Buckingham Pi Theorem
Matthew Chandler
Abstract
Dimensional analysis is a method of reducing the number of variables in an experiment in order to simplify the process of describing a relationship among those variables. This paper discusses the use of the method.
Introduction
In many applications of science and engineering, the mathematical relationships between variables in a system are unknown. In order to find the relationship between these variables, one would be required to incrementally change the value of one variable while holding all of the other variables constant. This process would then be repeated for each variable until the relationships were discovered. This would be a difficult and sometimes impossible procedure to perform. By combining the terms to reduce the number of variables, the process becomes less complicated and more reasonable to perform. This process of dimensional analysis is based on the Buckingham pi theorem.
Theorem
The Buckingham pi theorem is stated as follows: If an equation involving k variables is dimensionally homogeneous, it can be reduced to a relationship among k - r independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables. The combinations formed by the variables are dimensionless products and are often called pi terms. The minimum number of reference dimensions needed to describe the original list of variables is represented by r. The number of pi terms required to describe the system is r fewer terms than the original list of variables. The process of dimensional analysis is relatively simple to apply.
In order to perform dimensional analysis, one must first list all of the variables that are defining a problem, including dimensional and nondimensional constants. These variables must then be expressed in terms of basic dimensions such as mass, length, and time or force, length, and time. The dimension matrix provides a convenient way to summarize the dimensions and reduce the problem into a problem of linear algebra. In the dimension matrix, the variables are represented by the columns of the matrix and the reference dimensions are represented by the rows of the matrix. The numbers within the matrix represent the exponent on the reference dimension in the units of the corresponding variable, as demonstrated here: .
The dimension matrix can then be row reduced to find the number of pivot columns which is equal to the rank of the matrix. The number of variables minus the rank of the matrix gives the number of independent dimensionless combinations that must be formed. The basis for the null space must then be found by solvingthe system, where A is the dimension matrix. The solution can be written in parametric form. The vectors in the parametric solution provide the combinations, and powers, of the original variables that produce nondimensional groups. The entries in these 1 x n vectors correspond, from top to bottom, to the variables listed in the dimension matrix, from left to right. The problem of finding the combinations and powers of the repeating variables is therefore reduced to finding thenull space of the dimension matrix.
Finally, the pi terms can be written in relation to one another to describe the problem in the following form: . In this case, would contain the dependent variable. The actual functional relationship, , must be determined by experiment. It can clearly be seen that the number of terms used to solve for has been reduced by r terms, thus simplifying the process.
References
Lay, David C., "Linear Algebra And It's Applications," third edition, Pearson Education, Inc., 2003
Young, Donald F., ET. Al., "A Brief Introduction To Fluid Mechanics," second addition, John Wiley & Sons, Inc, 2001
Hanche-Olsen, Harold, "Buckingham's pi-theorem," version 2001-09-15, web resource, 1998
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