Buckingham II theorem

Dimensional analysis, often referred to as the II-theorem is based on the fact that every system that is governed by m physical quantities can be reduced to a set of m - n mutually independent dimensionless groups, where n is the number of basic dimensions that are present in the physical quantities. The II-theorem was introduced by Buckingham [1] in 1914 and is therefore known as the Buckingham II-theorem. The II-theorem is a procedure to determine dimensionless numbers from a list of variables or physical quantities that are related to a specific problem. This is best illustrated by an example problem. 

Dynamic similarity ensures that the ratios of all forces, on the fluid flow and boundaries, in the prototype and scale model are the same and can be expressed as constants. Ratios of forces in fluid flows are often expressed in terms of dimensionless numbers. These dimensionless numbers are derived using what is called dimensional analysis using the Buckingham II theorem. 

To establish the necessary dimensionless groups, a systematic dimensional analysis needs to be carried out where Buckingham II theorem is used to reduce the number of dimensionless groups (4). Assuming that a process can be described by k variables, we can express one variable as a function of the other k — variables, i.e.. 

Consider a fluidic system with surface tension effects. The characteristic scales for length and velocity are L and U. The physical parameters are density /O, viscosity v, gravity g, and surface tension y. By using the Buckingham II theorem one can obtain three independent nondimensional groups from these six variables. An option for a set of three... 

The need for dimensional consistency imposes a restraint in respect of each of the ftinda-iiMntals involved in the dimensions of the variables. This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each of the fundamentals. A generalisation of tiiis statement is provided in Buckingham s II theorem " which states that the number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions. In mathematical terms, this can be expressed as follows ... 

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