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.
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...
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...
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)...
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
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...
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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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...