62 Continuum mechanics and transport
Field laws for stress, flow, heat, and diffusion
Heat leaves a wall, stress moves through a beam, solute spreads through a river, and pressure drives flow through a pipe. Different materials, different scales, different equations on the page. Underneath, the same move repeats: write what is conserved, write how the material responds, and let the field describe the rest.
This chapter is where upper-year engineering starts to feel less like a list of subjects and more like a single modelling language. Structural analysis, heat transfer, porous flow, and contaminant transport do not share the same units or applications. They do share the same mathematical architecture.
The architecture has two parts. First, a conservation law says something cannot appear or disappear except through flow across a boundary or production inside the region. Second, a constitutive law says how the material reacts: how heat flows down a temperature gradient, how solute diffuses down a concentration gradient, how stress relates to strain in a simple elastic solid.
62.1 The local balance law
Take a small control volume inside a body, pipe, soil column, or fluid region. Let \(u\) be the amount of something stored per unit volume. Depending on the problem, \(u\) might be thermal energy, mass concentration, momentum density, or another conserved quantity.
Let \(\mathbf{q}\) be the flux of that quantity. Flux is a vector quantity: its magnitude measures how much passes through a unit area per unit time, and its direction is the direction of net transport.
The generic local balance law is
\[\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{q} = s\]
where:
- \(\partial u/\partial t\) is the local rate of storage
- \(\nabla \cdot \mathbf{q}\) is the net outward flux
- \(s\) is a source term
This one line is the template behind a large fraction of continuum modelling. Change the meaning of \(u\), \(\mathbf{q}\), and \(s\), and you get a different physical system. The bookkeeping structure stays the same.
The divergence term matters because a continuum model is local. It does not ask “how much crossed the outer boundary of the whole pipe?” first. It asks what is happening in an arbitrarily small piece. Once the local law is right, the global law follows by integration.
62.2 Constitutive laws
A balance law alone is not enough. It tells you what must be accounted for, not how the material actually behaves. That extra relation is the constitutive law.
Two standard examples are:
Fourier’s law of heat conduction
\[\mathbf{q} = -k\nabla T\]
Heat flux points down the temperature gradient. The constant \(k\) is thermal conductivity. The minus sign means heat flows from hot to cold.
Fick’s law of diffusion
\[\mathbf{J} = -D\nabla c\]
Mass flux points down the concentration gradient. The constant \(D\) is the diffusivity.
The structure is the same in both. A gradient creates a driving force. The material constant tells you how strongly the medium responds.
In elementary elasticity, the constitutive law is often written as a linear stress-strain relation. In one dimension,
\[\sigma = E\varepsilon\]
where \(\sigma\) is stress (force per unit area, Pa), \(\varepsilon\) is strain (fractional change in length, dimensionless), and \(E\) is Young’s modulus (the material stiffness).
Again, the structure is familiar: one field quantity is related to another by a material response parameter.
62.3 Why signs and directions matter
Continuum laws are full of minus signs, outward normals, and directional derivatives. These are not decorative details. They are the physics.
If temperature increases to the right, then \(\nabla T\) points right. Fourier’s law says the heat flux points left. Miss the minus sign and you literally model heat flowing from cold to hot.
The same warning applies everywhere in this chapter. Continuum mathematics is geometry with units attached. Direction is part of the answer.
Conservation laws come from bookkeeping on a region. Constitutive laws come from empirical or theoretical descriptions of how a material responds. The PDE appears when you combine them.
For example, if thermal energy density is proportional to temperature and heat flux obeys Fourier’s law, then storage plus divergence of heat flux produces the heat equation. The PDE is not the starting point. It is the compressed result of conservation plus material behaviour.
62.4 The core method
A first pass through a continuum or transport problem usually goes like this:
- Identify the conserved quantity: mass, energy, momentum, or solute.
- Write the local balance law for storage, flux, and source.
- Choose the constitutive law that links flux or stress to the driving field.
- Substitute the constitutive law into the balance law.
- Inspect the resulting PDE or field equation and connect each term back to a physical mechanism.
- Add boundary and initial conditions only after the governing structure is clear.
That last point matters. Students often see the PDE first and the modelling logic second. In upper-year work, reversing that order helps. If you know where every term came from, the equation stops feeling arbitrary.
62.5 Worked example 1: heat conduction in one dimension
Let \(T(x,t)\) be temperature in a rod. Suppose there is no internal heat source. The local energy balance says:
\[\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{q} = 0\]
In one dimension, and with thermal energy density proportional to temperature, this becomes
\[\rho c_p \frac{\partial T}{\partial t} + \frac{\partial q}{\partial x} = 0\]
where \(\rho\) is mass density (kg m\(^{-3}\)) and \(c_p\) is specific heat capacity (J kg\(^{-1}\) K\(^{-1}\)) — the two material parameters relating energy storage to temperature change.
Use Fourier’s law in one dimension:
\[q = -k\frac{\partial T}{\partial x}\]
Substitute:
\[\rho c_p \frac{\partial T}{\partial t} - \frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) = 0\]
If \(k\) is constant, this simplifies to
\[\rho c_p \frac{\partial T}{\partial t} = k\frac{\partial^2 T}{\partial x^2}\]
or
\[\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}, \qquad \alpha = \frac{k}{\rho c_p}\]
This is the heat equation. The second spatial derivative tells you how much the temperature field differs from its local linear trend. Where the field is strongly curved, temperature changes quickly in time.
62.6 Worked example 2: contaminant diffusion in groundwater
Let \(c(x,t)\) be solute concentration in a one-dimensional porous medium. Ignore advection for the moment and assume no chemical reaction source.
The balance law is
\[\frac{\partial c}{\partial t} + \frac{\partial J}{\partial x} = 0\]
Use Fick’s law:
\[J = -D\frac{\partial c}{\partial x}\]
Substitute:
\[\frac{\partial c}{\partial t} - \frac{\partial}{\partial x}\left(D\frac{\partial c}{\partial x}\right)=0\]
For constant diffusivity:
\[\frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2}\]
Mathematically this is the same diffusion structure as the heat equation. Physically the field means something different. That is exactly why this chapter matters: it trains you to recognise when two different-looking systems share the same mathematical skeleton.
62.7 Worked example 3: linear elasticity in one dimension
Take a slender bar under axial load. In the simplest small-strain setting, strain is
\[\varepsilon = \frac{du}{dx}\]
where \(u(x)\) is displacement. Hooke’s law gives
\[\sigma = E\varepsilon = E\frac{du}{dx}\]
If body forces are negligible and the bar is in static equilibrium, force balance requires axial stress to satisfy
\[\frac{d\sigma}{dx} = 0\]
Substitute the constitutive relation:
\[\frac{d}{dx}\left(E\frac{du}{dx}\right) = 0\]
For constant \(E\):
\[E\frac{d^2u}{dx^2} = 0 \qquad \Rightarrow \qquad \frac{d^2u}{dx^2} = 0\]
This is much simpler than the heat and diffusion examples, but the logic is the same. Balance law plus constitutive law gives the governing equation.
62.8 Where this goes
The most direct continuation is Computational methods for engineering models. Once the governing equations are written, most upper-year work shifts from derivation to solution. Real geometries, mixed materials, and realistic boundary conditions usually push these problems toward numerical methods, meshes, and solver design.
This chapter also sets up a habit that carries far beyond classical engineering. Environmental transport, remote sensing inversion, and geophysics all use the same structure. The quantity and units change. The control-volume logic does not.
- heat flow through walls, fins, and machine components
- diffusion and mixing in process equipment
- contaminant transport in groundwater and rivers
- stress and deformation in structural members
- porous flow and subsurface energy transport
- continuum models in earth and environmental systems
62.9 Exercises
These are project-style exercises. State the physical meaning of each term as well as the final equation.
62.9.1 Exercise 1
Starting from the one-dimensional heat balance
\[\rho c_p \frac{\partial T}{\partial t} + \frac{\partial q}{\partial x} = 0\]
and Fourier’s law
\[q = -k\frac{\partial T}{\partial x}\]
derive the heat equation for constant \(k\).
62.9.2 Exercise 2
Write the diffusion equation for concentration \(c(x,t)\) from
\[\frac{\partial c}{\partial t} + \frac{\partial J}{\partial x} = 0, \qquad J = -D\frac{\partial c}{\partial x}\]
assuming \(D\) is constant. Then explain why the equation has the same form as the heat equation.
62.9.3 Exercise 3
For a one-dimensional elastic bar with
\[\varepsilon = \frac{du}{dx}, \qquad \sigma = E\varepsilon\]
and static equilibrium
\[\frac{d\sigma}{dx} = 0,\]
derive the governing equation for displacement when \(E\) is constant.
62.9.4 Exercise 4
Choose one field setting from your own area: a wall, pipe, aquifer, beam, soil column, battery, or river reach.
Prepare a one-page systems sketch naming:
- the conserved quantity
- the flux
- the constitutive law
- one plausible source term
- one important boundary condition
- one reason the resulting equation will probably need numerical solution