- •Preface
- •1.1 Introduction
- •1.2 Models and modelling
- •1.3 The learning process for mathematical modelling
- •Summary
- •Aims and objectives
- •2.1 Introduction
- •2.2 Examples
- •2.3 Further examples
- •Appendix 1
- •Appendix 2
- •Aims and objectives
- •3.1 Introduction
- •3.2 Definitions and terminology
- •3.3 Methodology and modelling flow chart
- •3.4 The methodology in practice
- •Background to the problem
- •Summary
- •Aims and objectives
- •4.1 Introduction
- •4.2 Listing factors
- •4.3 Making assumptions
- •4.4 Types of behaviour
- •4.5 Translating into mathematics
- •4.6 Choosing mathematical functions
- •Case 1
- •Case 2
- •Case 3
- •4.7 Relative sizes of terms
- •4.8 Units
- •4.9 Dimensions
- •4.10 Dimensional analysis
- •Summary
- •Aims and objectives
- •5.1 Introduction
- •5.2 First-order linear difference equations
- •5.3 Tending to a limit
- •5.4 More than one variable
- •5.5 Matrix models
- •5.6 Non-linear models and chaos
- •5.7 Using spreadsheets
- •Aims and objectives
- •6.1 Introduction
- •6.2 First order, one variable
- •6.3 Second order, one variable
- •6.4 Second order, two variables (uncoupled)
- •6.5 Simultaneous coupled differential equations
- •Summary
- •Aims and objectives
- •7.1 Introduction
- •7.2 Modelling random variables
- •7.3 Generating random numbers
- •7.4 Simulations
- •7.5 Using simulation models
- •7.6 Packages and simulation languages
- •Summary
- •Aims and objectives
- •8.1 Introduction
- •8.2 Data collection
- •8.3 Empirical models
- •8.4 Estimating parameters
- •8.5 Errors and accuracy
- •8.6 Testing models
- •Summary
- •Aims and objectives
- •9.1 Introduction
- •9.2 Driving speeds
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Rewritten problem statement
- •Obtain the mathematical solution
- •9.3 Tax on cigarette smoking
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •9.4 Shopping trips
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •Interpret the mathematical solution
- •Using the model
- •9.5 Disk pressing
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •Interpret the mathematical solution
- •Further thoughts
- •9.6 Gutter
- •Context and problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •9.7 Turf
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •Interpret the solution
- •9.8 Parachute jump
- •Context and problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •9.9 On the buses
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •9.10 Further battles
- •Discrete deterministic model
- •Discrete stochastic model
- •Comparing the models
- •9.11 Snooker
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •Interpret the mathematical solution
- •9.12 Further models
- •Mileage
- •Heads or tails
- •Picture hanging
- •Motorway
- •Vehicle-merging delay at a junction
- •Family names
- •Estimating animal populations
- •Simulation of population growth
- •Needle crystals
- •Car parking
- •Overhead projector
- •Sheep farming
- •Aims and objectives
- •10.1 Introduction
- •10.2 Report writing
- •Preliminary
- •Main body
- •Appendices
- •Summary
- •General remarks
- •10.3 A specimen report
- •Contents
- •1 PRELIMINARY SECTIONS
- •1.1 Summary and conclusions
- •1.2 Glossary
- •2 MAIN SECTIONS
- •2.1 Problem statement
- •2.2 Assumptions
- •2.3 Individual testing
- •2.4 Single-stage procedure
- •2.5 Two-stage procedure
- •2.6 Results
- •2.7 Regular section procedures
- •2.8 Conclusions
- •3 APPENDICES
- •3.1 Possible extensions
- •3.2 Mathematical analysis
- •10.4 Presentation
- •Preparation
- •Giving the presentation
- •Bibliography
- •Solutions to Exercises
- •Chapter 2
- •Example 2.2 – Double wiper overlap problem
- •Chapter 4
- •Chapter 5
- •Chapter 6
- •Chapter 8
- •Index
−2 and the (abbreviated) SI units for force are kg m s −2 (defined to be 1 N). In ordinary conversation, it is very common to use ‘mass’ and ‘weight’ as equivalent terms but do not confuse the two. The weight of an object is the force exerted on it by the Earth's gravitational field. It is an experimental fact that, near the Earth's surface, all bodies in free fall accelerate at the constant rate of g 9.806 65m s −2 or 32.174 ft s −2 . Consequently the weight of a mass of m kg is mg N. It may be helpful to think of a force of 1 N as the weight of a fairly large apple (mass of about 0.1 kg).
Any sensible equation must be dimensionally consistent, i.e. [left-hand side] = [right-hand side]. It is a good idea to carry out this check on all the equations appearing in a model. Modelling errors can often reveal themselves in this way. Note that any constants appearing in our equations can be dimensionless (i.e. pure numbers) or can have dimensions. For example, suppose that we are modelling the force on a moving object due to air resistance. If we assume the magnitude of the force F is proportional to the square of the speed v, this leads to a model of the form F = kv 2 . Checking the dimensions of this equation, we have [ F ] = [ kv 2 ], i.e.
For consistency, we require [ k ] = ML −1 and k will be measured in kg m −1 .
Note that if expressions involving exp( at ) or sin( at ) appear in our model, where t stands for time, the parameter a must have dimensions T −1 so that at is a dimensionless number.
If an equation involves a derivative, the dimensions of the derivative are given by the ratio of the dimensions.
For example, if p is the pressure in a fluid at any point, the pressure gradient in the direction of z is d p /d z and
Similarly, for partial derivatives,
and
4.10 Dimensional analysis
This is a method of using the fact that the dimensions of each term in an equation must be the same to suggest a relationship between the physical quantities involved. This will not give the exact form of a function but is still useful. Suppose that we are trying to develop a model which will predict the period of a swinging pendulum. A list of the factors involved might include the length l, the mass m, the acceleration g due to gravity, the amplitude θ, air resistance R and the rotation of the Earth. If we restrict this list to the first four factors, we can assume that the period t is given by some function of the four factors. Assume that t = kl a m b g c θ d ], where a, b, c, d and k are real numbers. Considering dimensions, we have
and so
( k and θ are dimensionless). Equating powers of M, L and T on both sides, we must have
This gives
In this expression, d could take any value and we could sum terms of this form to arrive at t = f (θ) l 1/2 g −1/2 . We have to find f (θ) some other way. For small θ, of course, f (θ) is approximately constant with value 2π.
If we also include the force R due to air resistance, our model becomes
and so
We require
and
We now have three equations with four unknowns. We could write any three of them in terms of the fourth. Suppose that we write everything in terms of b. We have
So
which generalises to
for some function f 1 .
Returning to the three equations with four unknowns, if we express all the parameters in terms of c, we have
giving
or
for some function f 2 .
In both of these expressions for t, we have a time factor multiplied by a function of the two dimensionless quantities, θ and mg/ R. The time factors are ( l/g ) 1/2 and ( ml/R ) 1/2 . The first of these gives a time scale associated with the period of oscillation and the second gives a time scale associated with the damping due to air resistance. The quantity mg/R is a dimensionless combination of physical quantities and is an example of a dimensionless group. This is a useful concept in modelling physical systems. In fluid mechanics a very useful dimensionless group is the Reynolds number ( Re ) = ρ ul /μ, where l is a characteristic length in the problem (e.g. the diameter of a pipe), ρ is the density of the fluid, μ is its viscosity and u is the fluid speed.
Example 4.9
The pressure p at a depth h below the surface of a fluid of density ρ is given by p = ρ gh, where g is the acceleration due to gravity. We can check that this equation is dimensionally correct as follows:
and
So as required.
Example 4.10
The fluid viscosity μ is defined by the relation force per unit area = μ × velocity gradient in the direction perpendicular to the area. If we take the dimensions of both sides of this relation, we have
So
and μ is measured in kilograms per metre per second (kg m −1 s −1 ). The dimensions of ( Re ) = ρ ul / μ are
i.e. ( Re ) is a dimensionless group as previously stated. Example 4.11
When a particle falls under gravity in a viscous fluid, the drag that it experiences counteracts the acceleration due to gravity and after a while the particle's speed stops increasing. When this happens, we say that it has reached its ‘terminal speed’. For a finite spherical particle, it is reasonable to suppose that the terminal speed v depends on the particle's diameter D, the viscosity μ of the fluid and the acceleration g due to gravity. Suppose also that v is directly proportional to the difference between the density ρ 1 of the particle and the density ρ 2 of the fluid. We assume that v = kD a (ρ 1 − ρ 2 )μ b g c .
Taking dimensions of both sides, we have
If k is a dimensionless number, then
and
So b = − 1, c = 1, a = 2 and our model is
Theoretical analysis using the laws of hydrodynamics leads to v = (ρ 1 − ρ 2 ) D 2 g /18μ, an expression known as ‘Stokes’ drag’. This clearly fits with our expression obtained from simple dimensional considerations.
Exercises
4.23 Newton's gravitational law states that F = Gm 1 m 2 / r 2 , where F is the magnitude of the
gravitational force of attraction between two masses m 1 and m 2 separated by a distance r. What are the units of the gravitational constant G ?
In a compressible fluid, the bulk modulus K is defined as
4.24
What are the dimensions of K ?
Which of the following equations contain an error and which are dimensionally correct?
1.
2.
4.25
3.
4.
p is a pressure, ρ is a density, u and v are velocities, F is a force, m is a mass, A is an area, and z and l are lengths.
The following models are meant to predict the volume flow rate Q of fluid through a small hole in the side of a large tank filled with fluid to a height H above the hole. Which one is dimensionally correct?
4.261.
2.
3.
A is the cross-sectional area of the hole and g is the acceleration due to gravity.
4.27 Check that the equation