66  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.

Structural analysis, heat transfer, porous flow, and contaminant transport do not share the same units or applications. They share the same mathematical architecture.

The architecture has two parts. A conservation law says something cannot appear or disappear except through flow across a boundary or production inside the region. A constitutive law says how the material responds: how heat flows down a temperature gradient, how solute diffuses down a concentration gradient, how stress relates to strain in a simple elastic solid.

66.1 What this chapter helps you do

Symbols to keep handy

These are the bits of notation you'll see a lot. If a line of symbols feels like a fence, read it out loud once, then keep going.

  • + = s: rate of storage plus divergence of flux equals source

  • = -Dc: J equals minus D grad c — Fick’s law of diffusion

  • = -kT: q equals minus k grad T — Fourier’s law of heat conduction

Definitions to keep handy

These are the words we keep coming back to. If one feels slippery, come back here and steady it before you push on.

  • field: A quantity defined everywhere in space (and sometimes time), like temperature, pressure, or concentration.

  • conservation law: A bookkeeping rule: change in storage equals what flows in/out plus what is produced inside.

  • flux: How much of something moves through an area per unit time, with a direction.

  • constitutive law: A material rule that links causes to responses (like flux to gradient).

  • divergence: A measure of net outflow from a point (how much a flux field spreads out).

Here is the main move this chapter is making, in plain terms. You do not need to be fast. You just need to keep the thread.

  • Coming in: Materials, fluids, and heat are described locally, even when the structure you care about is global.

  • Leaving with: Conservation laws and constitutive laws turn geometry, material behaviour, and transport into a common mathematical language.

66.2 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

What this line is saying

Read it out loud as: “rate of storage + net outflow = source.”

If you remember only one thing, remember this: it is bookkeeping. It is not trying to be clever. It is trying to keep you honest about where stuff can go.

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 most continuum modelling. Change what u means (heat, mass, momentum), change what \mathbf{q} means (heat flux, mass flux, stress transport), change what s means (heater, reaction, injection), and you get a different physical system. The bookkeeping structure stays the same. This means the conservation architecture is reusable: once you understand the balance law, many different physical theories become variations on one pattern.

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.

Use the controls to see the same balance law operate across several physical interpretations.

66.3 Constitutive laws

A balance law tells you what must be accounted for. It does not say how the material responds. 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).

The structure is the same: one field quantity is related to another by a material response parameter.

This 1D relation extends to three dimensions as a tensor equation relating the full strain tensor to the full stress tensor through a stiffness matrix. The scalar form here captures the logic; discipline courses in mechanics of materials and structural analysis build out the three-dimensional version.

66.4 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 applies to every constitutive law in this chapter. Continuum mathematics is geometry with units attached. Direction is part of the answer.

NoteWhere the PDE comes from

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.

66.5 The core method

A first pass through a continuum or transport problem usually goes like this:

  1. Identify the conserved quantity: mass, energy, momentum, or solute.
  2. Write the local balance law for storage, flux, and source.
  3. Choose the constitutive law that links flux or stress to the driving field.
  4. Substitute the constitutive law into the balance law.
  5. Inspect the resulting PDE or field equation and connect each term back to a physical mechanism.
  6. 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.

66.6 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, thermal energy per unit volume is u = \rho c_p T, where \rho is mass density (kg m^{-3}) and c_p is specific heat capacity (J kg^{-1} K^{-1}). Substituting into the generic balance law gives

\rho c_p \frac{\partial T}{\partial t} + \frac{\partial q}{\partial x} = 0

Constitutive law, in words

The balance law is bookkeeping, but it is not complete by itself. You also need a material rule that says how the flux responds to a driving difference.

For heat and diffusion, that rule is: stuff flows from high to low.

That is why Fourier’s and Fick’s laws have a minus sign: the flux points down the gradient.

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.

66.7 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}

This has the same structure as the heat equation. Physically the fields mean something different — but the mathematical skeleton is shared. Recognising that is most of what this chapter is for.

In real groundwater and river-transport problems, advection — the bulk movement of solute carried by flowing water — usually dominates or competes with diffusion. In the generic framework, advection enters through the flux term: the total flux is J = vc - D\,\partial c/\partial x, where vc is the solute carried by the bulk flow and -D\,\partial c/\partial x is the diffusive part from Fick’s law. Substituting into the balance law and rearranging gives

\frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} = D\frac{\partial^2 c}{\partial x^2}

The pure-diffusion case here applies when the pore velocity is low (dense clay, for example) or when setting up an initial model to isolate the diffusive contribution before adding flow. The full advection-diffusion equation is taken up in environmental and geotechnical engineering courses.

66.8 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{dw}{dx}

where w(x) is axial displacement (using w to avoid confusion with the generic stored quantity u used earlier). Hooke’s law gives

\sigma = E\varepsilon = E\frac{dw}{dx}

In the generic framework, stress plays the role of flux: it is the quantity that transmits the internal load across a cross-section. The “storage” term vanishes in static equilibrium, so the balance law reduces to a divergence-only statement with a source. The general 1D force balance in a body with distributed body force b (force per unit volume, e.g. self-weight, corresponding to the source term s in the generic law) is

\frac{d\sigma}{dx} + b = 0

If body forces are negligible (b = 0), this reduces to

\frac{d\sigma}{dx} = 0

Substitute the constitutive relation:

\frac{d}{dx}\left(E\frac{dw}{dx}\right) = 0

For constant E:

E\frac{d^2w}{dx^2} = 0 \qquad \Rightarrow \qquad \frac{d^2w}{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.

66.9 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.

Environmental transport, remote sensing inversion, and geophysics all use the same structure. The quantity and units change. The control-volume logic does not.

TipApplications
  • 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

66.10 Exercises

Each exercise asks you to derive a governing equation. State the physical meaning of each term as well as the final equation.

66.10.1 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.

66.10.2 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.

66.10.3 Exercise 3

For a one-dimensional elastic bar with axial displacement w(x),

\varepsilon = \frac{dw}{dx}, \qquad \sigma = E\varepsilon

and static equilibrium

\frac{d\sigma}{dx} = 0,

derive the governing equation for displacement when E is constant.

66.10.4 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:

  1. the conserved quantity
  2. the flux
  3. the constitutive law
  4. one plausible source term
  5. one important boundary condition
  6. one reason the resulting equation will probably need numerical solution