Dimensional analysis and nondimensionalisation

Learning outcomes

By the end of this week you should be able to:

• identify characteristic scales in a physical problem;

• nondimensionalise a first-order ODE by introducing dimensionless variables;

• interpret the resulting dimensionless parameters;

• explain why nondimensional solutions are universal (solution collapse).

Characteristic scales

When we write down a model, the variables and parameters carry units. For instance, in Newton’s cooling law \[\ddt{T} = -k(T - T_{\infty}),\] the temperature \(T\) is in kelvin, \(t\) in seconds, \(k\) in \(\mathrm{s}^{-1}\), and \(T_\infty\) is a reference temperature.

A characteristic scale is a representative value of a quantity that captures the natural “size” of that quantity in the problem. Common choices:

• For temperature excess: \(T_0 - T_\infty\), the initial excess above ambient.

• For time: \(1/k\), the natural relaxation time.

• For length: the size of the domain or a physical feature.

The idea is to make variables dimensionless by dividing by their characteristic scale. This removes units and reduces the number of parameters.

Nondimensionalisation: the procedure

1. Identify all dimensional variables and parameters.

2. Choose a characteristic scale for each variable.

3. Define dimensionless variables by dividing by those scales.

4. Rewrite the equation in terms of the dimensionless variables.

5. Read off the dimensionless parameters that appear.

Worked example: Newton’s cooling

Consider again \[\ddt{T} = -k(T - T_\infty), \qquad T(0) = T_0.\]

Step 1. Dimensional variables: \(T\) (temperature, K), \(t\) (time, s). Parameters: \(k\ [\mathrm{s}^{-1}]\), \(T_\infty\ [\mathrm{K}]\), \(T_0\ [\mathrm{K}]\).

Step 2. Characteristic scales: temperature scale \(\Delta T = T_0 - T_\infty\), time scale \(\tau = 1/k\).

Step 3. Define \[\theta = \frac{T - T_\infty}{T_0 - T_\infty}, \qquad s = k\,t.\] Both \(\theta\) and \(s\) are dimensionless.

Step 4. Rewrite the ODE. Note \(T = T_\infty + (T_0 - T_\infty)\theta\) and \(t = s/k\), so \[\frac{\dd T}{\dd t} = (T_0 - T_\infty)\,\frac{\dd\theta}{\dd s}\cdot k.\] Substituting: \[(T_0 - T_\infty)\,k\,\frac{\dd\theta}{\dd s} = -k\,(T_0 - T_\infty)\,\theta.\]

Step 5. Divide through: \[\boxed{\frac{\dd\theta}{\dd s} = -\theta, \qquad \theta(0) = 1.}\]

This is a universal equation — it contains no parameters at all. Its solution is simply \[\theta(s) = e^{-s}.\]

Solution collapse

To visualise collapse: suppose you solve the dimensional problem for three different values of \(k\) (say 0.1, 0.5, and 2.0 s\(^{-1}\)) and plot \(T(t)\) vs \(t\). You get three distinct curves.

Now plot \(\theta(s) = (T - T_\infty)/(T_0 - T_\infty)\) vs \(s = kt\) for all three. All three curves coincide exactly on \(e^{-s}\). This is the power of nondimensionalisation.

Dimensionless groups and physical meaning

When nondimensionalising a more complicated equation, not all parameters always cancel. What remains are dimensionless groups — combinations of the original parameters that have no units.

These groups carry physical meaning. For instance:

• The Reynolds number \(Re = \rho U L / \mu\) in fluid mechanics compares inertial to viscous forces.

• The Damköhler number \(Da\) in chemistry compares reaction to diffusion timescales.

A single dimensionless group governs the qualitative behaviour of the solution. Next week we will see a systematic method — the Buckingham \(\Pi\) theorem — for finding these groups from the dimensional parameters alone.