2.2.26 · D5Fluid Mechanics
Question bank — Dimensional analysis — Buckingham π theorem
True or false — justify
The theorem predicts the exact formula of the physical law.
False. It predicts only the number of independent dimensionless knobs and the form ; the actual function and constants like come from experiment or full theory.
If two problems have the same and , they must have the same physics.
False. Same means the same count of groups, but which variables appear and how the groups combine differ completely — pendulum and drag can share a count yet share no physics.
Adding an irrelevant variable to your list always increases by one.
True in the count ( rises by one, and if the variable brings no new dimension stays fixed, so rises), but it produces a spurious that the real law simply won't depend on — garbage in, garbage out.
The number of fundamental dimensions is always because the universe has .
False. is the number of dimensions actually present and independent in this problem; a purely geometric/kinematic problem may have (), and a thermal one (add temperature ).
equals the rank of the dimension matrix , not just the number of distinct base symbols you wrote down.
True. If two "dimensions" always appear in a fixed combination, they don't add an independent row, so can be less than the number of symbols — always use the rank.
Every set of variables can serve as repeating variables.
False. They must (a) together contain all dimensions and (b) be dimensionally independent (their sub-matrix must be invertible), otherwise they secretly form a and your groups collapse.
The dimensionless groups are unique.
False. Any product of powers of valid groups (e.g. , ) is another valid ; only the number and the span are fixed, not the particular basis you write.
A quantity that is already dimensionless (like an angle or a pure number) counts as a variable that raises .
Loosely true — it can enter the list, but it is its own group and adds nothing to , so it simply appears directly among the .
Spot the error
"For sphere drag I picked repeating variables , , and time ." — what's wrong?
is dimensionally dependent on and , so the repeaters' sub-matrix loses rank; they fail condition (b) and cannot span three independent dimensions.
"The pendulum period must depend on mass because heavy things feel different." — where's the flaw?
Mass is the only variable carrying dimension , so no other variable can cancel it into a dimensionless ratio; therefore cannot appear in the single group at all.
"I got , so drag ." — what's the mistake?
The theorem gives with an unknown function ; you cannot assume (or any specific form) from dimensions alone.
"My equation has variables and dimensions, so — dimensional analysis fails." — is this an error?
Not an error; legitimately means there are no free dimensionless groups, so dimensions fix the relation up to a pure constant — a very strong (not failed) result.
"Density and specific weight are both in my list, giving two mass-bearing variables." — trap?
Yes if is also listed: is a function of the others, so it is redundant and can silently reduce the rank you expected — drop derived duplicates.
"I chose (the drag force) as a repeating variable." — why is that risky?
The repeaters are the "building blocks" that appear in every ; putting the target quantity into them buries the thing you actually want to solve for inside all groups — keep the unknown out of the repeaters.
Why questions
Why does demanding "the law works in metres or miles" force the law into dimensionless groups?
Because rescaling units multiplies each term by a factor; only ratios that have no units are untouched by rescaling, so a rescale-invariant law must be expressible purely through such ratios.
Why is rather than minus the number of dimension symbols?
is the dimension of the null space of (the space of dimensionless exponent vectors), and by rank–nullity that equals ; the rank, not the symbol count, measures how many independent constraints the dimensions impose.
Why can dimensional analysis never hand you the constant in ?
is dimensionless, so it lives inside the unknown function/constant that dimensions cannot pin down; only solving the actual pendulum equation (or measuring) reveals it.
Why do we combine each non-repeating variable with the repeaters one at a time?
To guarantee we build exactly independent groups — each new non-repeater contributes exactly one fresh , and using the same repeater base keeps the groups a clean independent set.
Why does a bigger make a problem experimentally harder?
Each is a knob you must sweep in the lab; groups mean a -dimensional function to map out, so experiments grow combinatorially — drag with needs curves of vs , not a single number.
Why does matching all groups between a scale model and the real thing guarantee similar flow?
If every dimensionless group is equal, the two systems satisfy the same dimensionless law , so their behaviour is identical up to scale — this is the basis of Model testing and similarity.
Edge cases
What does (so ) tell you physically?
The variables can form no dimensionless ratio, so dimensions alone fix the relationship up to a single overall constant — the strongest possible dimensional prediction.
What if a variable's dimension is zero (already dimensionless, e.g. a strain or Mach number)?
It contributes to but adds no row/rank, so it stands as its own group and passes straight through into the final relation.
What happens if you forget a physically relevant variable?
Your count is too small and the "law" you get is incomplete — the missing quantity would have added a genuine that the real physics needs; dimensions can't warn you it's missing.
Can ever be less than the number of dimensions that visibly appear?
Yes — if those dimensions only ever occur in a fixed combination (e.g. and always as the ratio ), they collapse to fewer independent rows, so drops.
What if two chosen repeating variables happen to have identical dimensions?
Their rows are parallel, the sub-matrix is singular, and they cannot span an extra dimension — you must swap one out for a dimensionally distinct variable.
Is or its reciprocal the "correct" dimensionless group from the drag derivation?
Both are valid — the -group came out as ; since any invertible power of a is still a , convention just relabels it as . See Reynolds number.
Connections
- Reynolds number — the archetypal group and the reciprocal trap above
- Drag force and drag coefficient — where and the "assume " error bite
- Dimensional homogeneity — the unit-invariance principle behind every "why" here
- Model testing and similarity — the practical payoff of matching all groups
- Navier–Stokes equations — non-dimensionalising them derives without guessing
- Fundamental and derived units — how to correctly count