Dimensional analysis — choosing repeating variables correctly When applying the Buckingham Π method, how should the set of repeating variables be chosen?

Introduction / Context:
The Buckingham Π theorem compresses a physical relation among variables into fewer dimensionless groups. The success of the method hinges on selecting an appropriate set of repeating (or core) variables that define the dimensional foundation for constructing the Π terms.\n

Given Data / Assumptions:

• There is one dependent (response) variable and several independent variables.
• Basic dimensions (e.g., M, L, T, Θ) present in the problem are known.
• We must form dimensionless Π terms that span the dimensional space.

Concept / Approach:
Good repeating variables (1) do not include the dependent variable, (2) are dimensionally independent (no one can be formed by combination of others), and (3) collectively cover all basic dimensions in the problem. Typical choices in fluid problems include a characteristic length, velocity (or density/viscosity pair), and possibly gravity or surface tension depending on the physics of interest.

Step-by-Step Solution:
List all variables and identify basic dimensions involved.Select k repeating variables equal to the number of basic dimensions.Verify independence and coverage of all basic dimensions.Exclude the dependent variable from the repeating set.

Verification / Alternative check:
A dimension matrix with full rank (equal to the number of basic dimensions) for the repeating set confirms suitability; otherwise Π terms will be degenerate.

Why Other Options Are Wrong:
Including the dependent variable corrupts the Π construction.
Choosing derivable (dependent) variables causes rank deficiency.
Excluding a basic dimension prevents building fully dimensionless groups.
Picking many repeating variables increases complexity without benefit and violates the k = number of basic dimensions rule.

Common Pitfalls:

• Picking two velocities or two lengths as repeaters, which are not independent.
• Forgetting to include density and viscosity together when both inertial and viscous effects are present.

Final Answer:
Ensure the repeating variables collectively contain all basic dimensions present