Scaling and nondimensionalization

Nondimensionalization reveals dominant physics

Scaling turns quantities into ratios so one can see which terms matter before solving. The small parameter in an equation predicts a controlled approximation rather than a guess.

Choose reference scales

Let x = Lx*, t = Tt*, and u = Uu*. In the advection-diffusion equation u_t + c u_x = αu_xx, choose T = L/U. Dividing by U/T yields u*_t + u*_x = (α/(UL))u*_xx. The coefficient α/(UL) = 1/Pe compares diffusion with advection; its reciprocal is a Péclet number.

For U = 0.5 m/s, L = 0.02 m, and α = 1×10⁻⁷ m²/s, Pe = UL/α = 100,000. Advection dominates over that length scale, so pure diffusion is a poor first model.

Scaling cannot remove a mechanism without checking the ratio

A small coefficient may multiply a large derivative near a boundary layer, restoring its importance. Nondimensional terms predict where an approximation can fail as well as where it works.

Work it: Nondimensionalize y'' + (1/L²)y = 0 using x = Lx*. Then compare the resulting equation for two physical lengths and explain what remains invariant.