$$ \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\normalvec}{\boldsymbol{n}} \newcommand{\x}{\boldsymbol{x}} \renewcommand{\u}{\boldsymbol{u}} \renewcommand{\v}{\boldsymbol{v}} \newcommand{\w}{\boldsymbol{w}} \newcommand{\rpos}{\boldsymbol{r}} \newcommand{\f}{\boldsymbol{f}} \newcommand{\F}{\boldsymbol{F}} \newcommand{\stress}{\boldsymbol{\sigma}} \newcommand{\I}{\boldsymbol{I}} \newcommand{\U}{\boldsymbol{U}} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\ii}{\boldsymbol{i}} \newcommand{\jj}{\boldsymbol{j}} \newcommand{\kk}{\boldsymbol{k}} \newcommand{\ir}{\boldsymbol{i}_r} \newcommand{\ith}{\boldsymbol{i}_{\theta}} $$

 

 

 

Table of contents

Preface
Dimensions and units
      Fundamental concepts
            Base units and dimensions
            Dimensions of common physical quantities
            The Buckingham Pi theorem
            Absolute errors, relative errors, and units
            Units and computers
            Unit systems
            Example on challenges arising from unit systems
            PhysicalQuantity: a tool for computing with units
      Parampool: user interfaces with automatic unit conversion
            Pool of parameters
            Fetching pool data for computing
            Reading command-line options
            Setting default values in a file
            Specifying multiple values of input parameters
            Generating a graphical user interface
Ordinary differential equation models
      Exponential decay problems
            Fundamental ideas of scaling
            The basic model problem
            The technical steps of the scaling procedure
            Making software for utilizing the scaled model
            Scaling a generalized problem
            Variable coefficients
            Scaling a cooling problem with constant temperature in the surroundings
            Scaling a cooling problem with time-dependent surroundings
            Scaling a nonlinear ODE
            SIR ODE system for spreading of diseases
            SIRV model with finite immunity
            Michaelis-Menten kinetics for biochemical reactions
      Vibration problems
            Undamped vibrations without forcing
            Undamped vibrations with constant forcing
            Undamped vibrations with time-dependent forcing
            Damped vibrations with forcing
            Oscillating electric circuits
      Exercises
            Exercise 2.1: Perform unit conversion
            Problem 2.2: Scale a simple formula
            Exercise 2.3: Perform alternative scalings
            Problem 2.4: A nonlinear ODE for vertical motion with air resistance
            Exercise 2.5: Solve a decay ODE with discontinuous coefficient
            Exercise 2.6: Implement a scaled model for cooling
            Problem 2.7: Decay ODE with discontinuous coefficients
            Exercise 2.8: Alternative scalings of a cooling model
            Exercise 2.9: Projectile motion
            Problem 2.10: A predator-prey model
            Problem 2.11: A model for competing species
            Problem 2.12: Find the period of sinusoidal signals
            Problem 2.13: Oscillating mass with sliding friction
            Problem 2.14: Pendulum equations
            Exercise 2.15: ODEs for a binary star
            Problem 2.16: Duffing's equation
            Problem 2.17: Vertical motion in a varying gravity field
            Problem 2.18: A simplified Schroedinger equation
Basic partial differential equation models
      The wave equation
            Homogeneous Dirichlet conditions in 1D
            Implementation of the scaled wave equation
            Time-dependent Dirichlet condition
            Velocity initial condition
            Variable wave velocity and forcing
            Damped wave equation
            A three-dimensional wave equation problem
      The diffusion equation
            Homogeneous 1D diffusion equation
            Generalized diffusion PDE
            Jump boundary condition
            Oscillating Dirichlet condition
      Reaction-diffusion equations
            Fisher's equation
            Nonlinear reaction-diffusion PDE
      The convection-diffusion equation
            Convection-diffusion without a force term
            Stationary PDE
            Convection-diffusion with a source term
      Exercises
            Problem 3.1: Stationary Couette flow
            Exercise 3.2: Couette-Poiseuille flow
            Exercise 3.3: Pulsatile pipeflow
            Exercise 3.4: The linear cable equation
            Exercise 3.5: Heat conduction with discontinuous initial condition
            Problem 3.6: Scaling a welding problem
Advanced partial differential equation models
      The equations of linear elasticity
            The general time-dependent elasticity problem
            Dimensionless stress tensor
            When can the acceleration term be neglected?
            The stationary elasticity problem
            Quasi-static thermo-elasticity
      The Navier-Stokes equations
            The momentum equation without body forces
            Scaling of time for low Reynolds numbers
            Shear stress as pressure scale
            Gravity force and the Froude number
            Oscillating boundary conditions and the Strouhal number
            Cavitation and the Euler number
            Free surface conditions and the Weber number
      Thermal convection
            Forced convection
            Free convection
            The Grashof, Prandtl, and Eckert numbers
            Heat transfer at boundaries and the Nusselt and Biot numbers
      Compressible gas dynamics
            The Euler equations of gas dynamics
            General isentropic flow
            The acoustic approximation for sound waves
      Water surface waves driven by gravity
            The mathematical model
            Scaling
            Waves in deep water
            Long waves in shallow water
      Two-phase porous media flow
      The bidomain model in electrophysiology
            The mathematical model
            Scaling
            An alternative \( I_{\rm{ion}} \)
      Exercises
            Exercise 4.1: Comparison of vibration models for elastic structures
            Exercise 4.2: A model for quasi-static poro-elasticity
            Problem 4.3: Starting Couette flow
            Problem 4.4: Channel flow
      References