The concept:
A physical problem may be characterised by a group of dimensionless similarity parameters or variables rather than by the original dimensional variables.
This gives a clue to the reduction in the number of parameters requiring separate consideration in an experimental investigation.
For an example, if the Reynolds number Re = ρV Dh /µ is considered as the independent variable, in case of a flow of fluid through a closed duct of hydraulic diameter Dh, then a change in Re may be caused through a change in flow velocity V only. Thus a range of Re can be covered simply by the variation in V without varying other independent dimensional variables ρ,Dh and µ.
In fact, the variation in the Reynolds number physically implies the variation in any of the dimensional parameters defining it, though the change in Re, may be obtained through the variation in anyone parameter, say the velocity V.
A number of such dimensionless parameters in relation to dynamic similarity are shown in Table 5.1. Sometimes it becomes diffcult to derive these parameters straight forward from an estimation of the representative order of magnitudes of the forces involved. An alternative method of determining these dimensionless parameters by a mathematical technique is known as dimensional analysis .
The Technique:
The requirement of dimensional homogeneity imposes conditions on the quantities involved in a physical problem, and these restrictions, placed in the form of an algebraic function by the requirement of dimensional homogeneity, play the central role in dimensional analysis.
There are two existing approaches;
- one due to Buckingham known as Buckingham's pi theorem
- other due to Rayleigh known as Rayleigh's Indicial method
In our next slides we'll see few examples of the dimensions of physical quantities.
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