The main geometry under investigation was that of a rigid spherical particle sedimenting under gravity in an infinite cylindrical vertical pipe.  In Stokes flow, the sphere falls vertically with no sideways drift, due to reversibility.  In Newtonian flow with inertia, there is a sideways migration to a particular location (the Segré-Silberberg effect), whereas in viscoelastic flow there is a sideways drift towards the pipe wall, sometimes referred to as “negative lift.”  The traditional benchmark problem sets the sphere’s radius (\(a\)) to be half that of the pipe (\(2a\)), with the sphere falling along the pipe’s axis, and the asymmetric problem (as shown) has the sphere initially offset by \(2\varepsilon a\).

​The mesh reconnection algorithm was based primarily on properties of the Delaunay triangulation, which for any given set of vertices has the property that the circumsphere of each tetrahedron contains no vertex.  In practice, this involves considering local “flips” between the two configurations shown.  On the left there are two tetrahedra sharing a common face, \(ABC\).  The “2-3 flip” removes this face and adds a new edge \(DE\), forming instead three tetrahedra as shown on the right.  The “3-2 flip” is the same but in reverse.  Combining these with additional heuristic measures, the computational method was able to preserve mesh quality (in the sense of reducing the largest dihedral angle).

​Delaunay triangulation is optimal in 2D in the sense of maximizing the smallest angle in any triangle, but this property does not hold in 3D.  Consequently the Delaunay mesh in 3D can include some tetrahedral shapes with “bad” aspect-ratios for flow simulation purposes, known as “slivers.”  A sliver is a very flat tetrahedron without any particularly short edges.  In the example shown, the four vertices are nearly coplanar. On the left, a sliver is shown from two perspectives; on the right it is shown within its circumsphere.  The avoidance of slivers is considered a major goal of high-quality 3D mesh generation, and effective remedies involve assigning artificial weights to vertices, or adding/removing vertices in the neighbourhood of each sliver.

​To investigate dynamic viscoelastic particle drift, it is necessary to consider the build-up of polymer stretch within the flow (in the cross-sectional images, yellow is highly stretched, dark blue is unstretched, and between each image the sphere has fallen by one diameter).  At early times the polymer becomes highly stretched in the narrow gap first, and the sphere initially drifts inwards towards the axis. As the material in the wider gap becomes highly stretched, there is a transition to an outward drift, and this dominates at later times.  On the far side of the pipe there is a build up of stress due to shearing by the back flow through the wider gap between the sphere’s wake and the far wall.

​Using an argument which considers the radial pressure gradient due to normal stresses, in both the narrow gap and the wider gap on either side of the sphere, an analytical approximation for the ratio of outward drift velocity to vertical falling velocity was established.  In the case of a FENE-CR fluid this is approximately \(\frac{2c}{3\pi K_S} \min{(1, \mathrm{We})}\), where \(K_S\) is the Stokes drag coefficient, \(c\) is the polymer concentration, and \(\mathrm{We}\) is the Weissenberg number for the flow.  This approximation is linear in concentration (for the dilute case), and provides an accurate agreement to the numerical results (until \(c\) becomes large), constituting a qualitative and quantitative explanation of viscoelastic particle drift.

Illustration of the transformation process used to design metamaterials. Here we show an example of a ground cloak transformation where a virtual space with coordinates \(\mathbf{x}’=[x’,y’]\) is compressed upwards to form the physical space with coordinates \(\mathbf{x}=[x,y]\). This mapping from \(\mathbf{x}'\) to \(\mathbf{x}\) creates a triangular void in the physical space. We measure how much this transformation compresses each point in the virtual space using the Jacobian, denoted by \(\mathbf{F}\). From this we can determine the necessary properties for the metamaterial regions, for example, in my PhD we focus on steady-state thermal conduction and so we only need to engineer the thermal conductivity, which is given by
\begin{equation}
\mathbf{k} = \dfrac{\mathbf{F}k'\mathbf{F}^T}{\det\mathbf{F}}\tag{1}         ,
\end{equation}
where \(k’\) is the conductivity in the virtual domain. When the conductivity in the transformed, metamaterial regions satisfy (1), any object placed within the void is undetectable from the external field – making it invisible! The effects of a traditional ground cloak are simulated in image 2. In my PhD I use a modified version of this mapping to utilize it in a new, heat spreading context. A simulation of this application is provided in the image 3.

Simulation of a traditional ground cloak application:

Simulation of a modified ground cloak mapping being used in a new, heat spreading context.

​Ground cloaks are a particularly nice example of thermal metamaterials as their properties are anisotropic, but not homogeneous. Therefore we can accurately approximate them with a laminated design, which can be seen in image 4. 

The realisation process for the metamaterial in image 3(b), which can be approximated with a rotated laminate. 

​Although not shown here, we have extended this model (and the other models in our research) to include interfacial effects. These effects can dominate in multi-layered designs such as laminates.

Although PhD students obviously spend most of their time researching, this is not the only thing we do! A PhD can involve many responsibilities, for example, during my PhD I have spent time:

• Being a teaching assistant for undergraduate courses.
• Attending conferences to present work and network with other researchers in my field.
• Creating online material for outreach and widening participation – like writing blog posts and A-to-Z entries for this website!
• Helping at science festivals to promote the research focuses of the Mathematics of Waves and Materials group.

In the image below there is a photo of me helping at Bradford Science Festival earlier this year. The main aim of this weekend was to get primary school children excited about maths and science by explaining some simple concepts about sound. The next image shows a simple maze game that I designed on Scratch to help demonstrate the effects of different thermal metamaterials. The final image shows a photo I took of the Utah State Captiol when I attended a conference in Salt Lake City back in 2019.

Illustrating the homogenization process. Typically when dealing with materials on an everyday level we assume the “continuum hypothesis” so that the atomic and molecular scales of the medium are smeared out (as illustrated on the right of this image) and we can therefore usefully assign macroscopic properties such as density, viscosity, stiffness, etc. to the medium. Homogenization goes a step further and smears out the microstructure of a composite medium at a larger length scale. Here we illustrate a particulate composite, where inclusions are distributed inside a host matrix medium. The microstructural length scale is “q” and homogenization techniques work well when the macroscopic length scale is much bigger than the microstructural one. The inclusions could be introduced to provide improved conductivity or stiffness whilst reducing weight for example. Homogenization is useful because it means that we can assign a single effective property to the material in question, instead of one that varies in space.

​Visualisation of the microstructure of a specific type of composite material using X-ray Computed Tomography. These are cross-sections (a-c) and a 3d reconstruction (d) of a syntactic foam. This is a material consisting of thin spherical shells (the circles/spheres here) distributed inside a matrix, often a polymer. These materials are lightweight and stiff and frequently assist in absorbing sound.

​My PhD focussed mainly on how elastic waves propagate inside fibre-reinforced composites where the fibres are very long. Here their length is along the x3 axis. In such fibrous materials the microstructure (the fibres) can be considered to be characterised by a repeating periodic cell in the x1-x2 plane. The effective elastic properties of the material can then determined by applying homogenisation techniques. (At least) three different types of elastic waves can be considered to propagate – here illustrated as SH, SV and P waves.

​When fibres are placed on a hexagonal lattice, as illustrated here, the response when waves propagate in the x1-x2 plane is isotropic, so that the wave speed is independent of direction of propagation. Here we illustrate this by determining the associated effective “antiplane” shear modulus of the material as a function of the size (radius r) of the fibre. The fibre is ten times as soft as the matrix, so that the more fibre added, the softer the composite becomes (the effective shear modulus decreases). I considered a new approach to the implementation of the so-called asymptotic homogenization scheme, which is plotted here in orange. It can be seen that the results agree well with existing methods and energy bounds.

I also derived a completely new method, known as the integral equation method of homogenisation. This is an extremely efficient new technique, which can be used straightforwardly to model composites where the inclusions have rather arbitrary cross-section and also to geometries where fibres are placed on unusual lattices. Here we illustrate results when fibres are located on a rectangular lattice. This induces anisotropy, so that the elastic waves propagate with different speeds in different directions. This is illustrated here by plotting the associated effective shear modulus when waves propagate in the x1 and x2 directions. Our new integral equation method is plotted in comparison to the asymptotic method here, showing good agreement in both cases.