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
1
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
2
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 pre- sented 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.
3
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, differen- tial
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 com- pute on surfaces where I
don’t understand the results. For example, it can compute on
problems with variable dimen- sion 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 nu- merical 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.
5
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 exam- ple 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 par- allelism 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 pro-
posed numerical algorithm is the Sequential Regularization Method
[1]. An implemen- tation, 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 incom- pressible 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.
6
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 res- onance 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 re- cently,
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 nu- merical 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) in- painting.
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 prob- lems in
three-dimensions, for example, removing watermarks or subtitles
from video (2D + time). There are also likely ap- plications 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 algo- rithm (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 math- ematical 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-
8
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 tech- nique
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 prob-
lems 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.
9
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 fur- ther 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 decomposi-
tions 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.
11
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 ad- joint 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 tur-
bines.
(b) Computational domain with the tur- bine 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.
12
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 typi- cally 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
13
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
14
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
15
• 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
16
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
17
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 equa- tions (PDEs) and master equations for
probability distributions of biochemical species involved. These
equations include the chemical master equation (CME) and the chem-
ical 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].
18
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 differ- ential 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 dimen- sion 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 prob- lem. 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
19
20
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
21
[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 circa- dian 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 re- sulting 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.
22
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).
23
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 differentia- tion. 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.
24
One of the most popular models for gradient formation is based on
the localized produc- tion 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 incorpo- ration of further important
details from cell biology. These models will be analyzed us- ing 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.
25
Supervisors: Prof. Helen Byrne and Prof. Colin Please Contact:
[email protected] and
[email protected]
Background and problem statement: There is considerable interest in
understand- ing 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 under- stand 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.
26
[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.
27
Supervisors: Prof. Helen Byrne and Dr Chris Bell Contact:
[email protected] and
[email protected]
Background and problem statement: The lymphatic and vascular
systems are cou- pled: 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 in- teractions, 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 differ- ences 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 nu-
merical 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.
28
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
29
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 selec- tion.
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
30
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
behav- ior 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 vis- cous 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 im- prove 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.
31
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 au- tonomous 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 maintain- ing 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 in- teract 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 dis- sertation [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 col- lective 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 find- ing, 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
32
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
33
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,
34
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 in- teresting 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 com- bining 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.
35
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
36
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 concentra- tion 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
37
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 mem- brane 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
38
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 (in-
corporating buoyancy reversal) in the previ