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?!?