Confessions of an applied mathematician

Chris Budd

What is applied maths?

• Using maths to understand an aspect of the real world … usually through a simplified model and to predict or create new things

It is crazy that this works at all

• Learning NEW mathematics in the process

• Using this new mathematics to change the world

Some ways that maths has changed the modern world

Maxwell: Electromagnetism … radio, TV, radar, mobile phones

Linear algebra, graph theory, SVD ...Google

Error correcting codes

We live in interesting times with applied mathematics in a process of great transition!

20th century .. Great drivers of applied maths are physics, engineering and more recently biology

Expertise in ….

• Fluids

• Solids

• Reaction-diffusion problems

• Dynamical systems

• Signal processing

Usually deterministic Continuum problems, modelled by Differential Equations

Solutions methods

• Simple analytical methods eg. Separation of variables

• Approximate/asymptotic approaches

• Phase plane analysis

• Numerical methods eg. finite element methods

• PDE techniques eg. Calculus of variations

• Transforms: Fourier, Laplace, Radon

What are the drivers of 21st century applied mathematics?

• Information/Bio-informatics/Genetics?

• Commerce/retail sector?

• Complexity?

What new techniques do we need to consider?

• Discrete maths?

• Stochastic methods?

• Very large scale computations?

• Complex systems?

• Optimisation (discrete and continuous)?

Example 1: What happens when we eat?

Stomach

Small intestine:

7m x 1.25cm

Intestinal wall:

Villi and Microvilli

Process:

• Food enters stomach and leaves as Chyme

• Nutrients are absorbed through the intestinal wall

• Chyme passes through small intestine in 4.5hrs

Stomach

Intestinal wall

Colon,

illeocecal sphincter

Peristaltic wave

Mixing process

Objectives

• Model the process of food moving through the intestine

• Model the process of nutrient mixing and absorption

Basic flow model: axisymmetric Stokes flow pumped by a peristaltic wave and a pressure gradient

• Chyne moves at slow velocity: u(x,r,t)

• Nutrient concentration: c(x,r,t)

• Peristaltic wave: r = f(x,t)

x

r=f(x,t)

r

Wavelength:8cm

h = 1.25cm

• Navier Stokes

• Slow viscous Axisymmetric flow

• Velocity & Stokes Streamfunction

upuut

u 2).( 0. u

p ˆ

e

ru

ere )/( 0)( e

,ˆ eu

rr

L rrrxx

11

01 L

)/)(2cos(),( txhtxfr

),,( trx ),( rtxz FIXED FRAME WAVE FRAME

No slip on boundary

Change from

Impose periodicity

z

z

h

rr

0,0 ˆˆˆ rrr

ˆˆˆ1

ˆˆ ˆˆˆˆˆ2 rrrzz r

rrw ˆˆˆˆ ˆ

0ˆˆ1ˆˆ

ˆˆˆˆˆ2 rrrzz r

Axisymmetry

)ˆ2cos(1)ˆ( zzf

h

h

• Amplitude:

• Wave Number:

Small parameters

Flow depends on:

w ˆˆ

,6.0h

16.08

25.1

cm

cmh

Flow rate Proportional to pressure drop

Amplitude

Wave number

gives Poiseuille flow0

2Develop asymptotic series in powers of

• Reflux Pressure Rise Particles undergo net retrograde

motion

• Trapping

Regions of Pressure Rise & Pressure DropStreamlines encompass a bolus of fluid particles

Trapped Fluid recirculates

Distinct flow types

A

B

C D E

FG

0ˆ p

0ˆ p4/)1( 2

Flow regions

w

Poiseuille

A: Copumping, Detached TrappingA: Copumping, Detached TrappingB: Copumping, Centreline TrappingB: Copumping, Centreline TrappingC: Copumping, No TrappingC: Copumping, No Trapping

Illeocecal sphincter openIlleocecal sphincter open

D: Pumping, No TrappingD: Pumping, No TrappingE: Pumping, Centreline TrappingE: Pumping, Centreline Trapping

Illeocecal sphincter closedIlleocecal sphincter closed

4/)1( 2

Case A: Copumping, Detached Trapping

RecirculationParticle paths

x

Case C: Copumping, No Trapping

Poiseuille FlowParticle paths

x

Case E: Pumping, Centreline Trapping

Recirculation

Reflux

Particle paths

Calculate the concentration c(x,r,t)

oncDcuct2).(

oncKcnD a).(

1. Substitute asymptotic solution for u into

2. Solve for c(x,r,t) numerically using an upwind scheme on a domain transformed into a computational rectangle.

3. Calculate rate of absorption

Poiseuille flow Peristaltic flow

Type C flow: no trapping

Poiseuille flow Peristaltic flow

Type E flow: trapping and reflux

x t

Nutrient absorbedLocation of absorbed mass at final time

Peristaltic flow

Conclusions

• Peristalsis helps both pumping and mixing

• Significantly greater absorption with Peristaltic flow than with Poiseuille flow

Example 2: Mathematics can look inside you

Modern CAT scanner

CAT scanners work by casting many shadows with X-rays and using maths to assemble these into a picture

X-Ray

Object

Density f(x,y)

ρ : Distance from the object centre

θ : Angle of the X-Ray

Measure attenuation of X-Ray R(ρ, θ)

X-ray Source

Detector

Object

Attenuation R(ρ, θ)

Edge Edge

Edge Edge

If we can measure R(ρ, θ) accurately we can calculate

The density f(x,y) of the object at any point

Also used to

X-ray mummies

Radon 1917

Example 3: Finding land mines

Land mines are hidden in foliage and triggered by trip wires

Land mines are well hidden .. we can use maths to find them

Find the trip wires in this picture

Digital picture of foliage is taken by camera on a long pole

Effect: Image intensity f

••

•

•

Cause: Trip wires .. These are like X-Rays

Radon transform

x

y

f(x,y)

R(ρ,θ)

Points of high intensity in R correspond to trip wires

θ

ρ

Isolate points and transform back to find the wires

Mathematics finds the land mines!

Who says that maths isn’t relevant to real life?!?