AMIRAJ COLLEGE OF ENG & TECH.

FLUID MECHANICS GROUP NO : 9 PREPARED BY : ENROLLMENT NO:141080119050 141080119052 141080119059 151083119013. MECHANICAL BRANCH. Semester :4th (2nd year) TOPIC NAME : REYLEIGH’S

METHOD,BUCKINGHAM π-THEOREM.

INTRODUCTION It is method of dimensional analysis. This two methods is very important for

dimensional analysis. (1).Rayleigh`s method (2).Buckingham`s - theorem.

Rayleigh's Method In this method, the expression

for the variables in form of exponential equation and dimensionally homogeneous.

Let, Y is a variable, which depends on

variables, then functional relationship may be written as:

.............321 ,, xxx

,.....,, 321 xxxfY

Where , Y=dependent variable, =independent variables, f=function. This method is used for determining

expressions for a variables which depends upon maximum three or four variables only..

If the number of independent variables becomes more than four, then it is very difficulty to find expression for the dependent variables.

.............321 ,, xxx

Method involves the following steps (1).Gather all the independent variables

which govern variation of dependent variables.

(2).write the functional relationship with the given data

(3).write the equation in terms of a constant with exponents(power) a,b,c.....

,.....,, 321 xxxfY

,......,, 321cba xxxKY

where , K is a dimensionless co-efficient and a,b,c....are the arbitrary powers.

(4).Apply principal of dimensional homogeneity, and put the dimensions(M,L,T) of variables on both sides of equation.

(5).find out the values of exponents (a,b,c,...) by obtaining simultaneous equation.

(6).put the value of exponents (a,b,c...)

in the main equation and form the dimensionless parameter by grouping the variables with similar exponents..

Buckingham’s π-Theorem

This method is minimized difficulties of Rayleigh's theorem....

It states, "If there are n numbers of variables (dependent and independent variables) in the physical phenomenon and if these variables m numbers of fundamental dimensions (M,L,T), then the variables may be grouped into (n-m) dimensionless terms”

This dimensionless term is known as π. Let us consider a variable depends

upon independent variables then the functional equation can be

written as

The equation may be written in general form as

1xnxxx ,......, 32

),.....,,( 321 nxxxkx

cxxxxf n ),....,,,( 321

Where , c is a constant and f is a function. If there are n variables and m

fundamental dimensions, then according to Buckingham’s π-theorem

=constant.

The π-term is dimensionless and independent of the system.

mnF ,.....,,, 3211

Buckingham’s method involved following steps: (1).write the functional relationship of

given data, (2).write the equation in its general form

(3).Find the numbers of π-terms . If there are n variables and m is fundamental dimensions , numbers of π-terms=n-m

),.....,,( 321 nxxxfx

0),.....,,( 32 nxxxf

(4).select m number of repeating variables and write separate equation for each π-term. Each π-terms contain the repeating variables. The repeating variables are written in exponential form...

.......................................

...........................

43211111 xxxx cba

53212222 xxxx cba

6321333

3xxxx cba

ncba

mn xxxx mnmnmn 321

Where , are repeating variables.

(5).Each π-term solve by the principle of dimensional homogeneity, put the dimensions of variables in each π-term and find out the value of a,b,c,,, by solving simultaneous equations..

(6).Now put the values of a,b,c,... In the π-terms.

(7).write the functional relation in the required form

321 ,, xxx

0,.....,,, 3211 mnF

Procedure for selection of Repeating variables: Number of repeating

variables= no . of fundamental dimensions=m

The repeating variables should not be dependent variable.

It should not be dimensionless. No two variables should have

the same dimensions.

The repeating variables together must have the dimensions as MLT.

The repeating variables should be selected in such a way that

(1).one variables contains geometric property as length, diameter, height, width, etc.

(2).other variables contains flow property as velocity ,acceleration etc.

(3).Third variables contains fluid property as dynamic viscosity, density, etc.

In most of common problems of fluid mechanics , the pair of repeating variables as (1).

(2). (3). (4).

,,Vl,,Vl

,,Vd

,,Vd