processing on surfaces, something developed over several Oxford M.Sc. theses [1, 2, 3]. We know some things about surface integrals thanks to Tom Marz. How do we formulate Euler– Lagrange equations for these constrained expressions?
In either case, the project would involve Runge–Kutta methods, finite difference schemes, some numerical linear algebra, and some geometry. A project would involve a mixture of theory and practical computation.
In any of these projects, we would hope to improve our understanding of the constrained problem and more generally of numerical techniques for solving PDE problems on curved surfaces.
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References
[1] H. Biddle. Nonlinear diffusion filtering on surfaces. M.Sc. dissertation, Oxford Uni- versity, 2011.
[2] H. Biddle, I. von Glehn, C. B. Macdonald, and T. Marz. A volume-based method for denoising on curved surfaces, 2013. To appear in Proc. ICIP13, 20th IEEE International Conference on Image Processing.
[3] E. Naden. Fully anisotropic diffusion on surfaces and applications in image processing. M.Sc. dissertation, Oxford University, 2013.
[4] S. J. Ruuth and B. Merriman. A simple embedding method for solving partial differential equations on surfaces. J. Comput. Phys., 227(3):1943–1961, 2008.
[5] I. von Glehn, T. Marz, and C. B. Macdonald. An embedded method-of-lines approach to solving partial differential equations on surfaces, 2013. Submitted.
2.7 Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
Supervisor: Prof. Jared Tanner Contact: [email protected]
Matrix completion concerns recovering a matrix from few of its entries. For a general matrix with independent entries this task is not possible, but for matrices with further structure the intercorrelation of entries may allow the full matrix to be recovered. The prototypical assumption of structure is low rank, in which case algorithms have been shown to be able to recover low rank matrices from, asymptotically, the optimally fewest number of measurements; that is, the number of degrees of freedom in the low rank matrix. This is an active area of research including development of fundamental theory, algorithms, and their application from online recommendation systems to image processing.
This topic can accommodate a variety of questions, ranging from: a) developing an understanding of fundamental theory such as the embedding constants of inner matrix inner products with low rank matrices; b) implementing and benchmarking competing simple algorithms in a parallel infrastructure such as graphical processing units; c) repro- ducing and if possible extending some of the more complex algorithms such as random projection divide and conquer algorithms; or d) review literature in tensor decompositions and completions. These projects involve a good understanding of numerical linear algebra, some familiarity with probability, and computer programming.
References
[1] Lester Mackey, Ameet Talwalker, and Michael I. Jordon. Distributed Matrix Com- pletion and Robust Factorization. http://arxiv.org/abs/1107.0789
[2] N. Halko, P. G. Martinsson and J. A. Tropp. Finding Structure with Randomness:
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions, SIAM Review, 53(2):217–288, May 2011.
[3] Benjamin Recht, Maryam Fazel and Pablo A. Parrilo. Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization, SIAM Review, 52(3):471–501, August 2010.
[4] Raghunandan H. Keshavan, Andrea Montanari and Sewoong Oh. Matrix completion from a few entries, IEEE Transactions on Information Theory, 56(6):2980–2998, June 2010.
2.8 Optimisation of Tidal Turbines for Renewable Energy
Supervisor: Dr Patrick Farrell Contact: [email protected]
Background and problem statement: Tidal stream turbines extract energy from the movement of the tides, in much the same way as wind turbines collect energy from the wind. The UK has abundant renewable marine energy resources, which could sup- ply reliable clean electricity, support a new local high-technology industrial sector, and reduce emissions of carbon emissions. The Carbon Trust predicts that the marine re- newables industry could be worth tens of billions of pounds by 2050, and support tens of thousands of UK-based jobs; if the UK moves quickly, it could become the world leader in this technology, as Denmark has become in wind turbines. However, before this potential can be realised, the industry must solve a design problem. In order to extract an economically useful amount of energy, large arrays (up to several hundred) must be deployed on a given site. How should the turbines in an array be placed to extract the maximum possible energy? The configuration makes a major difference to the power extracted, and thus to the economic viability of the installation.
Description of the approach planned and techniques needed: For a given turbine configuration (the control), a set of partial differential equations (the nonlinear shallow water or Navier-Stokes equations) is to be solved for the resulting flow configuration, and the power extracted computed (a functional involving the cube of the flow speed). The optimisation problem is to maximise the power extracted subject to the physical constraints and that the design is feasible (e.g., that the turbines satisfy a minimum distance constraint, that they are deployed within the site licensed, etc.). In a recent publication [1], I solve this optimisation problem using the adjoint technique, which solves an auxiliary PDE that propagates causality backwards, allowing for the very efficient computation of the gradient of the power extracted with respect to the turbine locations. These adjoint PDEs are automatically derived from the forward problem using the hybrid symbolic/algorithmic differentiation approach presented in [2]. With this adjoint technique, optimisation algorithms that rely on first-order derivative information may be used, such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm [3]. However, without second-order information, the use of more powerful optimisation algorithms such as Newton’s method is precluded.
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What you’d hope to achieve: In this project, the research student will extend the tidal turbine optimisation solver to use (variants of) Newton’s method for PDE- constrained optimisation. It is anticipated that this will greatly accelerate the conver- gence of the optimisation algorithm. This will rely on incorporating recent (unpublished) advances in the extremely efficient computation of tangent linear and second-order adjoint solutions, which enable the computation of the necessary second-order derivative information very quickly.
(a) Satellite image of Stroma Island and Caithness; MeyGen Ltd. have licensed this site to deploy a 398MW array of tidal turbines.
(b) Computational domain with the turbine site marked pink.
(c) Initial turbine positions (256 turbines). (d) Optimised turbine positions. In this idealised case, the optimisation improved the farm efficiency by 32%.
References
[1] S. W. Funke, P. E. Farrell and M. D. Piggott. Tidal turbine array optimisation using the adjoint approach, Renewable Energy, 63:658–673, 2014.
[2] P. E. Farrell, D. A. Ham, S. W. Funke and M. E. Rognes. Automated derivation of the adjoint of high-level transient finite element programs, SIAM Journal on Scientific Computing, 35(4):C369–C393, 2013.
[3] J. Nocedal and S. J. Wright. Numerical Optimization, Second Edition, Springer Verlag, 2006.
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Supervisor: Dr Patrick Farrell Contact: [email protected]
Background and problem statement: The response of the world’s ice sheets to a changing environment is a key ingredient in the understanding of past and future global climate change, due to their potential for rapid contributions to sea level change [1]. However, there remain significant gaps in our understanding of the dynamics of fast-flowing glacial ice, in part because computer models of ice sheets must take as input physical properties which are unknown or difficult to measure. These unknown properties, such as bedrock topography and ice temperature, are often spatially variable, and hence there are an extremely large number of unknown inputs which affect the predictions derived from computer simulations. A popular technique for determining these unknown values is to use available observations, such as satellite-derived altimetry and surface velocities, to invert for these values; i.e. to find the values such that the model output best fits the observations. However, a question remains: to what degree are the estimated values constrained by those observations?
Description of the approach planned and techniques needed: Practitioners typically take a deterministic approach to model inversion: a single point in parameter space is sought that best minimises the misfit functional. Adopting a Bayesian perspective, this is equivalent to minimising the negative log of the posterior density. However, the Bayesian approach offers additional insight: the covariance of the posterior distribution can be locally characterised by computing the eigendecomposition of the misfit Hessian evaluated at that minimiser (see, e.g., [2]). This allows for the identification of directions in parameter space that are well-constrained or poorly-constrained by the available data.
What you’d hope to achieve: The student will implement a simple discretisation of the higher-order Blatter-Pattyn ice sheet model [3], and apply it to steady isothermal simulations of the Greenland ice sheet (see Figure 7). The student will then generate synthetic observations from known input data, and use it in solving the deterministic inverse problem, taking care to avoid “inverse crimes” [4]. The student will then apply matrix-free eigendecomposition algorithms to characterise the covariance of the posterior distribution at that misfit minimiser.
References
[1] S. Solomon D. Qin, M. Manning, Z. Chen, M. Marquis, K. Averyt, M. Tignor and H. L. Miller (Editors). Climate Change 2007: The Physical Science Basis, Cambridge University Press, 2007.
[2] W. C. Thacker. The role of the Hessian matrix in fitting models to measurements, Journal of Geophysical Research, 94(C5):6177–6196, 1989.
[3] M. Perego, M. Gunzburger and J. Burkardt. Parallel finite element implementation for higher-order ice-sheet models, Journal of Glaciology, 58(207):76–88, 2012.
[4] J. Kaipio and E. Somersalo. Statistical and Computational Inverse Problems, Volume
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Figure 7: The velocity solution of the steady isothermal Blatter-Pattyn equations, dis- cretised using finite elements on a 10km mesh of the Greenland ice sheet.
160 of Applied Mathematical Sciences, Springer-Verlag, 2004.
2.10 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor: Dr David Hewett Possible Collaborator: Prof. U. Peter Svensson, NTNU Trondheim Contact: [email protected]
Wave scattering problems arise in many applications in acoustics, electromagnetics and linear elasticity. However, exact solutions to the (apparently simple) wave equations modelling these processes are rare. One special geometry amenable to an exact analysis is the exterior of a wedge. For the wedge scattering problem with sound-soft (Dirichlet) or sound-hard (Neumann) boundary conditions, Svensson et al. [1] have recently shown how the “diffracted” field component of the exact closed-form solution can be written as a superposition of point sources distributed along the diffracting edge. As well as being appealing from a physical point of view, this “edge source” formulation has also been used by Svensson and Asheim [2] to develop a new integral equation formulation for scattering problems, which may offer a promising alternative to existing tools such as the boundary element method.
The aim of this project is to investigate whether or not an edge source solution rep- resentation is possible for the problem of diffraction by a wedge on which impedance (absorbing) boundary conditions are imposed. The impedance boundary condition is more realistic for acoustic modelling of many reflecting surfaces, but is more challenging to analyse than the Dirichlet and Neumann cases. The project will focus on the special case of a right-angled impedance wedge, for which Rawlins [3] has shown that the exact
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solution can be expressed in terms of the (known) solution for the Dirichlet wedge.
A general background in wave propagation would be useful but is not essential. Knowl- edge of complex analysis (in particular complex contour integral manipulations) would also be valuable.
References
[1] U. P. Svensson, P. T. Calamia and S. Nakanishi. Frequency-domain edge diffraction for finite and infinite edges. Acta acustica united with acustica, 95(3):568–572, 2009.
[2] U. P. Svensson and A. Asheim. An integral equation formulation for the diffraction from convex plates and polyhedra. Tech. Report TW610, KU Leuven, 2012.
[3] A. D. Rawlins. Diffraction of an E- or H-polarized electromagnetic plane wave by a right-angle wedge with imperfectly conducting faces. Q. J. Mech. Appl. Math., 43(2):161– 172, 1990.
2.11 What to do with DLA
Supervisors: Dr Dmitry Belyaev and Dr Alan Hammond Contact: [email protected]
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2 (analog of Kesten’s theorem stating that the dimension is at least 3/2). I propose to study the development of DLA cluster started from the large disc.
To fix scale, we fix the size of the particles to be equal to one. We start from the disc of radius N (i.e. dense cluster of N2 particles). We would like to study how fast the fractal structure will appear (will it happen in N2 steps, or may be faster or slower).
What should be computed:
• We start with the disc of size N (denoted by D0). First of all we should compute Di for i = 1 . . . N2.
• For some of Di we should compute the distribution of harmonic measure (probably in the form of dimension spectrum). The best choice would be compute this for all values of i, but this is will demand too much computer time and each step introduces very localized change in harmonic measure. I propose to find (experimentally) the time step δ such that within this time change of measure distribution is globally significant but small.
There are several things we could do with the data
• Check how fast the spectrum approaches the spectrum of DLA cluster (i.e how fast the smooth structure of the disc will be forgotten).
• Compare obtained spectra with the spectra computed by Mandelbrot et al
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• Compute correlations (rate of their decay) of harmonic measure. Namely we start with N sectors, such that they all carry 1/N of harmonic measure (strong cor relations, uniform distribution). After some time the distribution should start resembling DLA and correlations should weaken.
2.12 Random Plane Wave and Percolation
Supervisor: Dr Dmitry Belyaev Contact: [email protected]
The main goal of this project is to explore connections between two important and interesting physics models: plane waves and percolation. The first model appears in the study of quantum systems and other problems related to the eigenfunctions of the Laplacian, the second model is used to describe porous media, spread of forest fires as well as many other phenomena.
Random plane waves. There are many problems in physics that are related to the study of eigenfunctions of Laplace operator and their zero level sets. (For example the sand on a vibrating plate concentrates on the zero level set of a Laplace eigenfunction). It is conjectured that for a very large class of domains, the typical behaviour of a high energy eigenfunction is the same as that of a random superposition of simple plane waves (RPW). This motivates our interest in RPW and their nodal lines (curves where the plane wave is equal to zero).
(a) Random spherical harmonic: an ana- logue of RPW on a sphere (figure by A. Barnett).
(b) Sand on a vibrating placte (photos from MIT Physics TSG site).
There are several ways to think about RPW, they lead to different approximations that could be used for simulations. The simplest way is to say that a random plane wave is a random linear combination of plane waves. A standard plane wave in R2 is
ReAekθ·z
where A is the amplitude, E = k2 is the energy and θ is a unit vector which gives the
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direction of the wave. Simple computation shows that plane wave solves the Helmholtz equation f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplace operator with eigenvalue −k2). Informally we can define RPW as
Re
N∑ n=1
Cne ikθn·z
) where Cn are independent random (complex) normal variables and θn are directions. One can take θn to be equi-distributed θn = e2πin/N or to be N independent uniformly distributed random unit vectors.
The problem is that it is not clear in what sense this series converges and how to prove it, on the other hand it might be used for some simulations since it only uses very simple trigonometric functions. For rigorous definition is is better to use another formula
F (z) =
Re ( CnJn(kr)einθ
) where z = reiθ, Cn as before and Jn is the Bessel function of the first kind. Check that this function is a solution of Helmholtz equation.
We want to study the nodal lines (set where F = 0) of the function F , namely we are interested in their behaviour inside of a fixed domain as k → ∞. This is the same as fixing k = 1 and studying nodal lines in expanding domains.
Percolation. This is one of the most studied models in statistical physics, this year alone there were 12000 papers on the subject. Yet, there are many aspects of this model that are not known rigorously and even not well understood numerically. There are many versions of this model, but we will need the simplest one: edge percolation. We fix some graph, possibly infinite, the best example is Z2 grid and probability p ∈ [0, 1]. After that we independently keep each edge with probability p or remove it with probability 1 − p. Alternatively we can think that edges are “open” or “closed” with probabilities p and 1− p. We are interested in connected components of this random graph and how their structure depend on p.
Connection between models. It was conjectured that the nodal domains of the random plane wave are well described by percolation on the square lattice Z2 with p = 1/2. Recent careful numerical experiments have shown that this is not quite true and the model should be corrected. I propose to study percolation on a random graph which is generated by RPW in the following way: its vertices are given by all local maxima of the plane wave and edges are encoded by the saddle points. For each saddle there are two directions of steepest ascent (following gradient flow lines) that terminate at two local maxima. The edge corresponding to the saddle connects two vertices that correspond to these two local maxima. I believe that the percolation with p = 1/2 on this lattice is a good model for nodal domains.
The main parts of the project will be
• Sample RPW on a relatively large domain. The size of the domain should be of the order 100λ where λ = 2π/k is the wavelength. For these samples we will have
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to locate all critical points, and establish how they are connected by the flow lines. This will require very high precision computations since the flow lines can pass very close to the other saddle points. The result of this stage will be the sampling of the random graph that was described above.
• In this stage we are going to study the percolation on the random graph that was sampled in the first stage. We will have to sample sufficient number of the percolation realizations on each of the graph samples and study their statistics. Finally it will be compared with the known statistics for the nodal domains of RPW.
2.13 Numerical Solution of Equations in Biochemistry
Supervisors: Dr Tomas Vejchodsky and Dr Radek Erban Contact: [email protected] and [email protected]
Biochemical processes in living cells can be described in terms of partial differential equations (PDEs) and master equations for probability distributions of biochemical species involved. These equations include the chemical master equation (CME) and the chemical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]. Numerical solution of these equations is challenging due to the structure and size of particular problems.
The CME is an infinite system of linear differential equations which has to be truncated to a finite size for computational reasons. The CFP equation is a linear evolutionary PDE of convection-diffusion type which can be solved by standard numerical methods such as the finite difference method (FDM), finite volume method (FVM), and finite element method (FEM) [2,3]. In any case, approximate stationary solution of both the CME and CFP is determined by a null-space of a large, sparse, and nonsymmetric matrix. Computing a null-space is not straightforward, however, it can be solved by standard methods of numerical linear algebra [3,4,5].
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In this project, we will first investigate the properties of CME and CFP equations and then concentrate on their efficient numerical solution. This is particularly challenging in the case of a higher number of chemical species (more than three) in the system, because this number corresponds to the dimension of the resulting problem. Solving high-dimensional problems is difficult due to the so called “curse of dimensionality”, meaning that the number of degrees of freedom grows exponentially with the dimension. However, this can be countered by using tensor methods [6].
This project is suitable for students interested in numerical methods for partial differential equations. The focus will be on numerical methods, rather than on applications in biology.
References
[1] Special Topic Course C6.4b: “Stochastic Modelling of Biological Processes”
[2] H. C. Elman, D. J. Silvester, A. J. Wathen: “Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics”, 2005
[3] Core Course B2: “Finite Element Methods and Further Numerical Linear Algebra”
[4] L. N. Trefethen, D. Bau: “Numerical Linear Algebra”, 1997
[5] Core Course B1: “Numerical Solution of Differential Equations and Numerical Linear Algebra”
[6] V. Kazeev, M. Khammash, M. Nip and C. Schwab: “Direct Solution of the Chemical Master Equation using Quantized Tensor Trains”, 2013
2.14 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor: Dr Kathryn Gillow Contact: [email protected]
The basic idea of an electrochemical experiment is that a known potential is applied to a working electrode in a solution. This causes oxidation or reduction to take place at the electrode and in turn this means that a current (which can be measured) flows. The current depends on a number of physical parameters of the solution including the diffusion coefficient, the resistance of the solution and the rate of reaction.
Mathematically the concentration of the chemicals is modelled using a reaction-convection- diffusion equation and for a rotating disc electrode we can assume that one space dimension is enough for the model. The current is then a linear functional of the concentration. Solving for the current with given values of the parameters is known as the forwards problem. Of more interest is the inverse problem where an experimental current is given and and the parameters are to be calculated.
The idea of this project is to first develop an efficient solver for the forwards problem and then use it as the basis of a solver for the inverse problem. It is then of interest to find out how the experimentally controllable parameters (reaction rate, applied potential
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3.1 Circadian Rhythms and their Robustness to Noise
Supervisors: Dr Tomas Vejchodsky and Dr Radek Erban Contact: [email protected] and [email protected]
Biochemical processes in living cells typically involve chemical species of very low copy numbers (e.g. one molecule of DNA and low numbers of mRNA molecules). Therefore the classical description based on concentrations is not applicable. The intrinsic noise is crucial in these systems, because it yields substantial and important effects such as stochastic focusing, stochastic resonance and noise-induced oscillations [1].
In this project we will try to explain how regular circadian rhythms can robustly persist in biochemical systems that are highly influenced by the intrinsic noise. There are many mathematical models of circadian rhythms based on gene regulation [2,3,4]. This means that concentrations of certain proteins within the cell cytoplasm oscillates within a 24 hour period due to positive and/or negative feedback loops. The feedback is caused by the protein molecule binding to the promoter region of a gene which activates or represses the gene expression. The intrinsic noise can have strong effects on these biochemical reactions, because of low copy numbers of interacting biomolecules involved. In spite of this fact, robust circadian rhythms are observed in many types of cells.
In this project, we begin with model reduction of a mathematical model of circadian rhythms, using the quasi-steady state assumptions [5]. Then the reduced system will be analysed for bifurcations to understand details of its dynamics and sensitivity to noise. We will use numerical methods to solve systems of ordinary differential equations and stochastic simulation algorithms to sample trajectories of stochastic systems. We will aim to explain details of the circadian model dynamics in both deterministic and stochastic regimes. This will yield the understanding of the observed noise robustness
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[1] Special Topic Course C6.4b: “Stochastic Modelling of Biological Processes”
[2] D. B. Forger and C. S. Peskin: A detailed predictive model of the mammalian circadian clock, PNAS, 100(25), 14806-14811, 2003
[3] J. Villar, H. Kueh, N. Barkai, S. Leibler: Mechanisms of noise-resistance in genetic oscillators, Proc. Nat. Acad. Sci. USA 99, pp. 5988-5992, 2002
[4] Z. Xie, D. Kulasiri: Modelling of circadian rhythms in Drosophila incorporating the interlocked PER/TIM and VRI/PDP1 feedback loops, J. Theor. Biology 245, 290-304, 2007
[5] L. A. Segel and M. Slemrod: The quasi-steady-state assumption: a case study in perturbation, SIAM Review 31(3), 446-477, 1989
3.2 The Analysis of Low Dimensional Plankton Models
Supervisor: Dr Irene Moroz Contact: [email protected]
The increasing exploitation of marine resources has driven a demand for complex bio- geochemical models of the oceans and the life they contain. The current models are constructed from the bottom up, considering the biochemistry of individual species or functional types, allowing them to interact according to their position in the food web, and embedding the ecological system in a physical model of ocean dynamics. The resulting ecology simulation models typically have no conservation laws and the ecology often produces emergent properties, that is, surprising behaviours for which there is no obvious explanation. Because realistic models have too many experimentally poorly defined parameters (often in excess of 100), there is a need to analyse simpler models.
A recent approach by Cropp and Norbury (2007) involves the construction of complex ecosystem models by imposing conservation of mass with explicit resource limitation at all trophic levels (i.e. positions occupied in a food chain). The project aims to analyse models containing two “predators” and two “prey” with Michaelis-Menten kinematics. A systematic approach to elicit the bifurcation structure and routes to chaos using parameter values, appropriate to different ocean areas would be adopted. In particular the influence of nonlinearity in the functional (life) forms on the stability properties of the system and the bifurcation properties of the model will be comprehensively numerically enumerated and mathematically analysed.
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3.3 Individual and Population-Level Models for Cell Biology Processes
Supervisor: Dr Ruth Baker Collaborator: Dr Mat Simpson, Queensland University of Technology, Bris- bane Contact: [email protected]
Modelling the individual and collective behaviour of cells is central to many areas of theoretical biology, from the development of embryos to the growth and invasion of tumours. Methods for modelling cell processes at the individual level include agent- based space-jump and velocity-jump processes, both on- and off-lattice. One may include biological detail in these models, taking volume exclusion into account by, for example, allowing a maximum occupancy of lattice sites, or modelling adhesion by allowing cell movement rates to depend on the local cell density. However, it is often difficult to carry out mathematical analysis of such models and we are restricted to computational simulation to generate statistics on population-level behaviour. Models derived on the population level, whilst more amenable to analysis, are often more phenomenological, without careful regard given to the detail of the cell processes under consideration. The rigorous development of connections between individual- and population-level models is crucial if we are to accurately interrogate biological systems.
A project in this area could investigate a number of phenomena in relation to these processes, not limited to the following.
• The links between exclusion processes (where a lattice site may be occupied by at most one agent) and those that allow multiple agents to occupy the same site.
• The extent to which limiting PDEs describing the evolution of cell density can be derived from different underlying motility models.
• The effects of cell shape on motility and proliferation.
• The effects of crowding upon cell processes, and the possibility for anomalous diffusion.
• The potential of exclusion processes to give rise to patterning by a Turing-type mechanism.
• The extension of velocity-jump and off-lattice models to include domain growth, and comparison of results with those already put forward in the literature.
References
[1] M. J. Simpson, R. E. Baker and S. W. McCue. Models of collective cell spreading with variable cell aspect ratio: A motivation for degenerate diffusion models. Phys. Rev. E, 83(2), 021901 (2011).
[2] R. E. Baker, C. A. Yates and R. Erban. From microscopic to mesoscopic descriptions of cell migration on growing domains. Bull. Math. Biol. 72(3):719-762 (2010).
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x
y
Figure 8: Comparing the motility of agents of aspect ratio L = 2 undergoing a random walk with rotations.
[3] M. J. Simpson, K. A. Landman and B. D. Hughes. Multi-species simple exclusion processes. Physica A, 388(4):399-406 (2009).
3.4 A New Model for the Establishment of Morphogen Gradients
Supervisor: Dr Ruth Baker Collaborator: Professor Stas Shvartsman, University of Princeton Contact: [email protected]
During embryonic development, a single cell gives rise to the whole organism, where cells of multiple different types are arranged in complex structures of functional tissues and organs. This remarkable transformation relies on extensive cell-cell communication. In one type of cell communication, a small group of cells produce a chemical that in- structs cells located nearby. Cells located close to the source of the signal receive a lot of it, whereas cells located further away receive progressively smaller amounts. In this way, a locally produced chemical establishes a concentration profile that can “organize” the developing tissue, providing spatial control of gene expression and cell differentiation. Starting from the late 1980s, such concentration profiles, known as “morphogen gradients” have been detected in a large number of developing tissues in essentially all animals, from worms to humans.
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One of the most popular models for gradient formation is based on the localized production and spatially uniform degradation of a diffusible protein. In this model, molecules move to the cells, which are “waiting” for the arrival of a signal that tells them what to do. Mathematically, the system is often modelled using reaction-diffusion equations, which can be readily solved and used to fit to experimental data. Recently, however, experiments in a number of systems suggest that the mechanisms underlying morphogen gradient formation can be more complex. Instead of passively waiting for the arrival of the signal, cells can form long-range dynamic projections that reach out in space and are used to transport a signal back to the cell. The resulting concentration profile is the same, but the mechanism of formation is very different.
This project is concerned with formulating a theory of morphogen gradient formation by these dynamic projections, known as “cytonemes”. The start point will be a one- dimensional model of cytoneme-mediated chemical transport based on the theory of two interacting random walks, which describe both moving cytonemes and moving molecules. Further extensions will include the extension to two spatial dimensions and the incorporation of further important details from cell biology. These models will be analyzed using a range of computational and analytical tools, from stochastic simulations to Greens functions techniques.
In parallel with this theoretical work, the Shvartsman Lab are investigating the existence and potential roles of cytonemes in cell communication mediated by the Epidermal Growth Factor signaling pathway, which controls developmental processes in multiple animals. Our experimental system is Drosophila, where the EGF pathway is involved in patterning of essentially all tissue types.
Figure 9: Cells from the anterior region of a Drosophila wing disc projecting cytonemes in an in vitro experiment.
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Supervisors: Prof. Helen Byrne and Prof. Colin Please Contact: [email protected] and [email protected]
Background and problem statement: There is considerable interest in understanding how human skin reforms after damage and how its thickness is controlled. Such understanding is necessary, for example, for improving methods of skin grafts for burns victims, for identifying methods of controlling skin diseases such as psoriasis, and for as- sisting in the creation of artificial skin to test the safety of household products, cosmetics and new drugs. Substantial experimental evidence comes from groups in Brisbane and Utrecht who are examining the behaviour and viability of artificial skin. They remove all the cells from skin samples and then deposit a few cells in the tissue, before incubating it in a well-defined medium. Examples of the type of regrowth that they observe are shown in the diagram below. Three, distinct layers can be detected: the lower, de-epithelialised human dermis (DED) layer which is the extracellular material from the original sample with no cells in it; the viable epidermal layer (TAL) in which the deposited cells divide, move and grow; and the cornified layer (KL) which contains dead cells that still retain some structure. Interesting behaviour can be observed: for example the DED region has an undulating surface, yet the KL layer is very flat. The aim of this project is to understand the dynamics of the growing layers and the mechanisms that might be controlling the observed behaviour. There is considerable discussion and controversy about how the cells communicate and how the layers are formed: the models developed in this project will be used to test and compare the alternative hypotheses.
Description of the planned approach and the techniques needed: Models will be examined and extended which involve transport of chemicals and cell motion and require the introduction of moving boundaries to account for the various interfaces separating the different skin layers and their changing thickness. Mechanisms to be considered could include growth, motion and death of cells, transport of nutrient and other signalling molecules, mechanical stresses in the layers. The behaviour of the resulting models will be examined using a combination of analytical and numerical approaches. The project will involve close collaboration with Dr Jos Malda, University of Utrecht.
References
[1] R. A. Dawson, Z. Upton, J. Malda, and D. G. Harkin (2006) Transplantation, 81(12): 1668-1676.
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[2] G. Topping, J. Malda, R. A. Dawson, and Z. Upton (2006). Primary Intention, 14: 14-21.
[3] M. Ponec (2002). Advanced Drug Delivery Reviews, 54: S19-S30.
[4] H. J. Stark, K. Boehnke, N. Mirancea et al (2006). Jl Invest Derm Symp Proc, 11(1): 93-105.
3.6 Modelling the Growth of Tumour Spheroids
Supervisors: Prof. Helen Byrne and Prof. Colin Please Contact: [email protected] and [email protected]
Background and problem statement A critical step in the dissemination of ovarian cancer is the formation of multicellular spheroids from cells shed from the primary tumour. There is increasing evidence that the mechanical properties of the tissue surrounding such tumour spheroids may influence their ability to grow and spread. In this project the student will develop mathematical models describing the growth of multicellular spheroids in established bioengineered three-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitro. The project will be supported by experimental work, being conducted at the Queensland University of Technology, Brisbane, Australia. The data obtained (see figure below) demonstrates that cells cultured in gels form spherical clusters of different sizes, and that their size depends on the mechanical properties of the tissue in which the cells are located.
The aim of this project is to develop and analyse new continuum models that can be used to investigate how cell-cell and cell-tissue interactions, and the mechanical properties of the gel in which the spheroid is embedded, influence tumour invasion. The project could have a mathematical or numerical focus (or involve a combination of the two).
References
[1] C. Gang, J. Tse, R.K. Jain and L.L. Munn (2009). PLoS ONE 4(2): e4632.
[2] C. Y. Chen, H .M. Byrne and J. R. King (2001). J Math Biol 43: 191-220.
[3] S. Krause, M. V. Maffini, A .M. Soto and C. Sonnenschein (2008). Tissue Eng Part C Methods. 14(3):261-71.
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Supervisors: Prof. Helen Byrne and Dr Chris Bell Contact: [email protected] and [email protected]
Background and problem statement: The lymphatic and vascular systems are coupled: fluid and nutrients are delivered by the vasculature while extracellular fluid flows from the capillaries into the lymphatic microvessels and is returned to the vasculature system via the thoracic ducts. Failure of the lymphatic system can result in conditions such as lymphoedema.
Although both transport systems interact and are similar, comprising large networks of vessels with an endothelial lining, experimental and theoretical research has focussed on the blood system. A variety of theoretical frameworks have been used to study aspects of angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahl et al., 2011]. Modelling of the lymphatic system is less advanced. Roose & Fowler (2008) considered the pre-patterning of lymphatic vessel morphology within collagen cells, via the establishment of a fluid flow network, while Friedman & Lolas (2005) considered a reaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli.
Recently, Swartz and coworkers have developed novel assays for the detailed investiga- tion of network formation from blood endothelial cells (BECs) and lymph endothelial cells (LECs). Ng, Helm & Swartz (2004) exposed ECs to interstitial flow in collagen gels, and found key differences between the two cell types in their cell-cell and cell-matrix interactions, and their responses to the local biophysical environment. Through combined experimental and theoretical work, Helm et al. (2005) and Fleury et al. (2006) showed that interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix- bound vascular endothelial growth factor (VEGF). Helm, Zisch & Swartz (2007) found that extracellular matrix composition (fibrin versus collagen) differentially influences the organization of the two endothelial cell types, with LECs showing the most extensive organization in fibrin-only matrix, and BECs preferring a collagen matrix. These differences are also observed in vivo and it is hypothesised that during dermal wound healing the tissue matrix remodels so that initially it is optimised for angiogenesis and at later stages for lymphangiogenesis.
The aim of this project is to generate a predictive tool that can be used to inform network formation from lymph endothelial cells in vivo (with applications to wound healing) and in vitro (with applications to tissue engineering for example).
Description of the Planned Approach and the Techniques Needed: In this project, we will use a discrete/hybrid modelling approach, similar to that developed in (Owen et al., 2009) and (Perfahl et al., 2011), to study lymphangiogenesis and the interplay between the lymph and vascular networks. In more detail, a discrete model that accounts for the evolving spatial structure of the vascular network will be coupled to reaction-diffusion equations describing the distribution of key growth factors. The behaviour of the model will be investigated using a combination of analytical and numerical approaches. The models will be informed by the experimental results of Swartz and co-workers, and once formulated, will be validated against the experimental data.
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Figure 10: 3D lymphangiogenesis assay. Cells sprout from dextran beads embedded in fibrin gel.
References
[1] Perfahl H, Byrne HM, Chen T, Estrella V, Alarcon T, et al. (2011) PLoS ONE 6(4): e14790.
[2] Roose T, Fowler AC, (2008) Bulletin of Mathematical Biology 70: 1772–1789.
[3] Friedman A, Lolas G, (2005) Math. Mod. Meth. Appl. Sci. 15: 95107.
[4] Ng CP, Helm C-LE, Swartz MA, (2004) Microvas. Res. 68: 258–264.
[5] Helm CL, Fleury ME, Zisch AH, Boschetti F, Swartz MA, (2005) Proc. Natl. Acad. Sci. U.S.A. 102: 15779–15784.
[6] Fleury ME, Boardman KC, Swartz MA, (2006) Biophys. J. 91: 113–121.
[7] Helm CL, Zisch A, Swartz MA, (2007) Biotechnol. Bioeng. 96: 167–176.
[8] Owen MR, Alarcon T, Maini PK, Byrne HM, (2009) J. Math. Biol. 58: 689–721.
3.8 Mathematical Modelling of the Negative Selection of T Cells in the Thymus
Supervisor: Prof. Jon Chapman Contact: [email protected]
Background: The thymus is the primary organ for the generation of naive T cells. During their maturation, T cells acquire an antigen-receptor with a randomly chosen specificity including reactivity to the body’s own proteins. To purge this pool of im- mature T cells from cells with a reactivity to self-antigens, specialised epithelial cells in the medulla of the thymus produce a broad range of proteins which are normally only detected in differentiated organs residing elsewhere in the body. The efficient and genome-wide transcription of these so called self-antigens secures the completeness by
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which these self-antigen reactive T cells are deleted. Hence, thymic medullary epithelial cells as a population provide a comprehensive “molecular library” of self-antigens that when recognized by developing, self-reactive T cells will initiate their death. This dele- tion of potentially harmful T cells is known as thymic negative selection and prevents the formation of a repertoire of effector T cells able to initiate an injurious autoimmune response.
The range of promiscuous genes expressed by each single medullary thymic epithelial cell (mTEC) is, however, thought to be limited to a selection of self-antigens. Consequently the library of self-antigens would only be representative in its entirety when a larger number of these medullary epithelial cells are concurrently available. However, a detailed quantitative and qualitative analysis of this concept has not yet been accomplished.
Description of the planned approach and the techniques needed: This project is to investigate mathematical models of T cell negative selection. Some existing models consider just one T cell-mTEC interaction, but include multiple receptors with some threshold criteria for whether the interaction “fires” [1]. Other models consider just one receptor-ligand binding, but in more detail, incorporating the sequence of the receptor peptide into the model, so that the strength of the interaction is determined by the similarity between the receptor sequence and the ligand sequence [2]. The goal of the project is to synthesis key components of existing models in such a way that they are suitable for the incorporation of gene expression data from individual mTECs.
The mathematics will involve stochastic models of reactions/interactions/binding-unbinding. An understanding of elementary probability theory and ordinary differential equations will help.
Reasonable expected outcome of project: A new model for T cell negative selection.
References
[1] Berg, H.A. van den; Rand, D.A.; Burroughs, N.J., “A reliable and safe t cell repertoire based on low-affinity t cell receptors,” Journal of Theoretical Biology, 209:465–486, 2001.
[2] Detours, Vincent; Mehr, Ramit; Perelson, Alan S, “A quantitative theory of affinity- driven t cell repertoire selection,” Journal of Theoretical Biology, 200:389–403, 1999.
3.9 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors: Dr Eamonn Gaffney and Dr Hermes Gadelha Contact: [email protected] and [email protected]
Background: The eukaryotic axoneme is a ubiquitous organelle found within cilia and flagella, which are filamentous cell appendages whose beating drives fluids in numerous physiologically important settings, including sperm swimming and egg transport in re- production, mucociliary clearance within the lung, circulation within the cerebrospinal fluid system, symmetry breaking in early developmental biology and the virulence of
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numerous medically important pathogenic parasites. Dynein molecular motors contract within the axoneme, exerting internal forces and moments; mechanically, these are bal- anced by a combination of viscous drag from the medium surrounding the cell and a passive elastic restoring response of the cilium or flagellum. The resulting dynamics gives rise to a propagating waveform which drives the surrounding fluid, and for free cells, results in swimming. Nonetheless, the subcellular details of the collective behavior of the dyneins and their regulation are poorly understood, suffering from numerous competing hypotheses. However consider the combination of cell videomicroscopy and mechanics. From movies of a swimming cell for example, one can determine rate of viscous dissipation associated with the flagellum using fluid dynamical theory. By energy conservation this is the time averaged rate of working of the dyneins, allowing energy expenditure to be measured as one example of how mechanics can be extracted from microscopy and how ultimately our understanding of the mechanics and biology driving cellular swimming may be improved.
Reasonable expected outcome of project: There are many possible projects and thus outcomes. One example would be to investigate image analysis techniques to improve flagellar extraction, another to explore sperm filament mechanics in detail using micromanipulator experiment data, a third would involve the use of fluid and filament mechanics to assess dynein behaviour from current videomicroscopy data and a final example would be to explore which waveforms are the most energetically efficient.
Techniques: Depending on the detailed choice of project, investigations in this field could rely on calculus of variations and the numerical solution of partial differential equations, for novel image analysis. Alternatively, the project could focus on viscous fluid mechanics and elastic micromechanics to assess dynein behaviours from microscope videos or combine mechanics and calculus of variations to find optimal waveforms for swimming.
References
[1] H. Gadelha, E. A. Gaffney and A. Goriely. PNAS 30:12180-85, 2013.
[2] E. A. Gaffney, H. Gadelha et al. Ann. Rev. Fluid Mech. 43:501-28, 2011.
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Supervisors: Dr Ulrich Dobramysl and Dr Radek Erban Contact: [email protected] and [email protected]
Much theory has been developed for the coordination and control of distributed autonomous agents, where collections of robots are acting in environments in which only short-range communication is possible [1]. By performing actions based on the presence or absence of signals, algorithms have been created to achieve some greater group level task; for instance, to reconnoitre an area of interest whilst collecting data or maintaining formations [2]. Algorithms of swarm (collective) robotics have often been motivated by collective animal behaviour [3]. Collective animal behaviour has been of interest for mathematical research throughout the last century [4]. In many of these mathematical approaches a model is proposed and then compared to the real-world behaviour of the animal groups under certain comparison measures. One example of this type of model was developed by Couzin et al [5] for individuals communicating through visual and contact interactions. Depending on parameter values, it can generate directed swarms, torus movement or weakly ordered groups of animals.
In the Mathematical Institute, we have a group of mobile e-puck robots [6] which can interact through a number of different channels (audio, video, bluetooth) — see Figure 11. These robots have sensors (resp. actuators) for contact and visual communication which can be used for mimicking the behaviour of animal models in [4,5]. In a previous dissertation [7], an M.Sc. student investigated an implementation of searching algorithms, similar to those used by flagellated bacteria, in a robotic system. A paper based on this M.Sc. dissertation [7] is currently being prepared for publication.
In this project, we will use a combination of experiments with robots and mathematical modelling. We will investigate accurate and efficient ways to mathematically model collective behaviour of individuals (robots) communicating through short-range (proximity sensors) and long-range (auditory and visual cues, bluetooth) means. We would like to understand the advantages of different types of hierarchies and strategies within a group of robots for the successful completion of a pre-defined group task, similar to the research that has been done in [8] for hens inside a barn and in [9] for pigeons during a flight. Depending on student interest, the robot tasks can also involve target area finding, maintaining formations in a complicated geometry or other assignments [10]. This project will involve analytical and numerical modeling, microcontroller programming, and efficient sensor data analysis.
References
[1] J. Reif and H. Wang, Robotics and Autonomous Systems 27(3):171194, 1999
[2] J. Desai, J. Ostrowski, and V. Kuma, IEEE Transactions on robotics of automation 17(6):905908, 2001
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Figure 11: A group of mobile e-puck robots
[3] Garnier, S., in Bio-inspired self-organizing robotic syst., eds: Meng and Jin, Springer, pp. 105-120 (2011)
[4] Sumpter, D. Collective Animal Behavior, Princeton University Press (2011)
[5] Couzin, I. et al, Journal of Theoretical Biology 218, pp. 1-11 (2002)
[6] Mondada, F. et al, Proc. of 9th Conf. on Autonomous Robot Systems and Competi- tions 1, pp. 59-65 (2009)
[7] J.T.King, “Hard-Sphere Velocity-Jump Processes: Applications to Swarm Robotics”, MSc dissertation, 2013
[8] Linquist, B., Bulletin of Mathematical Biology 71, pp. 556-584, (2009)
[9] Nagy, M. et al, Nature 464, pp. 890-893 (2010)
[10] Gazi, V. and Passimo K., Swarm Stability and Optimization, Springer (2011)
4.2 A Simple Model for Dansgaard-Oeschger Events
Supervisors: Dr Ian Hewitt and Dr Andrew Fowler Contact: [email protected] and [email protected]
Many northern hemisphere climate records show a series of rapid climate changes that recurred throughout the last glacial period. These “Dansgaard-Oeschger” (D-O) se- quences are most prominent in Greenland ice cores and consist of a very rapid (decades) warming, followed by an initial slow cooling and a final rapid temperature fall. They occurred somewhat periodically with a period of around 1500 years. What is respon- sible for this sequence is of course of great interest given current climate changes, and
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it has been hotly debated. Various factors point towards changes in ocean circulation being key. The suggestion is that the Atlantic meridional overturning circulation — the conveyor that transports warm equatorial water northwards and keeps the UK warm — underwent sudden changes in strength, and that this caused the rapid changes in air temperature over the Northern Hemisphere. A sudden injection of fresh water into the North Atlantic may be sufficient to cause such switches, but it is not fully understood what would cause this.
This project would explore the hypothesis that the Dansgaard-Oeschger events occur as a self-sustained oscillation of the ocean dynamics and the Northern hemisphere ice sheets. In this mechanism, the melting and growth of the ice sheets would be determined by the strength of the ocean circulation, but at the same time the melting itself provides the fresh water that drives the ocean circulation.
The project would consist of constructing simplified box models of the ocean and the ice sheets. These can be reduced to systems of non-linear ordinary differential equations that can be solved numerically and analyzed to examine steady states, stability, limit cycles, etc, under different assumptions. There are numerous levels of complexity — both in the modelling and mathematical analysis — that can be added sequentially depending on time and earlier success.
4.3 Modelling Snow and Ice Melt
Supervisor: Dr Ian Hewitt Contact: [email protected]
On ice sheets and glaciers snow builds up on the surface over the winter and melts during the summer. The quantity of the melt water that runs off from the surface is important because it is a large component of sea-level rise. Predicting this run off is not as straightforward as might be imagined, because as the snow melts the water infiltrates into the snowpack and some of it refreezes while the rest runs off along the ice surface. At lower altitudes, there is more melting than snow, so the snowpack is exhausted by mid summer and the underlying ice also melts. Almost all of the water runs off in this case. At high altitudes, the amount of melting is less than the accumulation of snow, so each year the older layers of snow are gradually compacted to form ice. Here it is very poorly understood how much of the water runs off and how much refreezes.
The aim of this project is to derive a mathematical model for the melting and compacting snow pack, and to solve some simplified problems to understand the roles of different physical parameters in influencing the amount of run off. The first task will be to develop a model. It will be a continuum model, describing snow as a deformable porous medium with Darcy flow through the pores. The interesting and unusual aspect of the model is the refreezing, which will require incorporating an energy conservation equation. The model will be simplified and then solved in one dimension (vertical) for some simple boundary conditions. This will almost certainly require a combination of numerical and asymptotic methods.
This project would require an interest in continuum modelling and fluid mechanics,
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a willingness to engage with and translate physics into mathematics, and some open- mindedness about using different techniques for solving partial differential equations.
There are various other potential projects associated with ice sheets and modelling interesting processes that affect them. Please discuss with me.
4.4 A Network-Based Computational Approach to Erosion Modelling
Supervisor: Dr Ian Hewitt Contact: [email protected]
Many erosive processes produce interesting geometrical structures, and a common result is the formation of branching channel networks. The obvious example is river networks, but similar processes occur in erosion of limestone caves, groundwater flow through soil, porous flow in oil and gas reservoirs, and the flow of molten rock inside the Earth. In all these situations, fluid flow over or through a porous substrate causes erosion that feeds back to alter fluid flow.
One of the interesting challenges of modelling this process mathematically is that the fluid flow transitions from an initially uniform state to an evolved state with a vastly different structure. For instance, a porous rock in which fluid flow is modelled by Darcy’s law may be eroded to form a cylindrical conduit for which Darcy’s law is no longer appropriate.
This project will explore a new numerical method to describe these processes by combining a distributed porous domain with a network of localized one-dimensional flow elements. The simplest generic problem involves an elliptic partial differential equation to describe fluid conservation, coupled to an evolution equation (ODE) for the erodi- ble material. The method to be developed will use a finite element method to solve the equations, incorporating elements of different dimension to account for the different flow structures. A similar approach has yielded realistic “looking” results for water flow beneath a glacier [1], but has raised a number of interesting questions that need to be explored.
The project will involve getting to grips with physical principles of fluid flow and erosion, developing and coding a numerical model, and exploring its behaviour.
Reference
[1] M. A. Werder, I. J. Hewitt, C. Schoof and G. E. Flowers. Modeling chanelized and distributed subglacial drainage in two dimensions. J. Geophys. Res. 118, 1–19, 2013.
4.5 Retracting Rims
These are actually three projects which share some common features.
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I. Bursting films. When a freely suspended film of a viscous liquid ruptures (for example in a bursting bubble), a rim forms around the expanding hole that grows as it is pushed deeper into the yet unperturbed film. Moreover, the rim forms ondulations in the spanwise direction. This problem has a long history, with the original Taylor-Culick formula describing the retraction velocity based on conservation of mass and momentum. (However, a systematic derivation of the long time evolution of the film profile has not been carried out for the case of large viscous dissipation.) The task in the thesis will be to (a) Rederive the underlying thin film model (b) investigate the evolution of the cross section using a (self-written) matlab code and asymptotic analysis (c) investigate the stability of the rim using a linear stability analysis. Extensions from the planar to the axisymmetric case may also be considered, and the effect of viscoelasticity.
II. Dewetting rims with strong slip. When a liquid is repelled from a flat surface, i.e. it is hydrophobic, holes will grow once they are formed, since this reduced the total energy of the liquid films (i.e. the sum of all interface energies is reduced by collecting the liquid into ridges or droplets with only a small liquid/solid interface area). The dynamics of this process has been carefully investigated for thin polymeric liquids, and it can show a surprisingly rich behaviour in particular in the case where there is also significant effective slip at the interface. In this project, we will (a) redrive the underlying thin film model (b) investigate the evolution of the cross section using a (self-written) matlab code and asymptotic analysis. Further steps could involve a stability analysis, and/or the inclusion of visco-elasticity.
III. Inertial dewetting. When a low-viscosity liquid such as water is deposited as a film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon, the opening of a hole leads to a very fast retraction of the liquid as it dewets from the substrate. The Reynolds numbers are large than one, suggesting inertia is important (more as in I. for low-viscosity films than as in II.). In contrast to the bursting suspended films, the normal component of gravity and friction at the liquid/solid substrate enter. Focusing first on the effect of the former, we will (a) derive a model for this situation (b) investigate the wave structure to identify the two fronts that are observed in the experiments (c) compare with the experiments. The step (b) will involve writing a matlab code to solve the model equations.
Extensions could be going from the planar- to the axisymmetric situation, or including the effect of friction at the substrate.
4.6 Modelling Spray Deposition for Applications in Manufacturing Su- percapacitors
Supervisors: Dr Andreas Munch and Dr Jim Oliver Contact: [email protected] and [email protected]
One of the main techniques to design nano-structured porous electrodes for use in super- capacitors and batteries is spray coating of colloidal droplets. When a colloidal droplet
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impacts a heated solid surface it will spread to a maximal extent, while the liquid evap- orates leaving behind nano-patterns of colloidal particles. A basic continuum model for the spreading and evaporating colloidal droplet may be found in [1]. The model involves a system of partial differential equations coupling the fluid motion with the concentration field of the particle distribution. Combining asymptotic and numerical techniques, the project aims to derive simplified models for the droplet profile and the evolution of the volume fraction that are used to predict the particle distribution under various impact conditions. (The project may involve collaboration with the research group of Professor Patrick Grant in Oxford’s Materials Sciences Department).
Reference
4.7 Mathematical Modelling of Membrane Fouling for Water Filtra- tion
Supervisors: Dr Ian Griffiths and Dr Andreas Munch Contact: [email protected] and [email protected]
Understanding membrane fouling is a key goal in separation science, and is an area in which detailed mathematical modelling can provide key insight for membrane design optimization. Historically, in a typical filtration set-up there are four key membrane fouling mechanisms:
• Standard blocking — small particles pass into the membrane pores and a finite number adhere to the walls causing pore constriction.
• Partial blocking — larger particles land on the membrane surface and partially cover a pore.
• Complete blocking — larger particles land on the membrane surface and cover a pore entirely.
• Caking — a layer of particles builds up on the membrane surface following complete blocking, which provides a further resistance in the form of an additional porous medium through which the feed must also permeate.
Recently new asymmetric membranes have been developed whose pore radius varies with depth. Such membranes have been demonstrated to possess novel filtration properties. For instance, a membrane whose pores constrict with depth can capture different particle sizes at different positions, while a membrane whose pores expand with depth may offer a mechanism to control the surface build-up associated with caking. This project aims to understand the role of each of the fouling mechanisms in the clogging of an asymmetric membrane, and in particular the interplay between the various mechanisms. Mathematical models based on stochastic simulations will be developed and continuum descriptions will be derived by examining various limiting time-averaged cases. The
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outcome will be a series of mathematical models that make predictions on the optimal operating regimes to filter a given contaminant.
References
[1] G. R. Bolton, D. LaCasse and R. Kuriyel 2006 Combined models of membrane fouling: Development and application to microfiltration and ultrafiltration of biological fluids. J. Memb Sci., 277, 75–84.
[2] G. R. Bolton, A. W. Boesch and M. J. Lazzara 2006 The effects of flow rate on membrane capacity: Development and application of adsorptive membrane fouling models. J. Memb. Sci., 279, 625–634.
[3] C-C. Ho and A. L. Zydney 2000 A combined pore blockage and cake filtration model for protein fouling during microfiltration. J. Colloid Interf. Sci., 232, 389–399.
4.8 Flow-Induced “Snap-Through”
Supervisors: Dr Dominic Vella and Dr Derek Moulton Contact: [email protected] and [email protected]
It is well known that in high-speed winds, umbrellas are forced from their initial state to an inverted state that is less efficient at keeping the rain off. This “snap-through” instability is an intrinsic feature of elastic systems with a natural curvature and is used in biology and engineering to generate fast motions. The proposed project looks at the combination of elasticity and fluid flows that produces this instability and aims to understand the critical properties of the transition.
The project will begin by reviewing previous work on the “snap-through” instability of an elastic arch subjected to static loads [1-3]. It will then move on to understand under which conditions fluid loading (e.g. in the limits of high and low speed flows) can cause
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such a snap-through to occur, the dynamics of this snapping and whether the snapped- through state is stable. We hope to include comparisons between models developed as part of this project and experiments conducted at Virginia Tech.
Expected outcomes involve the identification of a dimensionless fluid loading parameter and quantifying when snapping should occur as a function of this parameter.
References
[1] J. S. Humphreys, On dynamic snap buckling of shallow arches, AIAA J. 4, 878 (1967)
[2] A. Fargette, S. Neukirch and A. Antkowiak, Elastocapillary Snapping, http://arxiv.org/abs/1307.1775
[3] A. Pandey, D. E. Moulton, D. Vella and D. P. Holmes, Dynamics of Snapping Beams and Jumping Poppers, http://arxiv.org/abs/1310.3703
4.9 Plumes with Buoyancy Reversal
Supervisors: Dr Dominic Vella and Prof. John Wettlaufer Contact: [email protected] and [email protected]
One means by which the worst effects of climate change might be avoided is to pump large amounts of carbon dioxide into sub-surface aquifers: so-called carbon sequestration [1]. When pumped into aquifers, the carbon dioxide remains buoyant with respect to the ambient liquid and so rises back towards the surface. In practice, this rise is halted by layers of relatively impermeable rock, which trap the carbon dioxide until it has sufficient time to dissolve in the ambient water. However, once dissolved, the carbon dioxide/water mixture is unusual because it becomes denser than the water; it will therefore reverse direction and sink.
In an unconfined porous medium, it might be expected that the original plume may actually reach a steady height: sufficient mixing should occur over the course of its rise that the source of buoyancy becomes extinct at some critical height. This project will address the question: does an unconfined plume have a maximum rise height?
The project will begin by reviewing the classic analyses of buoyant plumes in a porous medium [2] and developing a numerical code to verify the similarity solutions presented there. Using a more realistic equation of state for carbon dioxide/water mixtures (incorporating buoyancy reversal) in the previ

m.sc. in mathematical modelling and scienti c computing

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