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Assignment # 1

Topic :Applications of

Buckingham-pi-theorem

for dimensional analysis

Instructor :

Mr.Shaid HussainAnsari

Submitted by : Sarwat Hameed

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Date :29/09/2010

Applications of Buckingham-pi-theorem fordimensional analysis

Dimensional analysisis a mathematical tool often appliedin physics, chemistry, and engineering to simplify a problem byreducing the number of variables to the smallest number of"essential" parameters. Systems which share these parameters

are called similarand do not have to be studied separately.

Buckingham Pi Theorem

The Buckingham Pi Theorem enables one to "discover" the non-dimensional numbers for a given process and is stated asIf a physical process has "n" variables and from these "k" areRepeating Variables, then the there are "n-k" independent non-dimensional numbers that can describe that process.For the example we have been considering there are 5 variablesthat govern the process. So n = 5. We have marked threerepeating variables, making k = 3. The theorem tells us that thereare two non-dimensional numbers.

Buckingham Pi Theorem relies on the identification of variablesinvolved in a process. Further, a few of these have to be markedas "Repeating Variables". This would seem to be a major difficultyin carrying out a dimensional analysis. Let us continue with ourexample of drag about a cylinder. We have identified thevariables involved - F (drag force), D(diameter of the

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cylinder), V (the fluid velocity), (density of the fluid), (viscosity ofthe fluid). Now we mark three of these as the Repeating Variables- V, D and density. The selection comes by practice. Some generalguidelines can however be given. These variables combine with

the non-repeating ones to form the non-dimensional numbers.Further, they do not form a non-dimensional number themselves.A general prescription is to consider velocity, linear dimensionand density to be the Repeating Variables. But by looking at thelist of variables one can often intuitively tell the RepeatingVariables from the rest.

Applications of Buckingham Pi theorem It is used in diversified fields such as Botany and Social Sciences

and books and volumes have been written on this topic.

By making use of this theorem it is possible to obtain thedimensionless groups more simply than by solving thesimultaneous equations for the indices.

Furthermore, the functional relationship can often be obtained ina form which is of more immediate use.

It is often useful to use one system to do problem, and thencheck that groups you obtain are dimensionless by converting toother system.

Typically pick variables which characterize the fluid properties,flow geometry, flow rate