Meet the Mathematicians Book 2009

mtm

Meet the Mathematicians 2009 1

Monday 6th april 2009 · the University of nottinghaM

meet the mathematicians

2009

2 3 Meet the Mathematicians 2009www.meetmaths.org.uk

mtm

WelcoMe Welcome to the 2nd ‘Meet the

Mathematicians’ (MTM) event, which is being run in parallel with the 51st British Applied Mathematics colloquium (BAMc), this year held at the University of Nottingham.

It may come as a surprise that mathematics is a living subject, with many new advances each year in novel as well as classical areas of mathematics. Almost all university mathematicians spend a good deal of time researching their favoured areas of mathematics, writing research articles and books, and attending conferences to discuss their latest discoveries.

The BAMc is the largest such annual gathering for applied mathematics in the UK. each year it attracts over 300 professors, lecturers and research students to speak about their latest mathematical discoveries in fields as diverse as mathematical biology, relativity, fluid dynamics and finance. Younger researchers are especially encouraged to attend and deliver their talks in a friendly and supportive environment.

The idea of MTM days arose because we wanted sixth formers interested in studying mathematics, or a related subject, at university to interact with mathematicians in the context of a research meeting.

That way they could have a chance to see the applications of mathematics beyond the ‘handle-turning’ constraints of the A-level syllabus. The MTM day is also designed to show students that university is not an extended sixth form, with lecturers playing the role of teachers, but is there for research and the advancement of knowledge. As a research conference, the BAMc forms a natural venue for this.

The talks you will hear today cover applications of mathematics to biological ordering, social interactions, the internet, slime and the entertainment industry. They are all driven by the underlying theme of the “unreasonable effectiveness of mathematics”. Some of the talks relate directly to the speakers’ main research interests; some relate ©

Th

e U

NIv

erSI

TY o

f N

oTT

INg

hA

M 2

008

BritishApplied MathematicsColloquium 2009


BAMC logo v2:bridging the gap 25/11/08 10:27 Page 1

to work they have done ‘for fun’. This illustrates the power of mathematics, in that it can be applied to describe, explain and predict phenomena in the real world. The talks will be recorded and you can catch them again on the web at www.meetmaths.org.uk, or watch the highlights on www.youtube.com.

Some of you will like mathematics because of the (school-inspired) belief that there is always a ‘unique answer’. one of the things you begin to learn at university is that this is not always true! When faced with real world problems, from black holes to complex industrial processes, in many cases a mathematician will start off having no idea what the ‘correct’ set of modelling equations should be. The major part of mathematical modelling is the understanding of the relevant and important processes underlying the behaviour of a system and their translation into equations. The methods for solving these equations are often secondary in importance. Sometimes the solutions of a model do not agree with experimental observations and modification of the assumptions, equations or approach is necessary. This is quite normal. It is not a failure. Insights are often gained through these dead ends. It makes the achievement of the final goal, of explaining something that no one else has done before, all the sweeter.

Many people who study mathematics traditionally go on to careers in finance.

With the country entering a recession, largely driven by failures within the financial sector, some may think that the future for mathematics is not bright. The opposite is true. In addition to the greater use of mathematics in the evaluation of financial risk, in the future the UK will need to trade on its high-technology sector. It will have to provide services or develop advanced science-based products that the rest of the world wants to buy. Mathematics will lead some of these fields, and will certainly underpin all of them.

The researchers at the BAMc will be playing significant roles in this future.We hope that the MTM day gives you a glimpse of what might be possible, so that, in time, wherever and whatever you choose to study, you can join them too.

have a great day!

Professor I David AbrahamsPresident of the Institute of Mathematics and its Applications

Dr Chris HowlsNational chair of the BAMc

4 5 www.meetmaths.org.uk Meet the Mathematicians 2009

mtm

ProgrAMMe10.30am – 11.00am

11.00am – 11.05am

11.05am – 11.45am

11.45am – 12.25pm

12.30pm – 2.00pm

2.00pm – 2.40pm

2.40pm – 3.20pm

3.20pm – 3.40pm

3.40pm – 4.20pm

4.30pm – 5.30pm

5.30pm – 5.40pm

Registration and Refreshments

Welcomeoliver Jensen (BAMc co-chairman)

The Geometry and Pigmentation of SeashellsStephen coombes

What do Hollywood, Sex and the WWW have to do with Maths?Keith hopcraft

Lunch and Activities

Designing Rides and RollercoastersJohn roberts

Slime Science – The Maths of Gloopoliver harlen

Refreshments

MTM Question Panel

Public Lecture: Some Thoughts on the Unreasonable Effectiveness of MathematicsJon Keating

FarewellDavid Abrahams and chris howls

11.05am (see page 7)

11.45am (see page 13)

2.00pm (see page 19)




www.meetmaths.org.uk6 7 Meet the Mathematicians 2009

mtm

The geoMeTrY AND PIgMeNTATIoN of SeAShellS

STePheN cooMBeSProfeSSor of APPlIeD MATheMATIcSUNIverSITY of NoTTINghAM

Seashells are beautiful objects that are admired for both their intricate

shapes and the patterns on their surfaces. Despite their complexity, these shapes are easily described using only elementary tools from geometry. Indeed a wide variety of natural shell shapes can be composed as surfaces in a 3-space and rendered using computer graphic imagery. Moreover, the pigmentation motifs that decorate many shells in the form of wavy stripes and checks, as well as chaotic and tent designs, can be generated by cellular automaton models, and in particular by the famous ‘rule 30’.

SeAShell AlgorIThMSThe photograph at the top of the next page shows some typical seashell shapes, all of which can be described in terms of a spirally coiled cone. from a mathematical perspective a natural description may be given in terms of a generating spiral and the shape of the opening or generating curve. There are now several algorithms for generating realistic seashell shapes, such as those described in the wonderfully illustrated book The Algorithmic Beauty of Sea Shells by hans Meinhardt.

To give the main idea behind these lUIS

flo

reS

8 9 Meet the Mathematicians 2009

mtm

www.meetmaths.org.uk

algorithms it is enough to consider the surface generated by rotating an expanding semi-circle in an upward direction as in the diagram opposite. other mathematical shell surfaces can be generated by rotating more realistic shell openings around helico-spirals. To learn about such mathematical shapes we first need to know more about circles, spirals and parametric descriptions of surfaces.

Planar spirals: Points in the plane may be specified with a pair of numbers, such as those of the cartesian coordinate system. Alternatively one may use the planar polar coordinate system as shown in the diagram at the top of page 9. A one-armed spiral is then described by the equation θ = f(r). A classic example is the Archimedean spiral with f(r) = r.

The parametric equation of a circle: A circle of radius R may be described in

terms of a single parameter θ as

x = R cos θ, y = R sin θ.If we let θ range over [0, 2π) then we generate a circle. If θ ranges over [0, π) then we generate a semi-circle.

3D spirals: A point in 3D may be described using the 3D cartesian coordinate system. In cartesian coordinates a point is specified with the triple (x, y, z). A helico-spiral may be expressed in terms of a single parameter θ by writing r = F(θ) and z = G(θ). for a 3D Archimedean spiral these functions are simply F(θ) = aθ and G(θ) = bθ for given constants a and b.

geNerATINg ShAPeSArmed with the above geometric notions we are now in a position to generate a simple seashell shape formed by the rotation of a generating curve along a

helico-spiral. As an example, let us take the generating curve to be a semi-circle and the helico-spiral to be a 3D Archimedean spiral, as in the diagram below.

To label a point on the surface we need to specify how far along the spiral we are (using θ) and how far round the semi-circle we are (using φ). This is easily calculated by letting

r → r + R sin φ, (1)z → z + R(1 − cos φ).

In terms of the 3D cartesian system the coordinates of the shell are given by

(x, y, z) = (r cos θ, r sin θ, z)

. hence, the surface of the shell is completely specified in terms of two parameters, θ and φ (where 0 ≤ φ < π):

x(θ, φ) = (aθ + R sin φ) cos θ,y(θ, φ) = (aθ + R sin φ) sin θ,z(θ, φ) = bθ + R(1 − cos φ).

To make more interesting shapes we can use different helico-spirals, make the radius of the semi-circle depend upon θ and φ (i.e. R = R(θ, φ)) or choose more complicated shapes for the generating curve. Some examples of shells generated

Some typical shells and their common names. from top to bottom and left to right: Soldier (california), Pelican’s foot (Mediterranean), Striped Whelk (Adriatic), honey Whelk (greece), Screw (Italy), Moon (Philippines), rock Snail (Mombasa), ring-Top cowry (Africa), Dove (Taiwan). These can all be described in terms of a spirally coiled cone (right)

θ

r

R

R sin φ

φ

Left: The generator of a simple seashell shape Right: A mathematical seashell obtained by rotating a semi-circular generating curve around an Archimedean generating spiral

y

xθ

r

(x, y)

(r, θ)

Left: cartesian and polar coordinate systems (where (x, y) = (r cosθ, r sinθ ) ) Right: An Archimedean spiral with f(r) = r

11

mtm

Meet the Mathematicians 200910 www.meetmaths.org.uk

with a logarithmic helico-spiral and various choices of the generating curve are shown in the diagram below.

SUrfAce PATTerNSAs well as having interesting shapes, many seashells also exhibit exotic patterns on their surfaces, such as that of the widespread species conus textile, shown in the photograph opposite. These patterns arise from the secretion of pigment from cells which lie in a narrow band along the shell’s lip.

each cell secretes pigments according to the activity of its neighbouring pigment cells and leaves behind a coloured pattern as the shell grows. In fact, these secretions are controlled in part by the mollusc nervous system and can be modelled with mathematical tools for describing dynamical systems.

however, many shell patterns can be described by simpler so-called cellular automaton (cA) models that do not track the details of the neurosecretory process.

A cellular automaton is a discrete model often studied in mathematics in the context of computability theory. It consists of a regular grid of cells, each in one of a finite number of states. Time is also discrete, and the state of a cell at time t+1 is a function of the states of the

cells in its neighbourhood at time t. The state update rules that define the creation of a new generation can be specified in terms of a simple table. for example, ‘rule 30’ can be listed in the form:

111 110 101 100 011 010 001 000↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓0 0 0 1 1 1 1 0

,

which tells us, for example, that if three adjacent cells in the cA currently have the pattern 100 (on-off-off ), then the middle cell will become 1 (on) in the next time step. The output 00011110 is

interpreted as an 8-bit binary number, equal to 30, and hence the name ‘rule 30’. Indeed, there are 28 = 256 possible cA rules of this type, though this rule is of particular interest because it produces complex, seemingly-random patterns like those in the diagram above.

here a black cell represents the state 1 (on) and a white cell the state 0 (off ). If we imagine colouring the seashell lip at time t of its growth with the pattern state obtained from the cA at time t then we would recover something like the texture of the cone snail conus textile, with its ‘cloth of gold’ pattern.

Mathematical seashell shapes. from top to bottom and left to right: conus, Nautilus, lyria, epitonium, Turritella, Planorbis, oxystele, Turbo, Struthiolaria

Stephen Coombes has a degree in Theoretical Physics from the University of exeter and a PhD in Neurocomputing from King's college london. he is currently a Professor of Applied Mathematics at the University of Nottingham, where he is applying the tools of dynamical systems theory to problems in neuroscience, including trying to figure out how his own brain works. he is actively engaged with experimentalists in Nottingham promoting the practical application of his recent research into travelling waves and spatially-structured activity in neural tissue.

When not doing mathematics Prof coombes enjoys running – though he is sadly getting slower every year!

coPY

rIg

hT

2005

rIc

hA

rD l

INg

conus textile (found in the waters of the Indo-Pacific) exhibits a cellular automaton pattern on its shell

‘rule 30’ cellular automaton (time decreasing down the page)

Space (shell lip)

Tim

e

Seashells exhbit exotic patterns on their surfaces


mtm

WhAT Do hollYWooD,

Sex AND The WWW hAve To

Do WITh MAThS?

KeITh hoPcrAfTASSocIATe ProfeSSor of MATheMATIcSThe UNIverSITY of NoTTINghAM

Most people’s initial perception of maths is of a dusty set of arcane

rules that are applied to describe properties of abstract quantities, whose importance and relevance is largely left unsaid. The common rejoinder is: “What’s the point?” Well, maths should rather be considered as a universal language that can describe, predict and unify

behaviours from the seemingly disparate – like the three topics appearing in the title of this talk, for example. There are two mathematical concepts that allow us to describe and establish the links between these topics – fractals and probability.

fractals are objects for which the notion of a characteristic scale-size is

14

mtm

15 Meet the Mathematicians 2009www.meetmaths.org.uk

meaningless. The answer to the question “how long is the coastline of Britain?” depends on the size of the smallest gradation of your measuring ruler. If it is a mile long, the ruler will fail to resolve any coves and bays that are smaller than this, and even if we reduce the gradation to a metre long then it will still be insensitive to the stones and pebbles on the sea-shore. The length of the coastline will get progressively longer with each increase in resolving power because you will be able to see more detail.

The Koch curve is a simple way of generating this kind of behaviour. replacing the middle third of a line segment by a triangle and repeating the procedure indefinitely produces an

object whose detail fails to be resolved by magnification. Moreover, the curve has a fractional dimension between one and two. These ideas go back a long way. Indeed, the 18th century poet and satirist Jonathan Swift once wrote:

So, naturalists observe, the fleahath smaller fleas that on him prey;And these have smaller still to bite ‘em;And so proceed, ad infinitum.

BAcoN NUMBerSThe notion of a scale size arises implicitly in probability distributions, too. These distributions tell us about the frequency at which events occur – from the accumulated score of a set of dice throws

to the number of atoms in a volume of gas at a given temperature and pressure.

The normal distribution that describes these has a rather narrow spread, which is characterised by the distribution’s standard deviation. This means that extreme events (for example, all the dice showing the same number) occur rarely. The spread of the distribution sets a scale-size on the frequency at which an event can occur. A profound and powerful mathematical result, called the central limit theorem, states that the normal distribution (or in the case of discrete objects, the Poisson distribution) will describe the fluctuations of an event occurring if there are many independent factors influencing the outcome of that event.

one can also construct probability distributions for social interactions. The ‘Kevin Bacon game’ works by picking an actor at random and counting how many co-stars it takes to get to Kevin Bacon. So, if actor A co-starred with actor B who co-starred with Kevin Bacon, then A has a Bacon number of two. If we carry out this procedure for a very large number of actors and form the probability distribution for the number of such connections, we obtain a result that is not a normal distribution. rather, it is described by a distribution for which a characteristic scale size does not exist – one that decreases like a power-law.

This tells us that the vast majority of actors only have one or two co-stars. These are the actors who only star in a

Step 1

Step 7

Step 2

The Koch curve is constructed from an initial straight-line segment of length 1 unit, from which the middle third is removed and replaced by a triangle, increasing the length to 4/3. repeating this procedure indefinitely obtains the crenellated curve, which has an infinite length, but can be drawn in a finite area. The fractional dimension of the curve is ln(4)/ln(3) ≈1.2618…

STA

rSTr

UcK

IMA

geS

(hTT

P://

creA

TIve

coM

Mo

NS.

org

/lIc

eNSe

S/BY

/2.0

/)

In the ‘Kevin Bacon game’ actors are assigned Bacon numbers depending on how many co-stars separate them from the prolific actor Kevin Bacon

17 16

mtm

Meet the Mathematicians 2009www.meetmaths.org.uk

single movie, i.e. extras. however, there are a small number of actors who have made many films and who consequently have a very high connectivity. This power-law pattern is also replicated in the number of sexual partners that a randomly selected individual has during a lifetime, as well as the number of hyperlinks into or out of a web-page.

The ScAle-free WeBWe can play the Kevin Bacon game for websites too, by letting robots randomly prowl the internet via the links into and

out of sites. on average it takes just six clicks to get from one randomly chosen site to another via hyperlinks. We can observe this relative closeness because the internet has the same scale-free and fractal character as the actor network.

This similarity is manifested by the probability distribution for the number of hyperlinks into a randomly selected site being described by a power-law. It is this property that enables one to navigate and traverse vast ‘distances’ between sites in the internet hyperspace. If this property were to change, then the connectivity

of the internet would become more local – it would inherit a characteristic scale size. As a consequence of this, the functionality of the internet would change radically.

The World Wide Web (WWW) currently has around 62 billion pages. But how

big can it get before it loses its fractal qualities and instead begins behaving according to the central limit theorem? By randomly connecting new sites to a scale-free network, we can determine how the probability distribution for the number of links evolves, and how many sites are required before it becomes close to being described by a Poisson distribution. The answer is about 1023 sites, a very big number! But how big? If we were to throw a die once a second and record the cumulative score, the answer would be described by the Poisson distribution after around ten seconds. on the other hand, suppose we were to add a new page to the WWW every second. Then the distribution of links between pages would be described by a Poisson distribution after around a billion times the age of the Universe – so don’t worry, you can keep on clicking!

The scale-free connectivity of the internet can be seen here. There is a cascade of hubs which enables sites to be connected across vast distances in internet hyperspace

Keith Hopcraft is a mathematician at the University of Nottingham. he is a mathematical physicist by training, and continues to work on the application of maths to physics. This includes studying how to make a sun on earth to help solve our energy needs through the power of nuclear fusion, how the complexities of social interactions can be described by principles that are very similar to the dynamics that describe the avalanches of a pile of sand, and how strange new materials can make lenses that can see smaller than a wavelength of light and make objects vanish from sight (without the aid of mirrors or trapdoors).

On average it takes just six clicks to get from one randomly chosen website to another via hyperlinks

IMA

ge

coPY

rIg

hT

2009

Th

e re

geN

TS o

f Th

e U

NIv

erSI

TY o

f cA

lIfo

rNIA

, All

rIg

hTS

reS

erve

D


mtm

DeSIgNINg rIDeS AND

rollercoASTerS

JohN roBerTSexecUTIve DIrecTor of oPerATIoNSJAcoBS eNgINeerINg groUP

Much of the thrill factor within a theme park ride is associated with

the forces exerted on the rider. generally our bodies experience the effect of acceleration due to a gravitational force of ‘1g’ (9.8 ms–2) acting vertically downwards, as a relative measure to our weight. We can also note a discernable change in forces when accelerating within a fast car (where we experience a g-force horizontally in the direction of travel) or in descending within a high-speed lift (where the normal vertical g-force decreases).

The magnitudes of these forces are generally modest – when a high performance car increases its speed from rest to 60mph in 4 seconds, it provides an

acceleration of around 6.6 ms–2, i.e. 0.67g. free fall, from sky diving say, guarantees an initial acceleration of 1g, although this is decreased by aerodynamic drag forces until a terminal velocity is attained. A feeling of weightlessness is experienced by riders on free fall rides at theme parks. In rollercoasters and other similar high speed theme park rides the g-forces are deliberately increased to a much higher level than the examples given above by rapidly changing the direction of the car through tight (small radius) curves and loops.

Designing a theme park ride or rollercoaster may sound fun, but to maximise the thrills and minimise the risks, mathematics is essential. for a

20

mtm

21 Meet the Mathematicians 2009www.meetmaths.org.uk

A DYNAMIcAl MoDelTo include forces into our model we can incorporate Newton’s second law mr = F for particles with fixed mass m and where F is the total force acting in the body. Thus, in the example of helical motion, the force on the body needed to maintain this motion is F = −maω2r , which would correspond to the force reaction exerted by the support track.

Now let’s consider the motion of a rollercoaster car as is moves through a circular loop (see diagram above right). It is natural to consider the motion in plane polar coordinates, where the coaster car maintains a fixed radial distance r = R from the centre of the loop and an angle θ from the bottom point of entry to the loop. The forces acting on the car are a vertical gravity force mg, where m is the mass of the car, a normal reaction force N and a tangential friction/aerodynamic resistance force F.

governing equations for this motion can be derived from Newton’s law in plane polar form. We end up with the following equations:

N = mg cos(θ) + mRθ2, (2)F = mRθ + mg sin(θ). (3)

To complete the formulation, an equation modelling the friction (and/or aerodynamic forces) would be needed, such as F = –μN, which then links the two equations. This leads to a second-order ordinary differential equation (oDe) requiring advanced analytic or numerical methods.

for the simplest case of neglecting frictional forces (so that F = 0), integration can be shown to give

12Rθ2 = 1

2v2

0R − g(1 − cos(θ)), (4)

N = mg cos(θ) + mRθ2, (5)where v0 is the initial velocity of the car. of practical interest is the case where the normal force N is zero, i.e. where the car

rollercoaster this means calculating the g-force that is applied to each passenger at every stage of the ride cycle. Too low a g-force will result in a boring experience with no thrills, but make the g-force too high and passengers will be at risk of nose bleeds – or worse. The maximum g-force needs to be more than about 3g and less than about 4g to give the necessary balance between thrills and required safety. But rollercoasters also have to work as a vehicle without an engine, without a driver and without brakes!

The mathematics required for the detailed calculations involved in theme park ride design is often complex and requires extensive computation, but simple models can be very effective for getting some understanding of the problem and providing good estimates. We can obtain a useful model by treating the motion of a rollercoaster car as a particle that can be described in terms of kinematics, dynamics and energy.

rollercoASTer KINeMATIcSIn many instances the 2-D and 3-D motion of the car is constrained to a curvilinear path and often a good starting point is to calculate the velocity

and accelerations of a body from the geometry of its path.

first, let’s consider the helical motion in time t, of a rollercoaster car with a position vector r through a corkscrew section of a ride (see diagram below) in terms of a time-varying angle θ(t).

In the cartesian coordinate system the position of the car can be expressed as

r = a cos(θ)i + a sin(θ)j + bθk,

where, j and k are unit vectors pointing along the coordinate axes. We can find the velocity components for the car by differentiating to give

r = −aθ sin(θ)i + aθ cos(θ)j + bθk,

and, assuming a constant angular speed (θ = 0), the acceleration is

r = −aθ2 cos(θ)i − aθ2 sin(θ)j = −aω2r.

here, r is a unit radial vector and ω = θis the angular velocity. We can deduce from the second equation that the speed of the car is v = |r| = ω

√a2 + b2 . for a

typical ride, taking a = 5m, b = 8m and v = 15ms–1, say, gives an inwards acceleration of 12.6 ms–2, i.e. 1.3g.

To maximise the thrills and minimise the risks, mathematics is essential

i

j

kr

geometry and axes for a ‘corkscrew ride’

entry

exitz

rF

N

mg

θ

configuration for a car undertaking a vertical loop

23 22 www.meetmaths.org.uk

mtm

Meet the Mathematicians 2009

SlIMe ScIeNce: The MATheMATIcS of

glooPolIver hArleNreADer IN PolYMerIc flUIDSUNIverSITY of leeDS

Many common liquids we find around us – for example pizza cheese, egg

whites or mayonnaise – don’t behave like ‘normal’ fluids such as water or golden syrup. Mayonnaise spreads easily on bread, but doesn’t flow under gravity; egg whites climb up a fork as you mix them and pizza cheese can stretch into thin filaments which snap like elastic bands. This funny behaviour isn’t confined to food stuffs – molten plastics behave in much the same way. What

makes these fluids different from ordinary fluids is that they contain long molecules (polymers) or other ingredients that get stretched out by the flow. In this talk we will look at some of these unusual flow phenomena and how we can describe them mathematically.

IS IT NeWToNIAN?Before considering how to describe such ‘funny fluids’, let us first consider the dynamics of normal fluids. consider

loses contact with the track. The limiting case is when the car is at the top of the loop (θ = π), and this will occur when v0

2 = 5gR. Thus, if v02 < 5gR at the start of

the loop, the car will lose contact with the track at some point on the trajectory. (In real rollercoasters, however, a circular loop is not used and the curve of the track is adjusted so that as the velocity changes the force remains relatively steady; also the velocity is increased so that a positive contact force with the track always applies.)

eNergY BAlANcINgIn some rides, such as the rollercoaster, the car is pulled up to an initial fixed height at the top of the ‘lift hill’ and the subsequent motion is driven by the initial potential energy associated with its mass and its height above the ground. Much insight and calculation for complex theme park rides can be achieved by simply considering the balance of energy as the rollercoaster moves from one point to another (assuming no friction losses).

Since energy is conserved, potential energy is transformed into kinetic energy, and vice versa, as the rollercoaster car travels around the track. When the car is going downhill, potential energy is being consumed as the car accelerates. conversely, when the car is travelling uphill potential energy is increased again as the car slows down. calculating the velocity from the energy balance is simple and this then allows the calculation of the centrifugal force generated as the car travels around the curves.

At a later stage in the design, a much more complicated calculation involving friction losses needs to be made. friction losses come from moving parts such as the rotation of the wheels and the bearings, and particularly from the air resistance of the car body and indeed the passengers themselves. They use up the initial potential energy given to the car – get the calculations on friction losses wrong and the car may come to a halt halfway round the track!

John Roberts is executive Director of operations at Jacobs, one of the UK’s largest engineering firms. he is one of the UK’s leading theme park engineers, with projects including the london eye and the ‘Big one’ at Blackpool Pleasure Beach. he is also a visiting professor of engineering Design at Manchester University and has appeared on television programmes including Scrapheap challenge, Battlefield Detective, richard & Judy, as well as in radio interviews with chris evans on DriveTime. like all engineers John makes use of mathematics in his day-to-day work, putting it to use for practical effect.

24 25 www.meetmaths.org.uk Meet the Mathematicians 2009

mtm

when sheared at a high rate with a knife, but mayonnaise in the jar retains the shape of the knife because it requires a minimum stress for it to flow at all. (So strictly speaking, mayonnaise is not a liquid but a weak solid.)

There are also fluids where the viscosity increases with strain-rate (shear-thickening fluids). A dramatic example is a solution made up of two parts cornstarch to one part water. This fluid flows easily if you stir slowly, but becomes rigid if you try to make it flow too fast.

flUID MeMorY A second way that a fluid can be non-Newtonian is if it exhibits memory or viscoelasticity. In such fluids, the stress doesn’t just depend on the current strain rate, but also on strain-rates applied at earlier times – the fluid ‘remembers’ some of what was done to it in the past. A good

example of this sort of material is the silicone polymer sold as ‘silly putty’. This can be made to bounce like a ball, but over longer time scales it can flow like a liquid. for small deformations we can describe this mathematically using an integral to add up the effect of all earlier strain rates:

σ =∫ t

s=−∞G(t − s)γ(s)ds.

here, the function G(t–s) tells us how much of the strain-rate from the earlier time s the fluid now remembers at time t. This function decays to zero over a period of time that characterises the memory time of the fluid. If you deform the fluid much more rapidly than this memory time it remembers almost the entire deformation, and behaves like an

an experiment in which we confine a layer of fluid of height h between two parallel plates of area A. If we now slide the top plate with velocity v relative to the bottom plate, we generate a flow in which different horizontal planes of fluid slide relative to each other. This shearing motion is resisted by interactions between the fluid molecules in adjoining layers, an idea which led Isaac Newton to postulate that the resistive force F, divided by the area A, should be proportional to the shear-rate in the fluid ( γ = ν/h ) so that

FA = µ ν

h = µγ.

The constant of proportionality µ is called the fluid viscosity and is a material property of the fluid. fluids that obey Newton’s postulate are called Newtonian fluids, and most ‘ordinary’ fluids such as air, water or golden syrup are Newtonian under normal conditions.

Any fluid that doesn’t obey Newton’s postulate is said to be non-Newtonian. Since this definition describes what a fluid isn’t rather than what it is, there are many different types of non-Newtonian behaviour and many ‘funny’ fluids exhibit a combination of these behaviours.

SheAr-ThINNINg flUIDSThe simplest type of non-Newtonian fluids are those where the force per unit area (which we call the stress σ) is still determined by the shear-rate, but in a nonlinear manner, so that instead of the viscosity µ being constant it depends upon the shear-rate, as follows:

σ = FA = µ(γ)γ.

fluids in which the viscosity decreases with increasing strain rate are called shear-thinning fluids. Shear-thinning is a desirable property in paint, for example, since you want the paint to be ‘thin’ enough to spread easily over the wall, but be sufficient ‘thick’ under the low stress of gravity that it doesn’t run before it dries. More extreme examples are soft-solids like mayonnaise. Mayonnaise spreads much more easily than honey on bread

v

h

Shearing flow between two parallel plates

Shear rate γ.

Stre

ss σ

Shear-thinning

Shear-thickening

Mayonnaise

Newtoniangraph showing stress versus shear rate for different non-Newtonian fluids

There are many different types of non-Newtonian behaviour in fluids

27 26

mtm

Meet the Mathematicians 2009www.meetmaths.org.uk

and strain-rate. Instead, we develop equations based upon an understanding of what the polymer molecules themselves are doing within the flow. for example, you can obtain many of these phenomena by thinking of each polymer molecule as acting like a little spring embedded within the fluid.

The study of these ‘funny fluids’ – which is known as rheology – is certainly a rich area of mathematics in combination with physics, chemistry and biology, but it is also a subject that has numerous applications in science and industry. After all, as we’ve seen, many of the fluids we come across in our daily lives are non-Newtonian – from mayonnaise to pizza cheese, and from plastics to paint.

Oliver Harlen read Mathematics at cambridge and then stayed on to do a PhD. he did postdoctoral research at cambridge and cornell Universities before taking up a lectureship at leeds in 1994, where he has been ever since.

Much of his research involves using mathematics to study the flow of fluids containing polymers. These strange fluids have elastic as well as viscous properties as shown, for example, by silly putty or molten cheese. This research has many applications in industry and he enjoys working on industrial problems with scientists and engineers.

Away from mathematics he enjoys family life with his wife and six year old son in the Yorkshire Dales.

elastic solid. however, deformations on timescales longer than the memory time are forgotten, so that in slowly varying flows it behaves like a viscous fluid.

NorMAl STreSSeSWhen you a stir a cup of tea, the level of the water surface falls near the spoon, due to the centrifugal force. however, when you stir egg whites the fluid rises up the stirrer. This ‘rod-climbing’ phenomenon is caused by protein molecules in egg whites being stretched out in the direction of flow. If you imagine the fluid as a suspension of elastic bands that are stretched in the direction of flow, the curvature of the flow lines causes an inward force which pushes the fluid up the stirrer.

Now think what might happen if you take such a suspension of elastic bands and start to stretch it. The more the fluid is stretched the greater it resists

further extension. This is called extension hardening. An everyday ‘fluid’ that exhibits this behaviour is pizza cheese – when you stretch it out, it forms long, elastic, filaments that can be cut with a pair of scissors. This resistance to stretching is also responsible for the ‘tubeless’ or ‘open’ siphon where a fluid can be drawn uphill without requiring a siphon tube.

MATheMATIcAl chAlleNgeSThe challenge for those of us working in this field is to be able to find systems of equations that describe the motion of these fluids. At the heart of the problem is the need to find an alternative to the Newtonian relationship between stress and strain-rate that can capture the different non-Newtonian phenomena of nonlinear viscosity, viscoelasticity and extension hardening. for complicated fluids, such as those containing polymers, there is no simple formula relating stress

coU

rTeS

Y o

f g

Are

Th M

cKIN

leY

fluid climbing up a rotating rod

coU

rTeS

Y o

f A

lexe

I lIK

hTM

AN

Mathematical simulation of a polymer melt

28 www.meetmaths.org.uk Meet the Mathematicians 2009 29

mtm

Philip Ingrey, PhD Student, University of NottinghamI have been studying at Nottingham for the past 5 years and during that time I have not only learnt a lot of maths, but have made some great friendships (I even met my wife here!) and have grown during my years at university.

I’m currently working towards a PhD studying how light interacts with materials. Specifically a new class of materials that could, one day, be used for invisibility. These materials, if configured correctly, bend light in strange ways and can be used to make lenses far better than anything we can manufacture currently,

high performance aerials, optical storage for computers and cloaks that can bend light round an object as if it wasn’t there. All this came from someone wondering what would happen if he put a minus sign in a 150 year old equation! Although seemingly different, all the examples above have their roots in the same mathematics. Mathematics allows us to alter and reconfigure these roots so that new and different applications can come out.

meet the mathematicians QUeSTIoN PANel

Rosemary Dyson, Postdoctoral Researcher, University of Nottingham My interest lies in understanding the mechanics of systems. I am currently a postdoctoral research fellow working at the centre for Plant Integrative Biology as part of a large team of experimentalists and theoreticians trying to understand how plant roots grow.

In contrast, my PhD work was about how you put shiny coatings onto pieces of paper, the same process by which, more interestingly, you put icing onto doughnuts. Along the way I have worked on questions

Chris Howls, Senior Lecturer, University of SouthamptonI should have studied history. I loved history. My dad was a historian. history was my best subject at school. Then I discovered applied mathematics. I was bowled over by the potential to apply mathematics and to explain what might happen, in the largest galaxies down to the smallest elementary particles. More so, that a geeky teenager could do that! I took a joint degree in mathematics and physics, for the best of both worlds. considering for about a second becoming a financial ‘master of the universe’ in the city, I instead took a PhD,

making the application of mathematics my career. I get frustrated by not having the time just to sit and think about a problem. I am

Liz Bentley, Head of Communications, Royal Meteorological SocietyI have always had an interest in maths and the weather, so what better than to combine them in a career in meteorology? have you ever wondered why it snows? or whether we can predict what kind of summer we’ll get? or how our lives will be affected by climate change? The answer to all these questions comes from observational data and mathematical modelling of the atmosphere.

A career in meteorology can lead you into many different areas such as research, modelling, field work, forecasting, consultancy, training and even a job on Tv if that takes your fancy! It’s great to work alongside the customer, who could be an rAf pilot, understanding their requirements and helping them to make decisions affected by the weather. There will be days when the forecast does not go to plan but you learn from these situations and move on.

as diverse as those brought by the oil industry and how to design pregnancy-testing kits. At first glance these areas seem completely unrelated, spanning a broad range of

disciplines, but they can all be described using the common language of mathematics. Being an applied mathematician allows me to study more or less anything; I get to learn about a whole host of things, and answer many different questions.

mtm

31 Meet the Mathematicians 200930 www.meetmaths.org.uk

Helen Byrne, Professor of Mathematics,University of NottinghamI wanted to study medicine. But I fainted at the sight of my own blood! Then I wanted to study physics. But I couldn’t get the experiments to work. I wanted to study law. But I didn’t want to learn lots of facts. And so, I studied maths. Afterwards, I didn’t want to get a job, so I took another maths degree. And then another. Still reluctant to get a ‘proper’ nine-to-five job, I stayed on to do some research. And then some more. As a graduate student, I became increasingly fascinated by (and drawn to) the idea of applying my mathematical knowledge to solve biomedical problems, and

modelling the growth and treatment of solid tumours, in particular.I enjoy my work – I am never bored or idle, I can choose what problems I want to

work on, and I am continually being intellectually challenged and stimulated. for example, in order to develop a realistic mathematical description of a new biological system I need to learn and understand enough of the biology to be able to identify the key processes involved. I may also need to master new mathematical techniques in order to solve the resulting mathematical equations. My job can also be extremely rewarding: it is great to see students grasp a new mathematical concept.

These days I sometimes I wish I had more time to do more mathematics, rather than teaching and supervising students and post-doctoral researchers. But on the plus side, I enjoy talking about maths and medicine with doctors and mathematicians from all over the world and the challenge of trying to use maths to help better understand and treat different diseases.

Stephen Coombes (Professor of Mathematics, University of Nottingham) will also be on the question panel. for a profile of Stephen and details of his talk: “The geometry and Pigmentation of Seashells”, see pages 7–11.

You are invited to put your own questions to the MTM panel in the question-and-answer session at 3.40pm.

SoMe ThoUghTS oN The UNreASoNABle

effecTIveNeSS of MATheMATIcS JoN KeATINgProfeSSor of MATheMATIcAl PhYSIcSUNIverSITY of BrISTol

Why is mathematics so successful in describing nature? In 1960, eugene

Wigner wrote a lovely article on “The Unreasonable effectiveness of Mathematics in the Natural Sciences”. In this, he points out how surprising it is that all of the fundamental laws of Nature can be expressed, extremely accurately (perfectly?), in terms of simple formulae that are mathematically natural and beautiful. often in these formulae quantities that were initially introduced within mathematics without any applications in mind turn out

to play a central role. And some quantities show up in rather unexpected places.

how surprising really is this? And does the apparently unreasonable effectiveness of mathematics tell us anything about the way we do science?

I hope to explain Wigner’s point of view, with examples, and to discuss some of my own thoughts on the subject, with excursions beyond the Natural Sciences. I also hope to discuss some of the limitations of current mathematical techniques and related challenges in Applied Mathematics.

Jon Keating did his first degree in Physics at oxford, then moved to Bristol to complete a PhD in Theoretical Physics, advised by Professor Sir Michael Berry. he took up a lectureship in Applied Mathematics at the University of Manchester in 1991. later he returned to Bristol, where for six years his position was sponsored by hewlett-Packard. his research is focused on the quantum properties of chaotic systems, random matrices, and connections with number theory.

inspired by the thrill of explaining something for the first time and seeing others make use of it. Maths entirely depends on people and the exchange of ideas between them. So, best of all, I just enjoy talking maths with (most of ) my colleagues, from all over the world.

PUBLIC LECTURE

www.meetmaths.org.uk32



BAMC logo v2:bridging the gap 25/11/08 10:27 Page 1

acknowledgeMentsMTM was created by Professor David Abrahams (Manchester) and Dr chris howls (Southampton). The MTM days are sponsored by the engineering and Physical Sciences research council (ePSrc). The ePSrc are one of the main civilian funders of mathematical research in the UK. We are indebted to the efficiency and professionalism of the local organising team at the University of Nottingham: Dr Stephen hibberd, Professor oliver Jensen, Dr Paul Matthews, Dr Joel feinstein, Dr ria Symonds (further Mathematics Network) and Sarah Shepherd (editor of iSquared Magazine). We also thank rich hewitt (Manchester) for the registration database and Wendy Sadler (Science Made Simple) for invaluable advice.

designed and edited by sarah shepherd

Documents

Meet the Mathematicians Book 2009