Scaling and Non-dimensionalization

Mathematical Modeling · Axiom Academy

LESSON Scaling and Non-dimensionalization Simplifying mathematical models by identifying the natural scales of a problem Scaling is the process of transforming variables by dividing them by characteristic scales - typical values that represent the natural size of each quantity in a problem. The result is a dimensionless (or non-dimensional ) variable that is typically of order one. Every physical problem has natural scales determined by its geometry, initial conditions, or physical parameters. Identifying these scales is the first step in non-dimensionalization. For a pendulum of length L in gravitational field g : Length scale: L (pendulum length) Time scale: T = sqrt(L/g) (natural period) Velocity scale: U = sqrt(gL) (characteristic speed) 3. Non-dimensionalizing Equations To non-dimensionalize an equation, substitute scaled variables and simplify. This process reveals the dimensionless parameters that govern the system's behavior. Identify all variables and their dimensions Choose characteristic scales for each dimension Define dimensionless variables (x* = x/L, t* = t/T, etc.) Substitute into the governing equations Simplify and identify dimensionless parameters 4. Example: Non-dimensionalizing the Pendulum Let's apply scaling to the pendulum equation. The governing equation is: Using the time scale T = sqrt(L/g) and defining t* = t/T:

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.