As suggested in the last section, if there are more than 4 variables in the problem, and only 3 dimensional quantities M , L , T , then we cannot find a unique relation between the variables. The best we can hope for is to find dimensionless groups of variables, usually just referred to as dimensionless groups , on which the problem depends.
In fact, this is quite a good approach, for reasons that will be discussed in the next section.
Buckingham Pi is a procedure for determining dimensionless groups from the variables in the problem. In general we can derive m - n dimensionless groups, often denoted 1 , F Select n of the variables - say A , B , C.
These are called the repeating variables , and will appear in all the terms. For instance, D , the pipe diameter, and r , the roughness height, both have dimension of L , and so cannot both be used as repeating variables. Select one other variable - say D. Some combination of A , B , C , D is dimensionless, and forms the first term or dimensionless group.
We can find the combination by dimensional analysis, by writing the group in the form Equating coefficients gives 3 equations for 4 unknowns, so we can express all the coefficients in terms of just one. Repeat this procedure with the repeating variables and the next variable, so use A , B , C , E.
Buckhingham's Pie Theorem
Continue until no variables are left. Having worked out all the dimensionless groups, the relationship between the variables can be expressed as a relationship between the various groups.
Typically we can write this as one group for example 1 as a function of the others 4. The head loss in a horizontal pipe in turbulent flow is related to the pressure drop p , and is a measure of the resistance to flow in the pipe.
It depends on the diameter of the pipe D , the viscosity and density , the length of the pipe l , the velocity of the flow v and the surface roughness. However one of the repeating variables is the diameter D , and so the ratio of the two is already dimensionless.
So the next dimensionless group is Finally, is already dimensionless anyway, being the ratio of the height of the roughness to the pipe diameter D. We can write this relation as a relation between the groups or in full This is as far as we can go using dimensional analysis.
In particular, the expression may depend on - 1 as easilly as on , so any dimensionless group can be inverted if that is more convenient. Any term may be expressed in terms of the others.
All of these short cuts were used in the previous example.