# What is Buckingham pi: Definition and 19 Discussions

In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
The Buckingham π theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by Boyle's law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold.

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1. ### Dimensional Analysis Buckingham Pi Theorem

This is the problem I'm currently working on: The pi groups I identified were h1, h2, d, D, g, t, and velocity, but when I looked at the solution, it selected Δh, D, t, ρ, d, ϒ, h1, with no explanation why those variables are needed. If I was solving with the Bernoulli equation, I wouldn't...
2. ### Is Exponential Needed for Non-Repeating Variable in Buckingham Pi Theorem?

Homework Statement https://projects.exeter.ac.uk/fluidflow/Courses/FluidDynamics3211-2/DimensionalAnalysis/dimensionalLecturese4.htmlaccording to this link , when we form the pi group , we need to put an exponent for the non-repeating variable ,( in this case , delta P is non-repeating variable...
3. ### Rules of choosing repeating variable in Buckingham pi theorem

Homework Statement i was told by my lecturer that when we choose the repeating variables in pi buckingham theorem , we can choose based on 3 property , which is geometry property which consists of (length , width and area) , then followed by flow property ( velocity , acceleartion, discharge)...
4. ### Why do we need to raise the whole pi_3 to power of -1/2?

Homework Statement in the third photo attached , why do we need to raise the whole pi _3 to power of -1/2 ? can we do so ? if we do so , the original pi_3 will be changed , right ? Homework EquationsThe Attempt at a Solution
5. ### Can Someone Explain Step 4 in the Buckingham Pi Theorem Homework?

Homework Statement http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/dimension/node9.html can somoene expalin about step 4 in the first photo attached ? What does it mean by each group has all the repeating variables and non-repeating variable ? Homework...
6. ### Buckingham Pi / Dimensional analysis

Homework Statement A capillary filled with water is placed in a container filled with a chemical of concentration C_{0} , measured in number of molecules per unit volume. The chemical diffuses into the capillary of water according to the following relation (where x is distance along...
7. ### Help with Buckingham Pi theory on turbine

I was wondering if someone is very well into the Buckingham Pi Theory and can assist me in understanding the outcome. I have two dimensionless equations and I need to get an understanding on how they were made by the Buckingham Pi theory. The situation is considered for the reynolds number...
8. ### Buckingham Pi Theorem: Choosing Common Variables

Homework Statement In buckingham pi theorem, you have p=n-k dimensionless groups (π1, π2,...) where n=number of total variables and k=number of total units among the variables For example, let's say we want to relate: ρ~m/L3 (density) μ~m/L*t (viscosity) v~L/t (velocity) d~L (distance) Da~L2/t...
9. ### Buckingham-Pi for "algorithmic" non-dimensionalization

I would like to use the Buckingham-Pi theorem in order to "algorithmify" non-dimensionalization of existing equations. I can get things to work for very simple problems, but am running into issues with a harder example. I posted my question on physics.stackexchange.com the day before yesterday...
10. ### How to Determine Pi Groups for Fluid Flow Through an Orifice

Homework Statement a) A orifice, diameter d (m), in the walls of a tank discharges water under a head, h (m), subjected to gravitational acceleration, g (m/s2). If the fluid has density, ρ (kg/m3)and viscosity, μ (Ns/m2), show that the quantity of fluid Q (m3/s) flowing out of the tank may...
11. ### Dimension Analysis and buckingham pi

Homework Statement can someone explain why we are interested in forming dimensionless products and why only n-j of them should be formed from the problem's variables? Homework Equations Step 1: List the variables in the problem Step 2: Express each of the variables in terms of basic...
12. ### Dimensional Analysis and buckingham pi

Homework Statement Hi Guys, I am a bit confused concerning one part of this topic. Specifically when trying to find non dimensional groups. My problem is a small thing in the finding of the indices. So, for example, your are trying to find a pi group when finding the drag on a car. so you...
13. ### Variable Set Distribution - Buckingham Pi Theorum

Background: I am trying to write a program for Buckingham Pi Groups. I need to find a way to list all the input varialbes as different sets. For example if I have 4 variables [V D p u] and I want to distribute them 3 ways I get 4 sets. Number of Sets = Binomial(Number of Variables...
14. ### Buckingham Pi - Understanding Trader Behavior in Stock Markets

Homework Statement A simple dynamical model for the price P (in £) of shares in a single stock or commodity traded in a stock market describes the behaviour of all the traders in the market with the same simple rule. All traders buy or sell shares at each "tick" (or time step \Delta t. To...
15. ### Buckingham Pi Theory: Propeller & Pipe Flow Analysis for Jay

Hi I feel I am competent enough at the Buckingham Pi theory regarding both Pipe flow and for Propeller analysis. Derive a relationship between the volume flow rate and rotational speed of propeller in terms of the diameter of the pipe and propeller and also fluid characteristics density and...
16. ### Buckingham PI Theorem proof - Dimensional Analysis

Homework Statement I am looking for a proof of Buckingham PI theorem in dimensional analysis, but can't really find one anywhere. I saw a proof involving posing the problem as a question in linear algebra, but it was quite unclear.
17. ### Buckingham Pi Theorem Explained: Understanding Variables and Parameters

Heya, I'm new here and really need help! So I'm having trouble with the *Buckingham Pi Theorem*. I think I've got the jist of it bar one thing...So you have a bunch of variables e.g a force, velocity, denisty, length, viscosity, speed of sound: f(F, V, roh, L, mu, a) Do the (N variables - M...
18. ### Dimensional Analysis & Buckingham Pi Theory

What can "Dimensional Analysis Techniques & Buckingham Pi Theory" be used for? GS
19. ### How Does the Buckingham Pi Theorem Apply to Homogeneous Functions in Physics?

This is more of a dimensional analysis or unit analysis problem than a basic analysis problem. So if I didn't post this thread in the right forum, please delete it. Moving on, I'm having trouble grasping the concepts behind the Buckingham Pi Theorem. After reading some textbooks and doing...