REYLEIGH’S METHOD,BUCKINGHAM π-THEOREM

REYLEIGH’S METHOD,BUCKINGHAM π-THEOREM.

1. AMIRAJ COLLEGE OF ENG & TECH.

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

3. INTRODUCTION It is method of dimensional analysis. This two methods is very important for dimensional analysis. (1).Rayleigh`s method (2).Buckingham`s - theorem.

4. 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:

5. 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.

6. 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.....

7. 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.

8. (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..

9. 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”

10. 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

11. 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.

12. 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

13. (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... ....................................... ...........................

14. 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

15. 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.

16. 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.

17. In most of common problems of fluid mechanics , the pair of repeating variables as (1). (2). (3). (4).